This page presents a list of 'the repeated factorisation of concatenated primefactors (in ascending order) of the composite numbers from 1 to 100'.
I know, this description, quite a mouthful. An easy example clarifies the procedure in the twinkling of an eye. Let me take the at first sight dull composite number 14. This startnumber 14 has two factors 2 and 7. Now paste these two factors together (from smallest to greatest) and you'll end up with the new number 27. Repeat the procedure with number 27.
27 = 3 x 3 x 3. Concatenation of these three factors gives the number 333.
So step 3 becomes 333 = 3 x 3 x 37.
Step 4 : 3337 = 47 x 71
Step 5 : 4771 = 13 x 367
Step 6 : 13367 = Prime ! Thus the expansion stops after six steps.
This approach is best described as breaking the number into its prime constituents. These latter then reassemble in ascending order quite willingly thereby forming ever greater numbers until miraculously a prime number is created which of course refuses to break down. The chainreaction stops.
It's a way to make composite number as interesting as prime numbers. Many composite numbers become prime in less than 6 steps but quite a few need many more steps. The following list shows the steps for composite numbers up to 100. Number 77 has the same expansion as number 49 because 49 equals 7 x 7 which is 77 ! and need therefore one step less to become prime. We haven't reached its 'home' prime number yet...
Can you help ?
It may become a very tedious job as the number of primes thins out. The chance that a randomly formed larger number is prime gets smaller and smaller. Yet there are infinite primes. Eventually it must happen !
Read also James G. Merickel's Wikipedia article about our Home Prime.
Messages
On [ April 24, 1999 ] Jeffrey Heleen (email) passed me the following information :
" On your webpage I see that your results match mine. I originally thought of the problem
around 1990 and using only a '386 computer' created the same table up to 1000. I wrote
an article published in the "Journal of Recreational Mathematics" a few years ago titled
'Family Numbers' describing my efforts.
Family Numbers: Constructing Primes by Prime Factor Splicing, JRM Vol. 28 #2,
199697, pp. 116119.
I notice that for a start number of 49 that it has been taken up to the 55th step,
the same place I was stopped. I was using the UBASIC program ECMX,
modified slightly, to do the search. I made it up over 1130 curves before
stopping. At present, the numbers smaller than 1000 which do not yet have endprimes are:
49, 146, 234, 242, 284, 300, 312, 320, 322, 326, 328, 336, 352, 360, 363,
372, 407, 412, 414, 460, 495, 548, 556, 558, 576, 592, 596, 642, 663, 665,
670, 693, 712, 714, 715, 744, 749, 762, 768, 782, 796, 800, 845, 847, 858,
861, 864, 866, 867, 896, 908, 925, 964, 969, 973, 978, 984 and 992.
Some numbers are left out as they can be categorised to smaller starting numbers,
for instance 77 and 711 belonging to initial number 49. Included are only the initial
starting numbers although in the article chart all numbers are referenced.
Some of these I have taken quite far but not found a solution. I know there are limits
using the ECM method. I have heard of the Number Field Sieve method but don't know
much about it or where to get it. Perhaps with that program some progress can be made.
I just found your page and thought it interesting. It surprised me to find others had worked
on this problem, (I had thought it original to me). I like all the material you've presented,
it gives lots of food for thought. I'm sure I'll be spending some time here. :) ..."
Thank you very much, Jeffrey, for sharing you work with us. I welcome your list of numbers
not having reached the endprime smaller than 1000 very much. It will provide new
'impetus' for likeminded persons to go further in exploring the subject. As 'palindromes' is
the main topic of my website I took the liberty to highlight the five palindromes in your list.
These became my favorites and would like to give them priority over the nonpalindromic
ones for numbercrunchers who are interested enough to accept the challenge.
On [ May 12, 1999 ] Jeffrey Heleen (email) sent me a first update of his work.
"... Now that I have somewhat more computing power than before (300 MHz vs.
25 MHz) I've started on the unsolved numbers less than 1000 once again. To date,
of the numbers I previously sent you that were unsolved, I have found endprimes
for three of them. Namely 360, 372 and 412. (PS. They are striked out in this list).
I have uncracked composite numbers for all those less than that and am still working
on the rest of the list. Attached is a text file created in Notepad of the uncracked numbers.
(PS. Available on simple demand). Hopefully someone else out there may be able
to take them further. I'll keep you posted as things develop."
On [ August 14, 1999 ] Jeffrey Heleen (email) sent me a second update of his work.
" Ok, as it now stands the unsolved numbers on the last list I gave you
(smaller than 1000) that have been solved are:
360, 372, 412, 558, 642, 693, 744, 796, 800, 847, 864, 867, 908 and 984.
Endprimes have been found for these."
On [ November 8, 1999 ] Paul Leyland (email) made a breakthrough by cracking
the c105 composite number from step 56 of HP(49) or step 55 of HP(77).
(Repeated Factorisation of Concatenated Primefactors of the Composite Numbers)
The factors are p41 * p65, or
21034137982005155236145561292210835084361
and
20679893341907378249919955987757483932846633610599367336113869677
He obtained these two prime factors by running his MPQS program for only a week !
The very same day  but alas(!) some 6 hours later  these results were already confirmed
when a second contender Eric Prestemon send me also the same factors p41 and p65.
Eric Prestemon used Paul Zimmermann's GMPECM software.
Well done, both of you !
go directly to table entry
On [ June 14, 2000 ] Paul Leyland (email) made a new breakthrough by cracking
the c137 composite number from step 74 of HP(49)  or step 73 of HP(77)  using ECM.
" After almost six months of computation on a PII300, Paul Leyland can reveal that
this number factors as p46 * p91. It was found after ~ 5000 curves at B1=3M and
3350 curves at B1=11M.
The two factors are respectively
3804796914905629947782783176497447433056300673
and
297575514989250201776144220639738150129498705/
0078840501836673559420025718611870754546792073 "
My compliments to you, Paul !
go directly to table entry
On [ August 6, 2000 ] Igor Schein (email) sent me some results of his investigation
of numbers greater than 1000.
" I've been looking at sequences for some N>1000.
Here's one which finishes after 54 steps: N = 2092.
2 * 2 * 523
101 * 223
...
and the home prime is formed from concatenating these last factors :
4007 * 9923 * 47303 * 636171471679 * 49567980853079631127541759358497387 *
11979321332839520964445116714814558542751169
I'm also looking at some sequences in bases other than base 10.
For example, base 2:
(100 = 4) > (1010 = 10) > (10101 = 21) > (11111 = 31)
The longest terminating sequence in base 2 I found so far is the one
starting at N = 1345, which terminates after 130 steps. There are 2 values
of N <= 3500 for which the sequence is longer than 130, but I haven't been
able to terminate them yet."
On [ November 25, 2000 ] Sander Hoogendoorn (email) informed me he started collecting
these Home Primes for numbers greater than 100. His database can now be consulted at the following address
Home Primes beyond 100
He reached already up to 312 and will whenever he has the time update his pages on a regular basis.
A great initiative indeed, Sander, for which we are all very grateful!
On [ April 5, 2002 ] Alex Kruppa (email) & Paul Leyland (email)
are delighted to announce that they have factored HP49(80).c138
( p69 * p70 ) with the general number field sieve.
The prime factors are
224690133218881151252602753388830692415125120972614888557571233246513
(69 digits) and
3344456987746930138631822411149806794710441073001296631385753707501227
(70 digits).
"Between us we had run enough ECM curves to be fairly sure that no factor would
have fewer than 45 digits. The result above shows why ECM was not successful.
We both searched for polynomials and each found about a dozen reasonable candidates.
The best polynomial was found by Alex and was
–117571849151668295576809408045 +
–483923124317877121410587521 * X
139472015921577159080103 * X^2
50167688067225969677 * X^3
1181017574467178 * X^4
355927036920 * X^5
with root 18403894182248001767571928 mod N.
Alex used Jens Franke's lattice siever; Paul used the CWI implementation of the line siever.
The factor bases used primes up to 3M and 10M on the rational and algebraic sides, and
large primes up to 500M and 1000M respectively. Elapsed time for sieving was around
three weeks on a variety of machines; cpu time used was around 2 cpu years though we
don't have accurate figures available at present.
At the end of the sieving phase, Alex had produced 49,256,164 unique relations (88% of the
total) and Paul 6,811,163 (12%). There were duplicates in the combined relations and we
finished the factorization with 53,687,196 unique relations.
The filtering, linear algebra and square root phases were performed at Microsoft Research
Cambridge because of the very heavy memory requirements of the first two phases. The first
filtering step took about 1.4 gigabytes of active memory and was run on a machine with 2GB
of RAM. The filtering stages resulted in a matrix with 2,540,557 rows and 2,535,422 columns
with a total weight of 119,862,073 set bits. The linear algebra was performed with CWI's
implementation of parallel block Lanczos running on all 32 PIII1000 cpus of MSRC's cluster.
It took close to 54 hours elapsed time, but only 14 hours cpu time per processor, to find 127
dependencies. Each process used abut 40MB of memory, for a total of almost 1.3GB.
The factorization given above was found on the first dependency by the square root phase,
which took 7.5 hours computation on a PIII500 machine.
We have not yet found the home prime. The next stage is composite, and we have already
reduced it to p1 * p7 * p16 * c136. Further ECM runs
are in progress and we'll let you
know if we find more factors."
Alex and Paul, that is top level programming, congratulations!
go directly to table entry
On [ May 16, 2002 ] Alex Kruppa (email) & Paul Leyland (email)
have factored HP49(81).c136
( p51 * p85 ) using the general number field sieve.
The prime factors are
448583441180516821621320259546896223296373907144113
(51 digits) and
3541865184678001629601182607226312880348900246844/
848900227068283166325369629573991313
(85 digits).
"We factored the c136 with GNFS using parameters very similar to those used for the
previous c138 and took a closely similar amount of processing power so I won't repeat
all the details I gave last time. On this occasion, Alex found the polynomial and performed
about 95% of the sieving. I did the filtering, linear algebra and square root phases. For
some reason, this factorization caused us many more problems than the previous one
and several bugs or inadequacies in our software came to light. One such problem forced us
