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121131141151161171
181191313323343353
373383717727737747
757787797919929949
959979989   

Smoothly Undulating Palprimes
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Smoothly Undulating Palindromic Primes (or SUPP's for short)
are numbers that are primes, palindromic in base 10, and the digits alternate,
but why smooth one might ask !
The smoothness was added to make a difference with the normal
undulating numbers. The description for normal undulating numbers
is that the next digits alternately go up and down (or down and up)
but the absolute difference values between two adjacent digits may differ.
(e.g. 906343609)
In a smoothly undulating number the absolute difference values
between two adjacent digits are always equal, therefore only two distinct
digits can appear in the number.
(e.g. 74747474747474747)

Actually, it was Charles Trigg who coined the term smoothly... in the following reference

C. W. Trigg, "Palindromic Octagonal Numbers",
Journal of Recreational Mathematics, 15:1, pp.41-46, 1982-83.

Sources were I found some SUPP's ¬

The Top Ten Prime Numbers by Rudolf Ondrejka
Palindrome prime number patterns by Harvey Heinz
Liczby pierwsze o szczególnym rozmieszczeniu cyfr (Polish PostScript file) by Andrzej Nowicki
In case one should discover more sources I will be most happy
to add them to the list. Just let me know.

SUPP's sorted by length

Some nontrivial combinations can never produce primes...
By Julien Peter Benney (email) [October 21, 2004]

(76)w7 is always composite because:

if w is of form 3n, then 7 is a divisor;
if w is of form 3n+1, then 13 is a divisor;
if w is of form 3n+2, then 3 is a divisor.

(71)w7 is composite in the following general cases:

if w is of form 3n, then 7 is a divisor;
if w is of form 3n+1, then 3 is a divisor.

(34)w3 is composite in the following general cases:

if w is of form 3n, then 3 is a divisor;
if w is of form 3n+1, then 7 is a divisor.

Thus in both last cases only for w of the form 3n+2
is there any chance of a prime !

Jeff Heleen wrote dd. [February 9, 2001]

"As far as I could see you didn't have a section on your site for these numbers.
While I'm sure someone somewhere must have done this before, I have done it also.
Within the limitations of the program I believe these are ALL the smoothly undulating
palindromic prime numbers with two distinct digits each, smaller than 843 digits long.
I used a modified APRT-CLE program in UBASIC to automate and perform the search
on a Pentium_II 300 MHz laptop."

That is indeed a very nice and interesting compilation, thanks Jeff. Great job!
At the same time it is a topic that might attract other dedicated number crunchers.
Perhaps you know a source where larger SUPP's are displayed.
Those are welcome as well! Send them in and I'll add them to the table.

Jeff Heleen wrote dd. [February 12, 2001]

"I have found the following website:
http://www.utm.edu/research/primes/lists/top_ten/topten.pdf
where, if you will look on page 43 (of 93) you will see the top ten
SUPP's as of February 24, 2001. The smallest two on this list are the same as my
highest two. It doesn't say whether these are ALL there are up to the
highest one shown. However, I suspect not, as they all start and end with
the digit 1. So perhaps there are more to discover in this range."

Message from Carlos Rivera dd. [February 14, 2001]

There are several extra terms:
(37)k3, is prime for k=424 & 946
(75)k7, is prime for k=539 & 707
(79)k7, is prime for k=838
(92)k9, is prime for k=428
(95)k9, is prime for k=647
(please verify them)
In the meanwhile I used PRIMEFORM to get the next pseudoprime
following my record from 1997:
(12)k1, is pseudoprime for k=3904 (7809 digits) far beyond the current
possibilities of rigorous primality testing of the speediest code (TITANIX)
Carlos argues 'As a matter of fact the real SUPP 's
are (for me) numbers (ab)ka, such that abs(a-b)=1'
explaining why he favours breaking records of the form (12)k1 above the others.

