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. :) .....Jeff

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 like-minded 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 GMP-ECM software.
Well done, both of you ! - goto 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 PII-300, 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! - goto 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 be consulted at the following address

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 PIII-1000 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 PIII-500 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! - goto 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 32-cpu cluster at MSR Cambridge.

HP49(82) is composite, with three small prime factors, one of moderate size (all found
by Alex) and a composite co-factor 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 math-fertile team. Well done again! - goto 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 GMP-ECM
and the largest one in the current year so far."

GMP-ECM 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

Richard Brent (email)'s reply

Paul Zimmermann (email)'s reply

goto directly to table entry

On [July 1, 2002 ] Alex Kruppa (email) & Paul Leyland (email)
factored HP49(85).c140 using GNFS into p68 * p73
The prime factors are

11825523118615173197502446002728268259724026363566923708566001149123
(68 digits) and
2056156715909184026132747632727779809352896881830754843064666748922417153
(73 digits).

" Alex Kruppa and I have taken the search for HP49 through several more
stages. The sticking point was the 140-digit cofactor of HP49(85). We
spent a lot of fruitless effort with ECM before turning to GNFS again.
As before, Alex found the polyomial and performed over 90% of the
sieving, whereas I did the filtering, linear algebra and square-root
phases. We found this factorization substantially harder than previous
ones in the series. The linear algebra took 65 hours on a 25-cpu
cluster; each square root took several hours on a single PIII-500 and
the factorization wasn't found until the fourth dependency. The
factorization finished on Saturday 29^th June. The result, p68 * p73,
shows why ECM was unsuccessful.

The next two stages were relatively easy and were factored by a fairly
small amount of ECM over the weekend. Stage 88 is now being attempted
with ECM. We found a few small prime factors and a c155 cofactor."

goto directly to table entry

On [July 22, 2002 ] Alex Kruppa (email) & Paul Leyland (email)
factored HP49(90).c176 using only ECM into p38 * p66 * p94
Alex found the p38 factor and Paul found the p44 factor.
The prime factors are

52260221223007872743384258441159930329
(38 digits) and
66076192121556175190138089562410746399109073
(44 digits).

" We are continuing to work on the c152. If ECM doesn't find a relatively
small factor, we should be able to finish it with GNFS though with some difficulty.
We're wondering if we may have to go beyond 100 steps..."

On [July 24, 2002 ] Alex Kruppa (email) & Paul Leyland (email)
factored the remaining HP49(94).c152 using GMP-ECM.
Alex got lucky again and found a p51 factor, the 101 digit cofactor being prime.
The prime factor is

681195332272435073462164213069789780765696608423481
(51 digits)

" The next step of the sequence HP49(95) is composite, so far the factors
3 * 7 * 26141 * 300119 * 1811141 * 1072782128567282855315039 * C153
are known. The c153 may yet yield another ~35 digits factor, and we will
do more ECM before we consider a GNFS factorization, which would seem
feasible if somewhat taxing."

Paul Zimmermann (email)'s reply

goto directly to table entries 90-95

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
153-digit 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
mid-November 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 151-digit cofactor is within range of
GNFS if we can't find any more ECM factors. "

goto directly to table entries 95-97

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 151-digit 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
GHz-year. The CWI line siever produced an additional 2M relations by
sieving over small b-values 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. "

goto directly to table entries 97-100

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
346369145517616832561580518436338147877062893457679622195929206654524672587613049343558394373396338194585783775269785675210636696425094776859733305947996048061499249566197147212934512427988113420226762897

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	7 7
2	7 11
3	3 3 79
4	31 109
5	13 2.393
6	3 44.131
7	17 31 653
8	7 11 43 523
9	11 11 577 1.019
10	311 35.742.029
11	7 17 261.644.891
12	11 19 3.431.873.899
13	11 613 4.799 345.907
14	3 204.751 189.066.719
15	3 1.068.250.396.355.573
16	621.611 49.980.213.343
17	3 3 6.906.794.442.245.927
18	73 4.615.161.567.701.999
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 105-digit composite number, was solved thanks to Paul Leyland 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 137-digit composite number p46 * p91, was also solved by Paul Leyland 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 & Paul Leyland 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 & Paul Leyland 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 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 & Paul Leyland 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 & Paul Leyland 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 & Paul Leyland 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 & Paul Leyland factored HP49(97).c151 = p55 * p96. [July 16, 2003] The quest for this elusive Home Prime continues...
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
100	3 37 2789 c204
PS	Please doublecheck the correctness of the above results before using them for continuing the search.

No homeprime reached after 99 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 98 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

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 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

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 82 
2 * 41
Homeprime 241 reached after 1 step

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 84 
2 * 2 * 3 * 7
Homeprime 2237 reached after 1 step

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 85 
5 * 17
11 * 47
31 * 37
Homeprime 3137 reached after 3 steps

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 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

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 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

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 88 
2 * 2 * 2 * 11
7 * 19 * 167
Homeprime 719167 reached after 2 steps

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 90 
2 * 3 * 3 * 5
5 * 467
7 * 11 * 71
Homeprime 71171 reached after 3 steps

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 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

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 92 
2 * 2 * 23
3 * 3 * 13 * 19
Homeprime 331319 reached after 2 steps

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 93 
3 * 31
Homeprime 331 reached after 1 step

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 94 
2 * 47
13 * 19
Homeprime 1319 reached after 2 steps

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 95 
5 * 19
3 * 173
19 * 167
3 * 6389
Homeprime 36389 reached after 4 steps

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 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 !

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 98 
2 * 7 * 7
Homeprime 277 reached after 1 step

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 99 
3 * 3 * 11
7 * 11 * 43
Homeprime 71143 reached after 2 steps

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 100 
2 * 2 * 5 * 5
5 * 11 * 41
3 * 17047
Homeprime 317047 reached after 3 steps

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Contributions and sources

I'm grateful for the work of Warut Roonguthai 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 196-reversal palindromic phenomenon... goto topic

• 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 (usr-time) :
  12 years 266 days 22 hours 40.9741 minutes.

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 entries to the table about palindromes. Click here to view some of the author's [P. De Geest] entries to the table.

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

Repeated Factorisation of Concatenated Primefactors of the Composite Numbers up to 100 and beyond...

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(All rights reserved) - Last modified : May 29, 2007.

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

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