to do the linear algebra twice, on two slightly different matrixes, and delayed our finding
the factors by several days.
Alex would like to thank the Research Unit for System Architecture of the Department
of Computer Sciences at the Technische Universität München for the use of their computers
in the sieving phase. I used two servers at Microsoft Research Ltd, Cambridge for my small
contribution to the sieving and for the square root phase; the linear algebra was performed
on the 32cpu cluster at MSR Cambridge.
HP49(82) is composite, with three small prime factors, one of moderate size (all found
by Alex) and a composite cofactor which will be much easier to factor than its two
predecessors. The current factorization is p1 * p3 * p9 * p30 * c118. We will be working
on the c118 in the near future. Regards, Paul & Alex"
Alex and Paul, you both are quite a mathfertile team. Well done again!
go directly to table entry
On [ May 22, 2002 ] Alex Kruppa (email) & Paul Leyland (email)
have also factored HP49(82) and HP49(83)
"We (Alex and myself) would like to report that although we've performed
the next two stages in the search for the home prime, HP(49) continues
to be elusive.
In our previous mail, we reported that HP49(82) contained a c118.
We factored that with GNFS. Alex did all the sieving this time and
preliminary filtering; I did the remaining filtering, linear algebra and
square root phases to discover the p55 * p64 factorization. Sieving
took a few days on Alex's machines (credit as before) and the phases I
did took about a day in total  about 40% of the time was spent in the
square root program because the first two dependencies gave the trivial
factorization and we found the true factors on the third.
The next step, HP49(83) was much easier. Several small and medium
factors were found by ECM, leaving a c83 cofactor which fell to MPQS in
a couple of hours or so.
The following step seems to be much harder. It is a c167 which contains
no very small factors. We've begun testing with ECM and if anything
turns up we will let you know. If ECM doesn't find anything this number
will be *very* hard."
On [ May 25, 2002 ] Alex Kruppa (email)
factored HP49(84) using ECM into p53 * p114
The prime factors are
81477382431617858607629654669086224895030590860856949
(53 digits) and
164853464798393151511356156289762200575122773801949600629/
560294634580313680599127053982894889995189794735466895119
(114 digits).
"Luck has prevailed. ECM found a 53 digit factor of the c167 of the 84^{th}
step of the HP49 sequence and thus completed this factorization.
The next step is composite, as of now its factorization is known to be
p1 * p2 * p5 * p6 * p16 * c140. ECM may yet find another factor, but
even if not, the c140 will be feasible by GNFS. The quest for this
elusive Home Prime continues...
This is also the fourth largest factor ever found by GMPECM
and the largest one in the current year so far."
GMPECM 4c, by P. Zimmermann (Inria), 16 Dec 1999, with contributions from
T. Granlund, P. Leyland, C. Curry, A. Stuebinger, G. Woltman, JC. Meyrignac,
A. Yamasaki, and the invaluable help from P.L. Montgomery.
Input number is
134318287965559312522053506677236571899578936128385172670/
615409359178147914132583891425336357638349050223174034385/
38106689190395089953894891189281091736983632645331931
(167 digits)
Using B1=11000000, B2=3890888820, polynomial x^60, sigma=219538831
Step 1 took 562640ms for 143669942 muls, 3 gcdexts
Step 2 took 255220ms for 63722040 muls, 118193 gcdexts
********** Factor found in step 2:
81477382431617858607629654669086224895030590860856949
Found probable prime factor of 53 digits:
81477382431617858607629654669086224895030590860856949
Report your potential champion to Richard Brent <rpb@comlab.ox.ac.uk>
(see Large Factors Found by ECM )
Probable prime cofactor
164853464798393151511356156289762200575122773801949600629/
560294634580313680599127053982894889995189794735466895119
(114 digits)
Paul Leyland (email)'s reply
<1801733, 1>, <2678887, 1>, <4384333, 1>, <3159028427, 1> ]"
go directly to table entries 9095
On [ January 20, 2003 ] Alex Kruppa (email) & Paul Leyland (email)
factored HP49(95).c153 into p68 * p85
starting with a search for GNFS polynomials
" Alex and I have completed two more stages of the search for HP49.
When we last mailed you, we were stuck at HP49(95) which had a
153digit composite cofactor.
We first saw the factors of this integer on Saturday 18th January
2003, after a computation that was started back in early September
2002 with a search for GNFS polynomials. Sieving began in
midNovember and ended on 28th December. The filtering phase took a
week or so, the linear algebra 5.5 days on 30 cpus of the cluster at
Microsoft Research, and the square root phase a few hours on a single
workstation.
We now know that HP49(95) equals 3 * 7 * 26141 * 300119 * 1811141 *
1072782128567282855315039 *
25381603104475027190830989059811875365234972412236/
253418807674044481 *
73122383885452606722686858220222364416902310830906/
44547260008735024402747827401799039
The last two factors have 68 and 85 digits.
ECM successfully factored the next iteration. HP49(96) = 17 * 937 *
2999 * 15011194746557 * 33716362272572345351978861258895209 *
15411267935624560746503422154192060735654946200813/
37305715946120222833565553933109196280811860424054/
1265521091783630320332376356814622960926093
where the last factor has 143 digits.
The next stage is composite, and we have a partial factorization by
ECM: 19 * 569 * 683 * 7450039 * 865252586740571 *
2971814878235479924213 * c151
We are fortunate in that the 151digit cofactor is within range of
GNFS if we can't find any more ECM factors. "
go directly to table entries 9597
On [ July 16, 2003 ] Alex Kruppa (email) & Paul Leyland (email)
factored HP49(97).c151 into p55 * p96
" Dear Patrick, we have expanded the HP49 sequence to the 100^{th} step,
but still had no luck in discovering the endprime.
The difficult part was the factorization of the 151digit composite
cofactor of HP49(97). After we had done enough ECM to be fairly sure
that no factors of less than 50 decimal digits remained, we decided to
complete this step with a GNFS factorization.
Polynomial selection was done with Thorsten Kleinjung's program and most
of the sieving was done by Alex with the Bahr/Franke/Kleinjung lattice
siever which produced 76M relations using approximately half an Athlon
GHzyear. The CWI line siever produced an additional 2M relations by
sieving over small bvalues for about one cpu week.
Duplicate removal, pruning and clique removal was done at the TU
München, leaving 18M relations which were then sent to Paul at MS
Research, Cambridge, and merged to form a (4.47M)^2 matrix. The matrix
took a week to solve on the MSRC cluster and the factors appeared on the
first dependency on Monday, 14th of July.
The factors are:
p55 = 1771052383785311834993979061208604132871538232335055323
p96 = 7160047222541505864838433131582635928178039748055681941 \
13667109128698119199668739510916110374031
As it turns out, the smaller factor arguably could have been discovered
with ECM, however the expected amount of cpu time for ECM to find it
would not have been much lower than what we spent on GNFS and, unlike
GNFS, would not have guaranteed to actually produce anything.
The 98^{th} and 99^{th} step of HP49 factored easily using ECM:
HP49(98) = c203 = 3 * 3 * 3 * 3 * 17 * 173 * 313 * 1104769163 *
366751448517289166567507 * p162
HP49(99) = c208 = 107 * 22861 * 3700198301407 * 581535317003127481 * p171
The 100^{th} step, however, appears to be not so easy. This far it is known that
HP49(100) = c210 = 3 * 37 * 2789 * c204
We have done enough ECM to be confident that no factors of less than 35
digits remain in the cofactor. Since a number of this size is far out of
reach for GNFS with today's technology, we will have to put all our
hopes in ECM. If that should fail to factor or at least substantially
reduce the size of this cofactor, then this 100^{th} step will mark the end
of the HP49 sequence expansion for a long time to come. "
go directly to table entries 97100
On [ January 3, 2004 ] Alex Kruppa (email) & Paul Leyland (email)
wrote the following
" Dear Patrick,
we're giving up. We have done 5500 curves at B1=11M, and 9000 curves at B1=44M,
but have been unable to find a factor of the c204 of HP49(100). Our resources don't
allow us to try ECM much further, so unfortunately we have to give up HP49 at this
point  after twenty steps and almost two years since we started working on it. Thank
you for keeping track of of the project record, and to all who'd like to take a shot
at this composite, we'd like to wish good luck!
Alex Kruppa and Paul Leyland "
The decimal expansion of this c204 is
3463691455176168325615805184363381478770628934576796221959292066545246725876130493435583943733963381/
9458578377526978567521063669642509477685973330594799604806149924956619714721293451242798811342022676/
2897
On [ February 9, 2010 ] Paul Leyland (email)
forwarded the following results from Nicolas Daminelli
" Nicolas Daminelli factored the c204 from HP49 yesterday.
His very nice result is below. [ go to table entry ]
> Subject: HP49(100) = prp62*prp143. Please review.
> Date: Mon, 8 Feb 2010 19:25:03 0800 (PST)
>
> All,
> Since a lot of effort was put into factoring HP49(100), I thought I
> would let you know that I got a lucky ECM for it at B1=260M.
> Could someone please doublecheck this result for primality and input
> correctness?
>
> GMPECM 6.2.3 [powered by GMP 4.3.1] [ECM]
> Input number is
> 3463691455176168325615805184363381478770628934576796221959292066545246725876130493435583943733963381/
> 9458578377526978567521063669642509477685973330594799604806149924956619714721293451242798811342022676/
> 2897 (204 digits)
> Using B1=260000000, B2=3178559884516, polynomial Dickson(30),
> sigma=1765817006
> Step 1 took 1320152ms
> Step 2 took 379725ms
> ********** Factor found in step 2:
> 13055369049845151421562673963068310715129211918218895785368571
> Found probable prime factor of 62 digits:
> 13055369049845151421562673963068310715129211918218895785368571
> Probable prime cofactor
> 2653078164203447671850231332797009603029004755483654727809454779301905374228983373655278628689069551/
> 1249799288531238767159656593259713779239907 has 143 digits
> Report your potential champion to Richard Brent (email)
> (see http://wwwmaths.anu.edu.au/~brent/ftp/champs.txt)
>
> Cheers,
> Nicolas
I'm working on extending the chain. For instance, the next value HP49(101) appears to be
3 * 3 * 179 * p209
[ go to table entry ]
The one after that HP49(102) is
3 * 7 * 12473 *
69442311247884384744096581 * 5504559912367454438295149601552867774551041 *
3313820999622709417530127238213785731734022602460004639101646954950942392321815618600363064109247466/
1686802611798393277995398615210685360809
[ go to table entry ]
The next stage HP49(103) has a c193. It was dealt with 8)
29 * 8627 * 89345257 * 8067774497 *
96513008244398921562099305449602682965323 *
91479820611369205267924863199536513832217953 *
233168124361466902944146099855900103168298336973117660251047536278410398151410570271601591803427840895029573
[ go to table entry ]