Start of April 2001 I noticed a new entry in G. L. Honaker, Jr.'s Prime Curios!
website of Landon Curt Noll. A beautiful SUPP, proved prime with special
hardware a few years ago (?), was introduced there, which immediately shattered
Carlos Rivera's previous record !
This SUPP has a length of 2883 digits 3(73)1441
See Prime Curios! 37373...37373
You can contact L. C. Noll through his home page at http://www.isthe.com/chongo/

The following link includes many details about Landon's proof of the SUPP
and why his proof got lost : Is (37*10^2883-73)/99 prime?
References:
Landon Curt Noll (37*10^2883-73)/99 is prime
Landon Curt Noll Misc prime numbers
Tom Magliery Prime numbers related to 37

Enters Hans Rosenthal, May 6, 2001, with new and more impressive data !

Here is a probable prime of length 10419 for your SUPP page:
3(13)5209 = 310*(1010418-1)/99+3
I don't know whether this one has been discovered by someone
else before (if you know of this, please send me a note).
Hans added that many more of the _abababa_ type will follow within not too long.
(i.e. a complete list of them up to 20001 digits... which arrived at June 4, 2001)
By doing so Hans no doubt brought this collection to the point where it will serve
as a standard reference work for this kind of numbers.
Many thanks for your excellent contribution, Hans!

Carlos Rivera writes, June 26, 2001, the following interesting observations.

1) Any smooth undulating palindrome number composed of two distinct digits
can be expressed in any one of the two forms:

a(ba)n = (ab)na

2) (ab)na = (ab)nx10+a
3) (ab)n = (ab)xR(2n)/R(2)
4) R(k) = (10k -1)/(10-1)

5) Consequently a(ba)n = (ab)na = (ab)x((102n-1)/99)x10+a

6) But:

(ab)x((102n-1)/99)x10+a =
[(ab)x102n+1 - (ab)x10 + 99a]/99 =
[(ab)x102n+1 - (ab)x10 + 100a - a]/99 =
[(ab)x102n+1 - (ba)]/99

7) a(ba)n = (ab)na = (ab)x((102n-1)/99)x10+a = [(ab)x102n+1 - (ba)]/99

8) The form a(ba)n = [(ab)x102n+1 - (ba)]/99 is the one used by you in your
page and formally is correct.

But the second form (ab)na = (ab)x((102n-1)/99)x10+a is a kind of more suitable
one form for primality test purposes, especially if:
° a = 1 &
° [(102n -1)/99] can be factorized until certain extent in order to use classical
  theorems like the Pocklington one.

Thanks Carlos for the interesting observations on the formula formats for the SUPP's.
Before Hans Rosenthal entered the stage I used the format you promote in entry 7
(highlighted in yellow).
But Hans convinced me to use to other one for the following reasons.
First the format [(ab)*102n+1-(ba)]/99 displays the exact digitlength
of the SUPP namely via the exponent (2n+1).
Secondly the (ab) and (ba) coefficients indicate straight away how the SUPP starts and ends !

Hans Rosenthal, September 4, 2001, broke Landon Curt Noll's old record
by prime proving the following SUPP of 3015 digits !

3(23)1507 = (32*103015-23)/99

Hans Rosenthal, October 19, 2001, sent in a list of five new records.
The largest one he prime proved is the following SUPP of 4859 digits !

9(89)2429 = (98*104859-89)/99
"All Primo certificates have been validated with Cert_Val. The proof of the above
largest known SUPP (second largest known ECPP prime) took exactly 11 weeks
on an Athlon 1.4 GHz, the full validation of this certificate took 20 and a half hours
on the same PC.
I believe that from now on it's a real challenge (also for myself) to complete/enlarge
the SUPP table."