The one after that HP49(104) is 23818343988967755319 * 54777437615079105991 * c178
and, of course, the c178 is having some ECM done on it.
[ go to table entry ]
The decimal expansion of this c178 is
2288848748830723756709840158262687862277624865467675188746398067991005709059160227428299787778938917/
125184983761148626168860248112075875959985895793291153197651933268308457583237
Paul "
go directly to table entries 100103
Nicolas Daminelli (email)
and Paul Leyland (email)
factored HP49(104).c178 = p88 * p90. [ January 11, 2011 ]
[ go to table entry ]
" Hi Patrick,
After a *large* amount of work Nicolas Daminelli and I can report
further progress on HP(49). Eleven months ago we reported the complete
factorizations of HP49(100) through HP49(103) and the partial
factorization of HP49(104) into two small primes and a 178digit
composite. We have now split that c178 with the general number field
sieve; the factors are
prp88 factor:
44834403720805672946171216333384339757993452434913200790674799323428\
10228530227200671373
prp90 factor:
51051169612601980880823648155455991928883186163936778023298602714308\
8905987751279464964569
[ go to table entry ]
So, HP49(105) is
23818343988967755319547774376150791059914483440372080567294617121633\
33843397579934524349132007906747993234281022853022720067137351051169\
61260198088082364815545599192888318616393677802329860271430889059877\
51279464964569 with the easy factorization:
7 * 11 * 79 * 197 * 499 * 4091 * 15121 *
64389786040953919965710874344888197677736518233910043151126969109698\
54529140544736144679860920889446959040344022226855890207568273687210\
41445591222744297916341457480713464115193376230466835244912640871
[ go to table entry ]
Likewise, HP49(106) is
71179197499409115121643897860409539199657108743448881976777365182339\
10043151126969109698545291405447361446798609208894469590403440222268\
55890207568273687210414455912227442979163414574807134641151933762304\
66835244912640871
with the partial factorization
43 * 991 * 4810307 * c210
[ go to table entry ]
Nicolas and I will continue to try to factor the remaining composite.
Best wishes,
Paul "
go directly to table entries 104105
David Cleaver factored HP49(106).c210 = p60 * p151. [ March 15, 2011 ]
[ go to table entry ]
" Hi Patrick,
I wanted to let you know that I have factored HP49(106) c210 with
gmpecm. The parameters used to find this factorization were B1=3e9
and lucky sigma=2191726882. This split the c210 into p60*p151, with:
p60 = 190452757734166693416188232333259334611734162845489390418059
p151 = 18232697103041058392848310072109296903722975431973052755627\
465217854545763653919927047996513309176643018205699833490198368212\
97297023820367595810298059
[ go to table entry ]
This leads us to HP49(107):
439914810307190452757734166693416188232333259334611734162845489390\
418059182326971030410583928483100721092969037229754319730527556274\
652178545457636539199270479965133091766430182056998334901983682129\
7297023820367595810298059
Which has a factorization of:
1753 * 390120509 * c211
I have managed to use gmpecm to break the HP49(107) c211 down into
p32*p41*c139, with:
p32 = 93072922824766566567768442402519
(B1=1e6, sigma = 4276099043 or 2869501145)
p41 = 37311795374684221788102577672620935022701
(B1=43e6, sigma = 2362621067)
I then used ggnfs and msieve to break the HP49(107) c139
into p48*p91, with:
p48 = 776608774003332977699738989914377031406165943489
p91 = 238515217510615583066827682091457775058209547751328924950233\
3328800859279231550379002369437
[ go to table entry ]
This leads us to HP49(108):
175339012050993072922824766566567768442402519373117953746842217881\
025776726209350227017766087740033329776997389899143770314061659434\
892385152175106155830668276820914577750582095477513289249502333328\
800859279231550379002369437
Which has a factorization of:
3 * 67 * 173 * 7043 * 3449252363 * 350737390831 *
50181679161380508176090501 * c170
I then used gmpecm to break the c170 into p33*c137, with
p33 = 426702672788176702435652976517619
(B1=1e6, sigma = 343265977)
I am currently working on the c137.
Best Wishes,
David C.
[ March 18, 2011 ]
Hello Patrick,
I wanted to let you know that I have factored the c137 from HP49(108)
with ggnfs and msieve into p56*p82, with:
p56 = 16940220143895123609020488909230648807347076448219872853
p82 = 163148117268223088893371751081490490332227703164667871910769\
4729251345099067292373
[ go to table entry ]
This brings us to HP49(109):
367173704334492523633507373908315018167916138050817609050142670267\
278817670243565297651761916940220143895123609020488909230648807347\
076448219872853163148117268223088893371751081490490332227703164667\
8719107694729251345099067292373
With partial factorization:
3 * 13 * 461 * 9919193 * c218
The decimal expansion of this c218 is:
205887366265114089846147577123320653125800725081307919448400281943\
987418697225435524701265304377790514135358007620424041194122984702\
926363070110507466952409055377392799800423695957762440386203758986\
71169513641669439559
Work is continuing on the c218. Thanks for your time.
[ go to table entry ]
David C."
go directly to table entries 106109
David Cleaver factored HP49(109).c218 = p53 * p166. [ April 20, 2011 ]
[ go to table entry ]
" Hello Patrick,
I have made a little more progress on HP49. I have factored
HP49(109).c218 with gmpecm. The parameters used to find this
factorization were B1=3e9 and lucky sigma=2191180896. This broke
the c218 into p53 * p166, with:
p53 = 10218004525815126545868469943487168366937255818391163
p166 = 20149469081262686435623240342745108022916072191842974708192\
355391999307609311781113754713832186190659994372136928622288341440\
04507405010566482975467510026391893737893
[ go to table entry ]
This brings us to HP49(110):
313461991919310218004525815126545868469943487168366937255818391163\
201494690812626864356232403427451080229160721918429747081923553919\
993076093117811137547138321861906599943721369286222883414400450740\
5010566482975467510026391893737893
Which had an easy factorization of:
3 * 7 * 619 * 23642578733 * c218
Upon seeing another c218, I thought this would take a while, but
after a little bit of work with gmpecm, I found a p17 and p21,
which brought us to c218 = p17 * p18 * c181
p17 = 10567889515208903
p21 = 138613953787999806719
[ go to table entry ]
I am continuing to work on HP49(110).c181. Its decimal expansion is:
696281547241055004524787014448111167755504896151096804844368096365\
759227064152650920222099941253589429607814503738760057809654611250\
7652368908194481257411644162747345175089477132247
David C."
go directly to table entries 109110
David Cleaver factored HP49(110).c181 = p79 * p103. [ September 03, 2012 ]
[ go to table entry ]
" Hello Patrick,
I have quite a few developments in the HP49 saga to report to you. I
spent about six to eight months trying to factor the HP49(110).c181 via
ecm. However, that never proved fruitful. Then earlier in the year I
started factoring the c181 via the General Number Field Sieve using
ggnfs and msieve to do the factorization. I started gathering
relations on 2012/05/17 and finished gathering relations on 2012/08/11.
On that day a 22M^2 matrix was built and linear algebra ran on it until
2012/08/30. 1.5 hours later, the square root step found the factors on
the first dependency. The c181 split into p79 * p103, with:
p79 = 13242639228855682038275386963913139191902992119830964965826611351\
44957500774771
p103 = 5257875980823060025161989259479167407618986741511789127217197204\
189147347509304829105884519047315609357
[ go to table entry ]
From here I have started using an excellent factoring utility called
yafu, which can very quickly find small factors and can even keep
working until it fully factors a number. In order to factor a number,
it checks for small factors, it tries the Fermat method, Pollard rho,
p1, ecm, the quadratic sieve, and it can try factoring via the number
field sieve. Some of the functionality depends on external binaries,
each of which are easy to find online. I typically use yafu to find
small factors of these numbers, and then I will manually run gmpecm
to try to find larger factors.
*** The above factorization leads us to HP49(111), which is a c236:
37619236425787331056788951520890313861395378799980671913242639228855682\
03827538696391313919190299211983096496582661135144957500774771525787598\
08230600251619892594791674076189867415117891272171972041891473475093048\
29105884519047315609357
Which had an easy factorization of:
3 * 7 * 3119 * 30168011 * 859257036259 * p212, with:
p212 = 2215672318292438329341551789093919668756597710700506491362229289\
49716840718670030689861282126551415737410604184248168739079523813765870\
35978526994468873432733002148738098978790716880067275297057598545539637\
098807
[ go to table entry ]
*** This leads us to HP49(112), which is a c238:
37311930168011859257036259221567231829243832934155178909391966875659771\
07005064913622292894971684071867003068986128212655141573741060418424816\
87390795238137658703597852699446887343273300214873809897879071688006727\
5297057598545539637098807
Which partially factored into:
131 * 2721660787 * 364148211209 * 4332696358733373457 * c196
I was able to factor HP49(112).c196 with gmpecm with B1=110e6 and lucky
sigma=426853020. This gives us the split c196 = p46 * p151, with:
p46 = 2871080232471495934021653967701541108613371057
p151 = 2310258942683190562148481349981529646166666457710725946445425378\
87492749301442403975219925042828842113740517603022067825908798556477692\
9828767588285591
[ go to table entry ]
*** This leads us to HP49(113), which is a c241:
13127216607873641482112094332696358733373457287108023247149593402165396\
77015411086133710572310258942683190562148481349981529646166666457710725\
94644542537887492749301442403975219925042828842113740517603022067825908\
7985564776929828767588285591
Which partially factored into:
3 * 13 * 23 * 521845650935569 * 868711762772471 * 319988447520300554621
* 28389161986882946018325701897 * c159
I was able to factor HP49(113).c159 with gmpecm with B1=110e6 and lucky
sigma=1608282488. This gives us the split c159 = p46 * p113, with:
p46 = 4476784590773507504219451975358661227634604289
p113 = 7937968436512120054007404759114714054246049623545813848950867773\
8334605570121814426485117922308620342259219122429
[ go to table entry ]
*** This leads us to HP49(114), which is a c244:
31323521845650935569868711762772471319988447520300554621283891619868829\
46018325701897447678459077350750421945197535866122763460428979379684365\
12120054007404759114714054246049623545813848950867773833460557012181442\
6485117922308620342259219122429
Which pretty quickly factored into:
19 * 983 * 2663 * 78607 * 9934389995249 * 21656051585046364524395089
* 45811515442003960460099942651 * p164, with:
p164 = 8128977805895626607033264670100486500595772941131584619352172246\
20002330247018292473999585316046931850891592168404345391443470839102446\
76679456195607708735639313427
[ go to table entry ]
*** This leads us to HP49(115), which is a c246:
19983266378607993438999524921656051585046364524395089458115154420039604\
60099942651812897780589562660703326467010048650059577294113158461935217\
22462000233024701829247399958531604693185089159216840434539144347083910\
244676679456195607708735639313427
Which pretty quickly factored into:
3 * 3 * 3 * 339257 * 256784956591 * 36693424661311252997
* 12089711795346540523800293 * p183, with:
p183 = 1915138220008002714613864800803463985954766228490424773556933617\
71517488657715528360851816690277903228983539095849848639342470604943315\
995243199368402520964016310098028081653466011863
[ go to table entry ]
*** This leads us to HP49(116), which is a c250:
33333925725678495659136693424661311252997120897117953465405238002931915\
13822000800271461386480080346398595476622849042477355693361771517488657\
71552836085181669027790322898353909584984863934247060494331599524319936\
8402520964016310098028081653466011863
Which took a short while to factor into:
227 * 52386283 * 39852303700003 * 34918470225660868578167
* 71390396918591830182237959705744641 * p169, with:
p169 = 2821594399022506045260907988881750768134579956275599251807250458\
64578242838340892740600945894595344446356435593917588135425058734571590\
0852529152005426789424094520021323
[ go to table entry ]
*** This leads us to HP49(117), which is a c252:
22752386283398523037000033491847022566086857816771390396918591830182237\
95970574464128215943990225060452609079888817507681345799562755992518072\
50458645782428383408927406009458945953444463564355939175881354250587345\
715900852529152005426789424094520021323
Which has the partial factorization:
3 * 23 * 99525233 * 12143755081 * 2844434001269627828783 * c210
The decimal expansion of HP49(117).c210 is:
95917046558938390327954019204739154761431263890783122604947504259838592\
10155458387786445163000407451624133752306936169505691875284896202298903\
67061416022719796052116843953349582116196928958606632045980053799913
[ go to table entry ]
I am continuing to work on HP49(117).c210. The search continues!
David C."
go directly to table entries 110117
David Cleaver factored HP49(117).c210 = p95 * p116. [ December 3, 2014 ]
[ go to table entry ]
" Hello Patrick,
I have some more progress to report! I have finally factored the c210
from HP49(117). I started trying to factor it via ecm with GMPECM from
August 31, 2012, through April 1, 2013. In that time I had completed
6*t60 (= 1.2*t65). Since it looked like ecm wouldn't crack it, I decided
to factor it via GNFS. I used Msieve, with gpu polynomial selection
capabilities, to run polynomial selection from March 16, 2013, through
July 15, 2013. I did some test sieving from July 2, 2013, through
July 17, 2013, to figure out which would most likely be the best polynomial
to use. The polynomial/parameters that I ended up using were:
type: gnfs
# norm 1.276810e020 alpha 9.238464 e 9.932e016 rroots 5
skew: 380780879.24
c0: 162062780465967901494709111193197313231897108452480
c1: 13812347733683567053764230784091491451134952
c2: 22102839514830809629173664060748362
c3: 520746958301071507022048503
c4: 168875771147350788
c5: 535643820
Y0: 17807540461984351122128386845862353501171
Y1: 108424014196861749931
rlim: 500000000
alim: 500000000
lpbr: 32
lpba: 33
mfbr: 64
mfba: 96
rlambda: 2.7
alambda: 3.7
I then collected relations from August 1, 2013, through May 19, 2014. It was
thanks to the help of the Amazon Elastic Cloud Compute (EC2) service that I
was able to finish collecting relations so quickly. I bought spot instances
to keep costs down, but there were several times when spot pricing would go
above my high bid and so my instances were shutdown. When I saw the prices
fall again I would start up several new instances. From August through
February I had 5 spot instances running. And from March through May I had
10 spot instances running. Each instance was a cc2.8xlarge instance, which
was a dual Xeon E52670 (total of 16 cores, 32 threads) with 60.5GB ram.
On each instance I ran the Amazon Linux AMI x86_64 HVM EBS OS and had 32
threads collecting relations. I wrote a simple web interface to hand out
work assignments. I wrote a simple script, that ran on each instance, which
got work from my web interface to keep all 32 threads busy. When a thread
finished processing a chunk of work, the script would gzip the relations and
send those over to my ftp server. Each thread used gnfslasieve4I16e and
would take anywhere from 9 to 12 hours to process a chunk of work. In
total, I ran 29882 instance hours x 32 threads = 956224 thread hours. I
collected a total of 859026466 relations with 520615500 unique, 338410717
duplicate, and 249 bad relations. From May 20, 2014, through May 29, 2014,
I tested different target densities in Msieve to make the matrix as small as
possible. I was able to use a target density of 125 which got the matrix
down to 71776202 x 71776379. I then ran Linear Algebra on a single 16 core
machine from May 29, 2014, through October 3, 2014, using 13 threads
(because more could actually slow the job down). At the end of that,
3052 hours, it had recovered 26 nontrivial dependencies. I then started
running the SquareRoot step, which took about 10.5 hours per dependency,
and the factorization was found on the 2nd dependency on October 4th, 2014!
So now, after 765 days (or 2 years, 1 month, and 5 days) of work, with
568 days (or 1 year, 6 months, 19 days) of that spent on GNFS on the c210,
we find that HP49(117) has the full factorization:
3 * 23 * 99525233 * 12143755081 * 2844434001269627828783 * p95 * p116
p95 = 579991850255395814930902290226590574070465619261577712182337361547894\
69010389546134055763746997
p116 = 16537654195779115452085989138575434836851289636159339542929128160183\
686876788663786928798769682882741367314660621029
*** This leads us to HP49(118), which is a c255:
323995252331214375508128444340012696278287835799918502553958149309022902265\
905740704656192615777121823373615478946901038954613405576374699716537654195\
779115452085989138575434836851289636159339542929128160183686876788663786928\
798769682882741367314660621029
Which had the partial factorization:
31 * 59 * 1559 * 89626966666659827 * c232
The c232 was factored by GMPECM after about 60 hours with the lucky
parameters: Using B1=43000000, B2=240490660426, sigma=1:1648399426
c232 = p47 * p185
p47 = 13239299873681306480869143752057086094694552407
p185 = 95758009093092999931626631706893838237446804628957426949545161439643\
850763983152977771070456646529606514514350669236880120878528179383004060084\
979184123521819930400481691041161410407051
*** This leads us to HP49(119), which is a c257:
315915598962696666665982713239299873681306480869143752057086094694552407957\
580090930929999316266317068938382374468046289574269495451614396438507639831\
529777710704566465296065145143506692368801208785281793830040600849791841235\
21819930400481691041161410407051
Which has the partial factorization:
73 * 16249 * c251
The decimal expansion of HP49(119).c251 is:
266330909267922634367369046305315204797687428494350971277546348221683954382\
507914865091802754788127799593469081315896606977094898528309347119787046816\
393993232632706978213255816917295388773177366265980367036319706797376648877\
20652086830617767029002763
I have been trying to factor this c251 with GMPECM since October 8th.
So far, I have run around 2*t60 with no luck. If this number ends up not
having a factor with less than, maybe, 80 digits, then HP49 may be stuck
here at step 119 for quite a long time. It will take a LOT of effort
to factor a c251 via GNFS. Here's hoping for a lucky ecm curve or a
dedicated effort to push this number further! The search continues!
David C."
go directly to table entries 117119
The Table
4
2 * 2
2 * 11
Homeprime 211 reached after 2 steps
( Sloane's A037919 )
6
2 * 3
Homeprime 23 reached after 1 step
8
2 * 2 * 2
2 * 3 * 37
3 * 19 * 41
3 * 3 * 3 * 7 * 13 * 13
3 * 11123771
7 * 149 * 317 * 941
229 * 31219729
11 * 2084656339
3 * 347 * 911 * 118189
11 * 613 * 496501723
97 * 130517 * 917327
53 * 1832651281459
3 * 3 * 3 * 11 * 139 * 653 * 3863 * 5107
Homeprime 3331113965338635107 reached after 13 steps
( Featured in Prime Curios! 3331113965338635107 )
( Sloane's A006919 and A037920 )
9
3 * 3
3 * 11
Homeprime 311 reached after 2 steps
( Sloane's A037921 )
10
2 * 5
5 * 5
5 * 11
7 * 73
Homeprime 773 reached after 4 steps
( Sloane's A037922 )
12
2 * 2 * 3
Homeprime 223 reached after 1 step
14
2 * 7
3 * 3 * 3
3 * 3 * 37
47 * 71
13 * 367
Homeprime 13367 reached after 5 steps
( Sloane's A037923 )
15
3 * 5
5 * 7
3 * 19
11 * 29
Homeprime 1129 reached after 4 steps
( Sloane's A037924 )
16
2 * 2 * 2 * 2
2 * 11 * 101
3 * 11 * 6397
3 * 163 * 6373
Homeprime 31636373 reached after 4 steps
( Sloane's A037925 )
18
2 * 3 * 3
Homeprime 233 reached after 1 step
20
2 * 2 * 5
3 * 3 * 5 * 5
5 * 11 * 61
11 * 4651
3 * 3 * 12739
17 * 194867
19 * 41 * 22073
709 * 273797
3 * 97 * 137 * 17791
11 * 3610337981
7 * 3391 * 4786213
3 * 3 * 3 * 3 * 7 * 23 * 31 * 1815403
13 * 17 * 23 * 655857429041
7 * 7 * 2688237874641409
3 * 31 * 8308475676071413
Homeprime 3318308475676071413 reached after 15 steps
( Sloane's A037926 )
21
3 * 7
Homeprime 37 reached after 1 step
22
2 * 11
Homeprime 211 reached after 1 step
24
2 * 2 * 2 * 3
3 * 3 * 13 * 19
Homeprime 331319 reached after 2 steps
( Sloane's A037927 )
25
5 * 5
5 * 11
7 * 73
Homeprime 773 reached after 3 steps
( Sloane's A037928 )
26
2 * 13
3 * 71
7 * 53
3 * 251
Homeprime 3251 reached after 4 steps
( Sloane's A037929 )
27
3 * 3 * 3
3 * 3 * 37
47 * 71
13 * 367
Homeprime 13367 reached after 4 steps
( Sloane's A037930 )
28
2 * 2 * 7
Homeprime 227 reached after 1 step
30
2 * 3 * 5
5 * 47
Homeprime 547 reached after 2 steps
( Sloane's A037931 )
32
2 * 2 * 2 * 2 * 2
2 * 41 * 271
Homeprime 241271 reached after 2 steps
( Sloane's A037932 )
33
3 * 11
Homeprime 311 reached after 1 step
34
2 * 17
7 * 31
17 * 43
3 * 7 * 83
3 * 13 * 97
Homeprime 31397 reached after 5 steps
( Sloane's A037933 )
35
5 * 7
3 * 19
11 * 29
Homeprime 1129 reached after 3 steps
( Sloane's A037934 )
36
2 * 2 * 3 * 3
7 * 11 * 29
Homeprime 71129 reached after 2 steps
( Sloane's A037935 )
38
2 * 19
3 * 73
PALINDROMIC homeprime 373 reached after 2 steps
( Sloane's A037936 )
39
3 * 13
PALINDROMIC homeprime 313 reached after 1 step
40
2 * 2 * 2 *5
5 * 5 * 89
3 * 3 * 3 * 3 * 3 * 23
7 * 7 * 59 * 1153
29 * 2675557
3 * 31 * 3147049
809 * 1019 * 4019
3 * 53639 * 502807
3 * 31 * 41 * 92745739
Homeprime 3314192745739 reached after 9 steps
( Sloane's A037937 )
42
2 * 3 * 7
3 * 79
Homeprime 379 reached after 2 steps
( Sloane's A037938 )
44
2 * 2 * 11
3 * 11 * 67
3 * 3 * 3463
13 * 113 * 227
173 * 229 * 331
11 * 15748121
541 * 2062381
11 * 607 * 810553
2281 * 5088913
Homeprime 22815088913 reached after 9 steps
( Sloane's A037939 )
45
3 * 3 * 5
5 * 67
3 * 3 * 3 * 3 * 7
17 * 37 * 53
239 * 727
3 * 41 * 1949
Homeprime 3411949 reached after 6 steps
( Sloane's A037940 )
46
2 * 23
Homeprime 223 reached after 1 step
48
2 * 2 * 2 * 2 * 3
71 * 313
3 * 11 * 2161
3 * 13 * 199 * 401
19 * 43 * 109 * 3517
11 * 17 * 109 * 877 * 1087
23 * 1481 * 7039 * 46591
3 * 3 * 7 * 53 * 67 * 1034726207
3 * 11251223678242069
23 * 4583 * 2952795526741
359 * 5782291 * 1130063089
835996339 * 43011938251
31 * 49123 * 54898161457127
467 * 79367 * 8496358995643
61 * 61 * 79 * 1591356884791277
Homeprime 6161791591356884791277 reached after 15 steps
( Sloane's A037941 )
49
STEP NUMBER  FACTORS OF [STEPNUMBER  1] 