Hans Rosenthal, October 27, 2002, sent a new SUPP record of 4885 digits !
( Announced at Walter Schneider's site at Undulants )

1(71)2442 = (17*104885-71)/99
The proof was done using Marcel Martin's Primo and took 2008 hours and 57 minutes
on a AMD Athlon 1.33 GHz. The Primo certificate was then validated with Cert_Val which took
on the same PC an additional 25 hours and 11 minutes.
See also the Top 20 ECPP records at http://www.ellipsa.net/pages/primotop20.html

Hans Rosenthal, December 10, 2002

PS: This might be an interesting link for the SUPP reference page:

http://www.lix.polytechnique.fr/Labo/Francois.Morain/english-index.html
It's very likely that François Morain will independently verify the Primo
certificate of the (former) "largest known SUPP"...

David Broadhurst, December 22, 2002
announced via a message (#2937) in the User group for the PrimeForm program that
the following SUPP is prime !

1(41)3171 = (14*106343-41)/99
Reaction from Hans Rosenthal, December 23, 2002
" Yes, David informed me, nice result, such a proof won't happen every day.
Jim Fougeron double-checked the primality of 1(71)_2442 = (17*10^4885-71)/99
by use of BLS (he only took about 24 hours for that). Both, David and Jim were pretty
lucky with finding enough factors in N-1 for their proofs. However, this can only
work for the SUPP's that start/end in 1 -- it will never work for the others.
I am really glad that I am no longer the only one to contribute new results to the
SUPP page. You should update it and also announce the new record on your main page."


Hans Rosenthal, July 13, 2003
announced via a message in Number Theory List (NMBRTHRY@LISTSERV.NODAK.EDU)
that the following SUPP is proven prime !

3(23)3479 = (32*106959-23)/99
He thereby also established a new Primo ECPP world record
performed on a single monoprocessor computersystem.
See also François Morain's websource at http://www.lix.polytechnique.fr/Labo/Francois.Morain/.
Congratulations Hans, an impressive achievement !
" I would like to inform you that I have certified the primality of
(32*10^6959-23)/99, a smoothly undulating palindromic prime (SUPP) [1]
having 6959 decimal digits, with the program Primo [2], Marcel Martin's
implementation of the elliptic curve primality proving (ECPP) algorithm.

The Primo certificate of primality is available at
http://www.ellipsa.net/primo/ecpp6959.zip (4457 KB)

The certification of this ordinary prime was started on 21 January 2002
with Primo 1.1.0 (tests 1 to 47) and completed on 7 July 2003 with
Primo 2.0.0 (tests 48 to 953) on an AMD Athlon 1.4 GHz. There was one
relevant interruption of the certification process from 29 March 2003,
6:47, until 3 April 2003, 22:45. So the total running time amounts to
approximately 527 days.

There is a kind of ECPP diary of the certification progress available,
which was started on 9 June 2002 at 21310 (of 23116) bits (so it is not
complete). This diary can be found at
http://www.ellipsa.net/primo/ecpp6959_diary.txt (9 KB)

I thank Marcel Martin for his help and advice, and most of all, for
making the ECPP algorithm available to the world of PC users in the
most comfortable form I can imagine: his marvellous Primo.