0 
49 
1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 
 3
 13
 18.836.286.194.043.641

20 
 3
 3
 3
 43
 14.369
 161.461
 11.627.309

21 
 3
 32.057
 1.618.455.677
 2.142.207.827

22 
 3
 1.367
 2.221
 5.573
 475.297
 1.376.323.127

23 
 7
 3.391
 51.263
 25.777.821.480.557.336.017

24 
 47
 67
 347
 431
 120.361.987
 12.947.236.602.187

25 
 3
 7
 7
 17
 12.809
 57.470.909
 57.713.323
 4.490.256.751

26 
 3.096.049.809.383
 121.823.389.214.993.262.890.297

27 
 7
 379
 62.363.251
 18.712.936.424.989.555.929.478.399

28 
 13
 1.181
 145.261.411
 33.089.538.087.518.197.265.265.053

29 
 3
 19
 521
 441.731.977.174.163.487.542.111.577.539.726.749

30 
 59
 5.415.617.656.474.189.392.601.483.764.603.009.147.911

31 
 13
 8.423
 1.466.957
 3.706.744.784.027.901.056.001.426.046.777

32 
 3
 12.919
 2.501.509.379
 96.709.539.317.201
 1.476.342.474.406.759

33 
 3
 2.039
 2.713
 3.121
 399.320.591
 151.296.378.525.102.203.388.346.189

34 
 13
 3.119
 651.853
 9.121.952.491
 13.288.820.301.002.347.322.382.772.769

35 
 11
 1.037.257.958.982.410.527
 11.667.094.407.302.642.807.490.159.301.277

36 
 11
 521
 947
 18.705.941
 109.372.661.574.127.837.007.959.097.317.735.411.121

37 
 63.186.539.723.577.497
 18.234.812.726.673.333.988.788.742.328.093.848.793

INFO 
From here on Warut Roonguthai from Bangkok Thailand
took over this factorisation task and completed the picture upto step 55 ! 
38 
 7
 11
 11
 131
 139
 197
 4.493
 5.335.335.211
 86.754.240.313.734.089.696.843.349.346.661

39 
 3
 7
 7
 11
 13
 71
 101
 6.948.131.790.459.407
 678.947.892.694.155.341.923.379.077.407.684.653

40 
 73
 101
 135.623
 125.831.783.320.571
 29.971.031.882.457.634.609.852.680.847.686.251.943.317

41 
 3
 11
 11.971
 72.271
 564.982.895.268.105.721.087
 453.190.074.064.393.495.190.773.755.017.652.247

42 
 3
 7
 31
 1.153.929.569.843
 4.141.591.345.095.168.649.790.005.875.768.086.611.455.076.505.611.
166.279

43 
 3
 3
 3
 3
 3
 7
 11
 151
 1.013
 16.117.103
 17.692.613
 68.878.841
 350.604.039.551
 18.931.001.222.053.567.659.972.075.047

44 
 379
 87.951.744.462.008.749.649.348.751.784.002.342.702.203.325.604.103.
216.176.784.227.054.268.232.116.293

45 
 7
 11
 13
 127
 223
 678.209.445.473.726.709.080.777
 1.975.783.627.239.622.801.952.043.181.949.336.676.523.721.088.629

46 
 1.187
 526.871
 1.137.089.799.261.311.878.547.509.623.397.801.472.835.395.151.221.
348.397.140.205.614.034.351.359.377

47 
 727
 43.189
 238.319.520.082.455.179.230.901
 1.586.997.602.061.240.784.213.075.478.539.371.666.587.718.903.373.
678.159

48 
 7
 17
 133.578.549.596.081
 30.755.003.310.552.165.121.241
 1.487.965.543.982.552.577.200.370.139.716.504.126.134.573.487.841

49 
 101
 31.408.991
 1.813.110.797
 315.942.563.737
 22.673.665.034.562.629.167.007
 17.404.858.255.585.899.140.058.606.434.748.914.537

50 
 7
 223
 4.097.425.722.814.870.501.750.027
 158.400.247.194.032.517.114.645.379.347.463.054.616.952.506.288.341.
051.992.127.524.371

51 
 13
 19
 43
 647
 7.129.763.692.644.907
 1.474.339.241.979.213.097.094.329.426.579.438.949.847.711.959.739.
988.164.520.170.173.919.543.019

52 
 7
 313.477
 1.123.279
 1.964.411.737
 3.641.599.001.219
 748.294.379.551.351.547.309.579.716.601.707.756.498.905.354.854.859.
516.129.556.133

53 
 7
 7
 1.163
 1.753
 1.666.965.059
 43.917.711.251.563.303.775.439.749.470.615.275.651.919.009.839.779.
126.156.077.932.235.038.449.963.463.079.517

54 
 3
 3
 3
 7
 167
 244.325.056.337.062.272.469.154.317.119.778.326.007.522.724.375.712.
551.901.020.784.140.917.502.333.620.946.432.057.740.074.247.159

55 
 95.261
 436.871
 49.566.201.248.508.475.546.723
 1.617.796.842.248.675.690.632.434.479.247.655.080.197.916.188.170.
687.651.874.396.287.605.885.943

INFO 
The next hard factorisation 3 * 73 * c105, where c105 is a 105digit composite number, was solved
thanks to Paul Leyland (email) efforts using his MPQS software. 
56 
 3
 73
 21.034.137.982.005.155.236.145.561.292.210.835.084.361
 20.679.893.341.907.378.249.919.955.987.757.483.932.846.633.610.599.
367.336.113.869.677

57 
 23
 1.987
 143.387.903
 72.143.526.680.202.408.596.508.799
 113.090.049.128.593.775.731.305.780.181
 69.806.150.833.129.469.702.241.285.634.418.469.092.261

58 
 3
 9.670.596.449.961.538.913
 327.033.806.488.198.658.171
 2.477.201.257.046.531.399.831
 98.704.160.856.892.726.169.859.475.666.929.224.642.658.253.490.099

59 
 13
 19
 160.609.702.226.564.934.871.769.367.637.604.043.150.490.881.457.998.
804.263.724.628.334.834.659.339.646.664.655.301.520.918.741.800.172. 705.479.717

60 
 157
 8.402.296.877.084.245.636.527.845.664.762.022.955.688.856.629.818.
203.814.039.896.335.212.323.787.639.106.029.711.180.264.464.450.584. 714.475.831.081

61 
 128.168.101
 172.477.653.991
 22.455.445.786.817
 65.332.794.124.035.517.737.631
 220.639.339.061.394.154.385.983.823
 220.581.485.388.414.661.593.519.979.370.771

62 
 3
 7
 4.939.695.780.628.938.754.847
 12.355.503.626.172.786.037.279.469.976.176.282.283.397.864.072.628.
868.373.427.595.840.785.303.013.447.717.379.263.356.481.433

63 
 7
 105.964.373
 24.350.824.393.100.367.598.521.157.441
 467.286.378.036.900.065.597.110.992.698.129.627
 4.442.288.093.165.186.933.753.059.824.843.132.778.546.864.729

64 
 2.412.283.239.859
 96.754.153.417.988.743.222.695.506.250.425.178.335.278.781
 30.445.640.085.060.252.589.094.308.546.324.740.821.837.930.438.456.
004.090.264.769.151

65 
 1.189.651
 11.038.123
 7.118.181.533
 5.792.688.085.739
 10.922.713.310.718.234.928.561
 4.078.812.653.263.891.607.034.642.347.735.526.263.222.971.482.322.
832.583.856.613.441

66 
 3
 191
 313
 5.441
 11.353
 65.629
 86.351.721.217.907
 189.480.492.770.041.397.975.778.744.916.436.090.908.111.415.209.216.
521.065.245.015.362.538.780.184.733.923.955.090.125.011

67 
 2.393
 21.757
 410.613.076.508.699
 3.876.849.118.743.036.123.169
 173.082.369.408.090.882.957.108.869.969
 2.224.658.011.938.594.415.822.497.545.769.437.225.933.713.909.741.
322.549

68 
 3.911
 6.119.196.047.318.155.881.071.321.597.670.976.075.264.953.130.076.
917.334.206.317.777.952.208.821.194.636.834.060.838.300.796.681.794. 309.836.215.092.286.809.999.933.859

69 
 31
 3.853
 327.487.748.935.034.436.140.092.523.811.338.219.703.752.124.051.891.
504.042.381.774.719.137.598.761.092.693.894.443.884.014.138.933.899. 804.173.004.823.156.541.697.713

70 
 3
 9.357.271.333
 578.256.191.549.058.763.967
 19.334.634.570.794.597.874.282.359.270.708.598.778.922.377.643.006.
273.044.425.599.731.834.360.178.326.725.701.290.113.727.046.644.961

71 
 1.759
 108.573.523
 111.815.087
 6.522.888.319
 16.655.258.595.913.603
 145.949.291.218.539.720.719
 11.623.622.370.200.394.561.800.507.172.511.384.828.996.177.438.754.
510.263.275.872.371.913

72 
 6.163
 4.174.200.952.681
 5.547.777.236.472.425.061.422.657
 1.232.560.116.019.241.948.254.195.923.713.574.444.418.269.087.218.
170.388.702.566.637.060.708.432.758.564.543.801.513.313.435.003

73 
 3
 3.061
 7.927
 3.477.193
 6.255.801.682.308.107.501.611.939
 3.892.390.194.218.645.736.560.211.718.067.925.517.775.155.309.393.
407.847.029.027.375.197.651.980.758.092.755.512.331.680.882.731.929

INFO 
The next incomplete factorisation 29201 * c137, where c137 is a 137digit composite number p46 * p91,
was also solved by Paul Leyland (email) after a six month search using ECM from [ Dec 9, 1999 ] to [ June 14, 2000 ] 
74 
 29.201
 3.804.796.914.905.629.947.782.783.176.497.447.433.056.300.673
 2.975.755.149.892.502.017.761.442.206.397.381.501.294.987.050.078.
840.501.836.673.559.420.025.718.611.870.754.546.792.073

75 
 7
 33.007.433
 129.882.136.063.656.181.313.509
 97.307.019.427.933.601.730.313.435.852.553.855.574.587.320.243.165.
126.450.409.979.970.503.810.321.948.072.508.820.320.776.101.506.693. 631.743.787

76 
 3
 3
 3
 3
 17
 1.418.896.363
 43.815.373.687
 10.900.494.140.674.939
 78.550.880.421.960.910.045.725.733.457.082.622.904.976.800.065.728.
761.972.806.264.115.655.795.567.291.460.808.947.772.426.698.634.661. 509

77 
 53
 113
 503
 1.158.841
 35.265.497.428.421.571.545.237
 27.074.564.821.365.993.301.638.001.206.958.026.410.300.450.845.667.
511.953.219.872.049.205.059.398.636.855.089.534.014.521.039.395.489. 209.448.269.731

78 
 7
 11
 18.913
 153.641
 23.738.139.268.537.590.206.883.527.079.655.170.564.630.066.989.225.
472.455.317.256.070.999.323.579.605.002.060.949.400.881.236.203.810. 641.821.862.903.434.018.510.186.009.610.511.699.191

79 
 67
 79
 599.342.540.099
 1.483.398.061.194.277
 2.532.349.728.015.299
 596.797.125.348.335.536.627.185.801.694.933.317.807.298.312.560.463.
320.551.394.686.643.698.469.578.957.649.460.916.482.413.383.929.168. 101.631

INFO 
Alex Kruppa (email) & Paul Leyland (email) are delighted to announce that they have factored HP49(80).c138 = p69 * p70. [ April 5, 2002 ] 
80 
 227
 878.737
 4.522.823.519
 224.690.133.218.881.151.252.602.753.388.830.692.415.125.120.972.614.
888.557.571.233.246.513  3.344.456.987.746.930.138.631.822.411.149.806.794.710.441.073.001.
296.631.385.753.707.501.227

INFO 
Alex Kruppa (email) & Paul Leyland (email)
have factored HP49(81).c136 = p51 * p85. [ May 16, 2002 ] 
81 
 7
 1.714.759
 1.194.888.576.072.091
 448.583.441.180.516.821.621.320.259.546.896.223.296.373.907.144.113
 3.541.865.184.678.001.629.601.182.607.226.312.880.348.900.246.844.
848.900.227.068.283.166.325.369.629.573.991.313

82 
 3
 569
 677.898.301
 917.546.126.718.667.312.210.004.739.359
 1.441.642.672.303.502.027.723.568.412.074.125.039.561.184.677.294.
613.349  4.685.165.561.827.338.796.028.717.898.771.987.572.324.741.651.023.
755.870.645.553.949

83 
 13
 43
 182.879.655.593
 1.252.205.350.667.723.657
 189.957.893.612.838.517.267.061
 540.935.917.814.791.413.258.389
 142.533.635.763.834.905.022.317.403.438.538.106.689
 190.395.089.953.894.891.189.281.091.736.983.632.645.331.931

INFO 
Alex Kruppa (email)
factored HP49(84) = c167 = p53 * p114. [ May 25, 2002 ] 
84 
 81.477.382.431.617.858.607.629.654.669.086.224.895.030.590.860.856.
949  164.853.464.798.393.151.511.356.156.289.762.200.575.122.773.801.949.
600.629.560.294.634.580.313.680.599.127.053.982.894.889.995.189.794. 735.466.895.119

INFO 
Alex Kruppa (email) & Paul Leyland (email)
factored HP49(85).c140 = p68 * p73. [ July 1, 2002 ] 
85 
 7
 23
 10.939
 329.941
 5.766.611.709.588.259
 11.825.523.118.615.173.197.502.446.002.728.268.259.724.026.363.566.
923.708.566.001.149.123  2.056.156.715.909.184.026.132.747.632.727.779.809.352.896.881.830.
754.843.064.666.748.922.417.153

86 
 7
 11
 152.423
 53.387.051
 29.792.742.241.519.577.123.723
 4.064.661.599.738.415.396.279.098.270.261
 9.529.979.513.297.872.090.944.082.176.810.466.905.146.436.569.031.
922.355.314.552.189.529.971.609.287.404.512.991.501.436.510.263.046. 431

87 
 7
 73
 139
 1.277
 2.210.107
 38.614.940.387.293
 881.196.447.153.797.210.617
 319.174.266.576.457.649.040.084.823
 326.638.085.447.941.270.182.143.980.995.395.782.788.476.970.814.706.
644.749.940.729.528.404.612.096.239.900.219.479.360.678.030.127

88 
 17
 89
 89
 509
 200.383
 6.246.073.153
 2.081.978.503.432.161.073.350.874.430.821
 43.287.923.588.860.077.733.151.721.662.275.296.467.194.175.854.379.
832.571.565.911.228.616.000.410.077.677.224.181.081.033.204.896.924. 979.589.411.918.456.508.547.201

89 
 3769
 474.899.201.911.338.881.950.956.011.726.678.255.032.132.484.767.164.
521.661.790.375.056.741.861.961.568.511.982.561.766.109.629.656.634. 221.234.431.298.994.754.209.604.778.476.433.330.905.036.092.651.869. 706.813.369.438.025.592.069.129

INFO 
Alex Kruppa (email) & Paul Leyland (email)
factored HP49(90).c176 = p38 * p44 * p94. [ July 22, 2002 ] 
90 
 7
 17
 19
 61
 52.260.221.223.007.872.743.384.258.441.159.930.329
 66.076.192.121.556.175.190.138.089.562.410.746.399.109.073
 7.914.695.431.991.081.178.555.838.906.948.006.630.214.230.434.758.
907.150.001.616.425.065.499.127.576.368.383.425.223.083.297

91 
 1.017.881
 1.276.262.783
 7.551.791.567
 258.690.803.851
 1.152.103.070.861.311.990.567
 245.289.251.993.982.856.094.156.277.577.608.404.269.873.793.786.599.
973.130.102.184.325.224.471.143.204.231.199.123.108.701.419.629.477. 860.118.533.938.997.808.860.701

92 
 19
 31
 41
 463
 1.623.701
 2.697.733
 292.207.779.163
 1.851.660.955.792.477
 485.240.944.406.043.416.435.941.190.113.333
 79.159.166.944.790.863.542.342.035.580.567.700.871.333.581.386.748.
195.916.535.938.401.239.061.672.133.983.522.915.259.717.378.301.248. 272.957

93 
 3
 7
 1.889
 486.882.611.516.222.054.948.027.230.527.090.755.632.007.249.470.562.
618.772.705.150.226.666.757.269.379.397.851.701.811.665.148.878.558. 631.224.270.351.349.942.558.007.042.949.108.773.010.056.679.050.116. 253.803.574.164.282.317.671.667.506.144.553

94 
 59
 4.111
 7.817
 239.760.658.585.807
 14.839.683.336.419.017
 681.195.332.272.435.073.462.164.213.069.789.780.765.696.608.423.481
 80.928.237.269.671.741.847.251.176.442.051.230.228.132.913.831.062.
450.594.358.151.265.525.588.922.400.325.147.005.752.739.661.876.019

INFO 
Alex Kruppa (email) & Paul Leyland (email)
factored HP49(95).c153 = p68 * p85. [ January 20, 2003 ] 
95 
 3
 7
 26.141
 300.119
 1.811.141
 1.072.782.128.567.282.855.315.039
 25.381.603.104.475.027.190.830.989.059.811.875.365.234.972.412.236.
253.418.807.674.044.481  7.312.238.388.545.260.672.268.685.822.022.236.441.690.231.083.090.
644.547.260.008.735.024.402.747.827.401.799.039

96 
 17
 937
 2.999
 15.011.194.746.557
 33.716.362.272.572.345.351.978.861.258.895.209
 15.411.267.935.624.560.746.503.422.154.192.060.735.654.946.200.813.
373.057.159.461.202.228.335.655.539.331.091.962.808.118.604.240.541. 265.521.091.783.630.320.332.376.356.814.622.960.926.093

INFO 
Alex Kruppa (email) & Paul Leyland (email)
factored HP49(97).c151 = p55 * p96. [ July 16, 2003 ] 
97 
 19
 569
 683
 7.450.039
 865.252.586.740.571
 2.971.814.878.235.479.924.213
 1.771.052.383.785.311.834.993.979.061.208.604.132.871.538.232.335.
055.323  716.004.722.254.150.586.483.843.313.158.263.592.817.803.974.805.568.
194.113.667.109.128.698.119.199.668.739.510.916.110.374.031

98 
 3
 3
 3
 3
 17
 173
 313
 1.104.769.163
 366.751.448.517.289.166.567.507
 647.762.862.045.807.583.832.036.089.265.238.362.727.899.898.079.339.
354.405.235.894.851.158.438.173.650.422.654.045.381.281.492.110.241. 570.912.663.690.359.389.778.909.529.774.668.266.908.592.416.498.348. 856.300.467

99 
 107
 22.861
 3.700.198.301.407
 581.535.317.003.127.481
 633.253.493.734.999.759.793.567.866.901.093.615.000.660.202.729.464.
171.721.664.981.941.672.355.364.244.979.546.033.697.842.584.415.925. 459.377.122.074.668.562.830.876.181.453.746.637.033.053.422.944.132. 164.520.381.030.890.363

INFO 
Nicolas Daminelli (email)
factored HP49(100).c204 = p62 * p143. [ February 8, 2010 ]
Paul Leyland (email)
promptly extended the chain by factorizing HP49(101), HP49(102) & HP49(103) 
100 
 3
 37
 2789
 13.055.369.049.845.151.421.562.673.963.068.310.715.129.211.918.218.
895.785.368.571  26.530.781.642.034.476.718.502.313.327.970.096.030.290.047.554.836.
547.278.094.547.793.019.053.742.289.833.736.552.786.286.890.695.511. 249.799.288.531.238.767.159.656.593.259.713.779.239.907

101 
 3
 3
 179
 20.935.997.085.994.354.428.625.165.832.568.216.205.357.586.289.833.
142.037.175.405.671.543.840.180.497.605.235.553.518.623.409.887.008. 129.131.154.591.902.443.542.692.796.119.052.153.354.278.268.053.219. 461.692.294.780.071.701.582.743.195.860.483.406.058.135.687.932.783. 227.337

102 
 3
 7
 12.473
 69.442.311.247.884.384.744.096.581
 5.504.559.912.367.454.438.295.149.601.552.867.774.551.041
 33.138.209.996.227.094.175.301.272.382.137.857.317.340.226.024.600.
046.391.016.469.549.509.423.923.218.156.186.003.630.641.092.474.661. 686.802.611.798.393.277.995.398.615.210.685.360.809

103 
 29
 8.627
 89.345.257
 8.067.774.497
 96.513.008.244.398.921.562.099.305.449.602.682.965.323
 91.479.820.611.369.205.267.924.863.199.536.513.832.217.953
 233.168.124.361.466.902.944.146.099.855.900.103.168.298.336.973.117.
660.251.047.536.278.410.398.151.410.570.271.601.591.803.427.840.895. 029.573

INFO 
Nicolas Daminelli (email)
and Paul Leyland (email)
factored HP49(104).c178 = p88 * p90. [ January 11, 2011 ]
They promptly extended the chain by factorizing HP49(105) completely and HP49(106) partially. 
104 
 23.818.343.988.967.755.319
 54.777.437.615.079.105.991
 4.483.440.372.080.567.294.617.121.633.338.433.975.799.345.243.491.
320.079.067.479.932.342.810.228.530.227.200.671.373  510.511.696.126.019.808.808.236.481.554.559.919.288.831.861.639.367.
780.232.986.027.143.088.905.987.751.279.464.964.569

105 
 7
 11
 79
 197
 499
 4.091
 15.121
 643.897.860.409.539.199.657.108.743.448.881.976.777.365.182.339.100.
431.511.269.691.096.985.452.914.054.473.614.467.986.092.088.944.695. 904.034.402.222.685.589.020.756.827.368.721.041.445.591.222.744.297. 916.341.457.480.713.464.115.193.376.230.466.835.244.912.640.871

INFO 
David Cleaver
factored HP49(106).c210 = p60 * p151. [ March 15, 2011 ]
Hereafter he extended the chain by factorizing HP49(107), HP49(108) completely and HP49(109) partially. 
106 
 43
 991
 4.810.307
 190.452.757.734.166.693.416.188.232.333.259.334.611.734.162.845.489.
390.418.059  1.823.269.710.304.105.839.284.831.007.210.929.690.372.297.543.197.
305.275.562.746.521.785.454.576.365.391.992.704.799.651.330.917.664. 301.820.569.983.349.019.836.821.297.297.023.820.367.595.810.298.059

107 
 1.753
 390.120.509
 93.072.922.824.766.566.567.768.442.402.519
 37.311.795.374.684.221.788.102.577.672.620.935.022.701
 776.608.774.003.332.977.699.738.989.914.377.031.406.165.943.489
 2.385.152.175.106.155.830.668.276.820.914.577.750.582.095.477.513.
289.249.502.333.328.800.859.279.231.550.379.002.369.437

108 
 3
 67
 173
 7.043
 3.449.252.363
 350.737.390.831
 50.181.679.161.380.508.176.090.501
 426.702.672.788.176.702.435.652.976.517.619
 16.940.220.143.895.123.609.020.488.909.230.648.807.347.076.448.219.
872.853  1.631.481.172.682.230.888.933.717.510.814.904.903.322.277.031.646.
678.719.107.694.729.251.345.099.067.292.373

INFO 
David Cleaver
factored HP49(109).c218 = p53 * p166. [ April 20, 2011 ]
Hereafter he extended the chain by factorizing HP49(110) partially.
The quest for this elusive Home Prime continues with HP49(110)... 
109 
 3
 13
 461
 9.919.193
 10.218.004.525.815.126.545.868.469.943.487.168.366.937.255.818.391.
163
 2.014.946.908.126.268.643.562.324.034.274.510.802.291.607.219.184.
297.470.819.235.539.199.930.760.931.178.111.375.471.383.218.619.065. 999.437.213.692.862.228.834.144.004.507.405.010.566.482.975.467.510. 026.391.893.737.893

INFO 
David Cleaver
factored HP49(110).c181 = p79 * p103. [ September 3, 2012 ]
Hereafter he extended the chain by factorizing HP49(111),
HP49(112), HP49(113), HP49(114), HP49(115) and HP49(116) fully and HP49(117) partially.
The quest for this elusive Home Prime continues with HP49(117)... 
110 
 3
 7
 619
 23.642.578.733
 10.567.889.515.208.903
 138.613.953.787.999.806.719
 1.324.263.922.885.568.203.827.538.696.391.313.919.190.299.211.983.
096.496.582.661.135.144.957.500.774.771  5.257.875.980.823.060.025.161.989.259.479.167.407.618.986.741.511.
789.127.217.197.204.189.147.347.509.304.829.105.884.519.047.315.609. 357

111 
 3
 7
 3.119
 30.168.011
 859.257.036.259
 22.156.723.182.924.383.293.415.517.890.939.196.687.565.977.107.005.
064.913.622.292.894.971.684.071.867.003.068.986.128.212.655.141.573. 741.060.418.424.816.873.907.952.381.376.587.035.978.526.994.468.873. 432.733.002.148.738.098.978.790.716.880.067.275.297.057.598.545.539. 637.098.807

112 
 131
 2.721.660.787
 364.148.211.209
 4.332.696.358.733.373.457
 2.871.080.232.471.495.934.021.653.967.701.541.108.613.371.057
 2.310.258.942.683.190.562.148.481.349.981.529.646.166.666.457.710.
725.946.445.425.378.874.927.493.014.424.039.752.199.250.428.288.421. 137.405.176.030.220.678.259.087.985.564.776.929.828.767.588.285.591

113 
 3
 13
 23
 521.845.650.935.569
 868.711.762.772.471
 319.988.447.520.300.554.621
 28.389.161.986.882.946.018.325.701.897
 4.476.784.590.773.507.504.219.451.975.358.661.227.634.604.289
 79.379.684.365.121.200.540.074.047.591.147.140.542.460.496.235.458.
138.489.508.677.738.334.605.570.121.814.426.485.117.922.308.620.342. 259.219.122.429

114 
 19
 983
 2.663
 78.607
 9.934.389.995.249
 21.656.051.585.046.364.524.395.089
 45.811.515.442.003.960.460.099.942.651
 81.289.778.058.956.266.070.332.646.701.004.865.005.957.729.411.315.
846.193.521.722.462.000.233.024.701.829.247.399.958.531.604.693.185. 089.159.216.840.434.539.144.347.083.910.244.676.679.456.195.607.708. 735.639.313.427

115 
 3
 3
 3
 339.257
 256.784.956.591
 36.693.424.661.311.252.997
 12.089.711.795.346.540.523.800.293
 191.513.822.000.800.271.461.386.480.080.346.398.595.476.622.849.042.
477.355.693.361.771.517.488.657.715.528.360.851.816.690.277.903.228. 983.539.095.849.848.639.342.470.604.943.315.995.243.199.368.402.520. 964.016.310.098.028.081.653.466.011.863 
116 
 227
 52.386.283
 39.852.303.700.003
 34.918.470.225.660.868.578.167
 71.390.396.918.591.830.182.237.959.705.744.641
 2.821.594.399.022.506.045.260.907.988.881.750.768.134.579.956.275.
599.251.807.250.458.645.782.428.383.408.927.406.009.458.945.953.444. 463.564.355.939.175.881.354.250.587.345.715.900.852.529.152.005.426. 789.424.094.520.021.323 
INFO 
David Cleaver
factored HP49(117).c210 = p95 * p116. [ December 3, 2014 ]
Hereafter he extended the chain by factorizing HP49(118) fully,
and HP49(119) partially.
The quest for this elusive Home Prime continues with HP49(119)... 
117 
 3
 23
 99.525.233
 12.143.755.081
 2.844.434.001.269.627.828.783
 57.999.185.025.539.581.493.090.229.022.659.057.407.046.561.926.157.
771.218.233.736.154.789.469.010.389.546.134.055.763.746.997
 16.537.654.195.779.115.452.085.989.138.575.434.836.851.289.636.159.
339.542.929.128.160.183.686.876.788.663.786.928.798.769.682.882.741. 367.314.660.621.029 
118 
 31
 59
 1.559
 89.626.966.666.659.827
 13.239.299.873.681.306.480.869.143.752.057.086.094.694.552.407
 95.758.009.093.092.999.931.626.631.706.893.838.237.446.804.628.957.
426.949.545.161.439.643.850.763.983.152.977.771.070.456.646.529.606. 514.514.350.669.236.880.120.878.528.179.383.004.060.084.979.184.123. 521.819.930.400.481.691.041.161.410.407.051 
119 

PS 
Please doublecheck the correctness of the above results before using them for continuing the search. There is a nice presentation of the sequence available, see the following link.
Factordb.com
This is a nice validation that the numbers are correct as listed. 
No homeprime reached after 118 steps !!
50
2 * 5 * 5
3 * 5 * 17
Homeprime 3517 reached after 2 steps
51
3 * 17
Homeprime 317 reached after 1 step
52
2 * 2 * 13
Homeprime 2213 reached after 1 step
54
2 * 3 * 3 * 3
Homeprime 2333 reached after 1 step
55
5 * 11
7 * 73
Homeprime 773 reached after 2 steps
56
2 * 2 * 2 * 7
17 * 131
37 * 463
Homeprime 37463 reached after 3 steps
57
3 * 19
11 * 29
Homeprime 1129 reached after 2 steps
58
2 * 29
Homeprime 229 reached after 1 step
60
2 * 2 * 3 * 5
3 * 5 * 149
Homeprime 35149 reached after 2 steps
62
2 * 31
3 * 7 * 11
3 * 1237
Homeprime 31237 reached after 3 steps
63
3 * 3 * 7
Homeprime 337 reached after 1 step
64
2 * 2 * 2 * 2 * 2 * 2
2 * 3 * 7 * 11 * 13 * 37
29 * 101 * 80953
853 * 3411701
3 * 181 * 367 * 42821
127 * 2505013723
Homeprime 1272505013723 reached after 6 steps
65
5 * 13
3 * 3 * 3 * 19
11 * 13 * 233
11 * 101203
3 * 3 * 23 * 53629
3 * 3 * 1523 * 24247
3 * 3 * 3 * 7 * 47 * 3732109
11 * 18013 * 16843763
151 * 740406071813
3 * 13 * 13 * 54833 * 5458223
3 * 3 * 97 * 179 * 373 * 7523 * 71411
1571 * 1601 * 1350675311441
3 * 3 * 13 * 33391 * 143947 * 279384649
11 * 23 * 204069263 * 6417517893491
7 * 11 * 1756639 * 83039633268945697
29 * 29 * 5165653 * 13503983 * 12122544283
228345060379 * 1282934064985326977
3 * 3 * 3 * 2979253 * 3030445387 * 9367290955541
1381 * 3211183211 * 75157763339900357651
Homeprime 1381321118321175157763339900357651
reached after 19 steps
66
2 * 3 * 11
Homeprime 2311 reached after 1 step
68
2 * 2 * 17
3 * 739
Homeprime 3739 reached after 2 steps
69
3 * 23
17 * 19
3 * 3 * 191
Homeprime 33191 reached after 3 steps
70
2 * 5 * 7
Homeprime 257 reached after 1 step
72
2 * 2 * 2 * 3 * 3
3 * 7411
11 * 19 * 179
Homeprime 1119179 reached after 3 steps
74
2 * 37
3 * 79
Homeprime 379 reached after 2 steps
75
3 * 5 * 5
5 * 71
Homeprime 571 reached after 2 steps
76
2 * 2 * 19
7 * 317
3 * 3 * 3 * 271
Homeprime 333271 reached after 3 steps
77
See expansion of number 49 from step 2 onwards
No homeprime reached after 117 steps !!
78
2 * 3 * 13
3 * 3 * 257
7 * 4751
3 * 24917
101 * 3217
3 * 17 * 19867
3 * 89 * 118801
3 * 129706267
Homeprime 3129706267 reached after 8 steps
80
2 * 2 * 2 * 2 * 5
5 * 5 * 7 * 127
3 * 3 * 103 * 601
23 * 1439287
3 * 43 * 53 * 33851
31 * 521 * 212701
11 * 29 * 83 * 1190513
24917 * 45343789
3 * 13 * 17 * 3758288603
47 * 109 * 211 * 289720051
521 * 90420751035931
3 * 7 * 13 * 28927 * 6608832661
13 * 293 * 974872480075829
11 * 131 * 4259 * 1290683 * 1678277
19 * 75253 * 45682591 * 170420821
3541 * 55782362519794174081
47 * 94253 * 1769473 * 45181157867
13 * 35801984243 * 10300789571213
24144697 * 1012307071 * 5465225099
7 * 344924244303295816495032157
25084266359 * 292810008440530123
3 * 103 * 187547 * 449917889 * 962054203309
3 * 17 * 1031 * 59017279006673853482475689
3 * 4091 * 84942079 * 1022090777 * 297603119071
3 * 7 * 10457 * 12329 * 16693 * 50392193 * 14969202179383
3 * 30259 * 71055159937 * 57524912931153279285967
3 * 89 * 13961402129 * 885962415125636289188463869
3 * 293 * 41233 * 11038436757548471 * 97266672953292277
3 * 7 * 11 * 1853767605161 * 7690929649893487130760222347
2887 * 128548159915501079906799324521471122535581
31 * 3169 * 1387271471 * 452100449741 * 46858220729781791489
Homeprime 313169138727147145210044974146858220729781791489
reached after 31 steps
81
3 * 3 * 3 * 3
3 * 11 * 101
7 * 7 * 7 *907
13 * 13 * 4603
3 * 4378201
11 * 13 * 19 * 12653
3 * 29 * 443 * 288833
3 * 61 * 83 * 21689597
19 * 3089 * 4597 * 13411
Homeprime 193089459713411 reached after 9 steps
82
2 * 41
Homeprime 241 reached after 1 step
84
2 * 2 * 3 * 7
Homeprime 2237 reached after 1 step
85
5 * 17
11 * 47
31 * 37
Homeprime 3137 reached after 3 steps
86
2 * 43
3 * 3 * 3 * 3 * 3
3 * 41 * 271
3 * 3 * 7 * 5417
3 * 1125139
47 * 607 * 1091
1453 * 327647
3 * 31 * 137 * 114067
23 * 14397265829
11 * 17 * 89 * 113 * 1230631
3 * 79 * 379 * 43721 * 284657
3 * 10457 * 1209331666867
3 * 13 * 73 * 283 * 3853254754967
7 * 7583 * 15642293 * 377850899
7 * 43 * 983 * 55228357 * 474772229
73 * 1019155551073390065373
601 * 290328047 * 4189529884459
Homeprime 6012903280474189529884459
reached after 17 steps
87
3 * 29
7 * 47
3 * 3 * 83
17 * 199
3 * 3 * 3 * 7 * 7 * 13
3 * 3 * 3 * 123619
17 * 19595507
7 * 14771 * 16631
37 * 1931813963
3 * 3 * 3 * 71 * 3779 * 51341
3 * 73 * 277 * 569 * 966803
3 * 3 * 31 * 89 * 15032724013
7 * 11 * 37 * 79 * 137 * 30853 * 35023
3 * 31 * 196838267 * 3886045633
4909 * 674672720039496037
73821863 * 66507054593299
3 * 3 * 17 * 1069 * 75833 * 595192748879
41 * 43 * 1881512748629379008933
Homeprime 41431881512748629379008933
reached after 18 steps
88
2 * 2 * 2 * 11
7 * 19 * 167
Homeprime 719167 reached after 2 steps
90
2 * 3 * 3 * 5
5 * 467
7 * 11 * 71
Homeprime 71171 reached after 3 steps
91
7 * 13
23 * 31
3 * 3 * 7 * 37
11 * 3067
3 * 3 * 17 * 739
3 * 1105913
23 * 61 * 22171
Homeprime 236122171 reached after 7 steps
92
2 * 2 * 23
3 * 3 * 13 * 19
Homeprime 331319 reached after 2 steps
93
3 * 31
Homeprime 331 reached after 1 step
94
2 * 47
13 * 19
Homeprime 1319 reached after 2 steps
95
5 * 19
3 * 173
19 * 167
3 * 6389
Homeprime 36389 reached after 4 steps
96
2 * 2 * 2 * 2 * 2 * 3
61 * 3643
19 * 32297
3 * 61 * 10559
3 * 12036853
17 * 43 * 426863
107 * 16293709
3 * 3 * 3 * 11 * 241 * 149717
1613 * 20651730409
3 * 23 * 2337980459861
7 * 13 * 103 * 211 * 1634389987
7 * 76771 * 13269579308471
7 * 3041 * 3649039943138033
73 * 607 * 24527 * 258733 * 2597533
3 * 11 * 31 * 71952341419928966371
59 * 52765626310871524219769
107 * 229 * 1349807 * 179981492001889
3 * 149 * 7267433 * 330083824660722439
9767 * 394909447 * 8166090096149911
6917 * 1463621591 * 9647886019564813
7 * 6091 * 16223342078849817010718849
1279 * 5949269916608349374368264831
37 * 739 * 467978980723278658134600017
7 * 19 * 19 * 47 * 6758847359447 * 47013224061319
7 * 227 * 1237 * 51179327 * 67321039 * 1061958490511
3 * 1437251 * 72118441 * 23241560648369540947007
13789 * 230597 * 88003588750669 * 1123460390190091
17 * 29 * 29 * 67 * 109 * 797 * 22263569 * 4660829203 * 1596899264419
Homeprime 172929671097972226356946608292031596899264419
reached after 28 steps !
98
2 * 7 * 7
Homeprime 277 reached after 1 step
99
3 * 3 * 11
7 * 11 * 43
Homeprime 71143 reached after 2 steps
100
2 * 2 * 5 * 5
5 * 11 * 41
3 * 17047
Homeprime 317047 reached after 3 steps
Contributions and sources
I'm grateful for the work of Warut Roonguthai (email) for the expansion of number HP( 49 ) upto step 55.
For the moment it seems this is becoming a similar unending case as the famous 196reversal palindromic phenomenon.  go to topic
Dave Rusin (email) tried with the means at his disposal to factor the number c105 of HP( 49 ) at step 56.
Here's why his program had to give up : the program he used has three phases :
 trial division checked for all small factors (up to around 10^6).
 The elliptic curve method is probabilistic, but it has a very high likelihood of finding any small factors, and decreasing
likelihood of finding larger ones. That program ran over 1200 times, which should have made it more likely than not that
it would find any prime factor of up to 30 digits. This morning [ June 15, 1997 ] it stopped without finding any.
 the quadratic sieve method will eventually factor the number completely, but requires a huge amount of partial data.
This morning, as it entered this phase, the computer gave the following message :
 number to factor [105 digits] :
 4349837300068295069063245879245079069305
 7918835011730882390173172445242099505108
 7537386538239252454821397
 estimated running time (usrtime) :
 12 years 266 days 22 hours 40.9741 minutes.
Dave said that it is not appropriate to wait that long. I totally agree.
It seems that we'll have to wait for more powerful computers and/or factorising programs.
This uncracked number c105 WAS a good 'testing case' thereto... read on !
Major progress came from Paul Leyland (email) who found the two prime factors p41 and p65
of HP( 49 ) at step 56 in a timespan of only one week (compare this with Dave Rusin estimate of 13 years!)
using his MPQS program.  go to topic
A new breakthrough came when he cracked the c137 of step 74 of HP( 49 ) into p46 * p91
after six months of computation using ECM.  go to topic
Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
The list has been poured into sequences by Neil Sloane, Jeff Burch and Michael Greenwald :
A006919
Write down all the prime divisors in previous term!.
A037271
Number of steps to reach a prime under: "replace n by concatenation of
its prime factors", when applied to nth composite number, or 1 if no such number.
A037272
Primes reached in A037271, or 1 if no such prime.
A037273
Number of steps to reach a prime under: "replace n by concatenation of
its prime factors", or 1 if no such number.
A037274
Home primes: for n >= 2, primes reached when you start with n, concatenate
its prime factors (A037276), and repeat until a prime is reached
(or 1 if no prime is ever reached).
A037275
Subsequence of record holders in A037274.
A037276
Replace n by concatenation of its prime factors.
A037919 up to A037941
Trajectory of 4 (up to 48) under prime factor concatenation procedure.
A056938
Trajectory of 49.
A064841
Working in base 2, replace n by concatenation of its prime divisors
in increasing order.
A065016
Working in base 2, replace n by the concatenation of its prime factors
(without repetition).
A048985
Working in base 2, replace n by concatenation of its prime divisors
in increasing order (write answer in base 10).
A064795
Home primes in base 2: primes reached when you start with n, and (working
in base 2) concatenate its prime factors (A048985); repeat until a
prime is reached (or 1 if no prime is ever reached).
A048986
Home primes in base 2: primes reached when you start with n, and (working
in base 2) concatenate its prime factors (A048985); repeat until a
prime is reached (or 1 if no prime is ever reached). [Answer is written in base 10.]
A049065
Record primes reached in A048986.
Click here to view some of the author's [P. De Geest] entries to the table.
Click here to view some entries to the table about palindromes.

Jeffrey Heleen (email) started working already on this topic problem many years ago
and even explored the topic up to the number 1000.  go to topic
Eric W. Weisstein maintains also an interesting 'Math Encyclopedia' page about
this topic under the heading Home Prime.
Prime Curios!  site maintained by G. L. Honaker Jr. and Chris Caldwell