Hans Rosenthal

[1] http://www.worldofnumbers.com/undulat.htm#

[2] http://www.ellipsa.net/primo/record.html

SUPP Factorization Projects
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The Table
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The reference table for
Smoothly Undulating Palindromic Primes
This collection is complete for
probable primes up to 40001
digits and for proven primes
up to  6343  digits.
CR = Carlos Rivera
DB = David Broadhurst
HR = Hans Rosenthal
JH = Jeffrey Heleen
LN = Landon Curt Noll
SUPPFormula
blue exp = # of digits		WhoWhenStatusProgram
Output Logs
 ¬
1(01)1 (10*103-01)/99
IMPORTANT NOTE		 JH Feb 09 2001 PRIME View
A062209 ¬
1(21)3 (12*107-21)/99 JH Feb 09 2001 PRIME View
1(21)5 (12*1011-21)/99 JH Feb 09 2001 PRIME View
1(21)21 (12*1043-21)/99 JH Feb 09 2001 PRIME View
1(21)69 (12*10139-21)/99 JH Feb 09 2001 PRIME View
1(21)313 (12*10627-21)/99 JH Feb 09 2001 PRIME View
1(21)699 (12*101399-21)/99 HR Jun 17 2001 PRIME View
1(21)798 (12*101597-21)/99 HR Jun 17 2001 PRIME View
1(21)989 (12*101979-21)/99 CR ___ __ 1997 PRIME View
1(21)3904 (12*107809-21)/99 CR ___ __ 2001 PROBABLE
PRIME		 View
1(21)7029 (12*1014059-21)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
 ¬
1(31)1 (13*103-31)/99 JH Feb 09 2001 PRIME View
1(31)12 (13*1025-31)/99 JH Feb 09 2001 PRIME View
A062210 ¬
1(41)5 (14*1011-41)/99 JH Feb 09 2001 PRIME View
1(41)138 (14*10277-41)/99 JH Feb 09 2001 PRIME View
1(41)239 (14*10479-41)/99 JH Feb 09 2001 PRIME View
1(41)291 (14*10583-41)/99 JH Feb 09 2001 PRIME View
1(41)815 (14*101631-41)/99 HR Aug 09 2001 PRIME View
1(41)3171 (14*106343-41)/99 HR
DB		 Dec 22 2002 PRIME View
1(41)7344 (14*1014689-41)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
A062211 ¬
1(51)1 (15*103-51)/99 JH Feb 09 2001 PRIME View
1(51)7 (15*1015-51)/99 JH Feb 09 2001 PRIME View
1(51)31 (15*1063-51)/99 JH Feb 09 2001 PRIME View
1(51)44 (15*1089-51)/99 JH Feb 09 2001 PRIME View
1(51)122 (15*10245-51)/99 JH Feb 09 2001 PRIME View
1(51)291 (15*10583-51)/99 JH Feb 09 2001 PRIME View
1(51)895 (15*101791-51)/99 HR Jun 17 2001 PRIME View
1(51)1061 (15*102123-51)/99 HR Aug 09 2001 PRIME View
1(51)3616 (15*107233-51)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
1(51)12393 (15*1024787-51)/99 HR Oct 19 2001 PROBABLE
PRIME		 View
A062212 ¬
1(61)3 (16*107-61)/99 JH Feb 09 2001 PRIME View
1(61)27 (16*1055-61)/99 JH Feb 09 2001 PRIME View
1(61)54 (16*10109-61)/99 JH Feb 09 2001 PRIME View
1(61)72 (16*10145-61)/99 JH Feb 09 2001 PRIME View
1(61)114 (16*10229-61)/99 JH Feb 09 2001 PRIME View
1(61)480 (16*10961-61)/99 HR Jun 04 2001 PRIME View
A062213 ¬
1(71)15 (17*1031-71)/99 JH Feb 09 2001 PRIME View
1(71)18 (17*1037-71)/99 JH Feb 09 2001 PRIME View
1(71)2442 (17*104885-71)/99 HR Oct 27 2002 PRIME View
A062214 ¬
1(81)1 (18*103-81)/99 JH Feb 09 2001 PRIME View
1(81)2 (18*105-81)/99 JH Feb 09 2001 PRIME View
1(81)38 (18*1077-81)/99 JH Feb 09 2001 PRIME View
1(81)81 (18*10163-81)/99 JH Feb 09 2001 PRIME View
1(81)739 (18*101479-81)/99 HR Aug 09 2001 PRIME View
1(81)1828 (18*103657-81)/99 HR Feb 11 2002 PRIME View
1(81)2286 (18*104573-81)/99 HR Aug 08 2002 PRIME View
1(81)4157 (18*108315-81)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
1(81)15129 (18*1030259-81)/99 HR Oct 19 2001 PROBABLE
PRIME		 View
1(81)15531 (18*1031063-81)/99 HR Oct 19 2001 PROBABLE
PRIME		 View
1(81)15927 (18*1031855-81)/99 HR Oct 19 2001 PROBABLE
PRIME		 View
1(81)18457 (18*1036915-81)/99 HR Oct 19 2001 PROBABLE
PRIME		 View
A062215 ¬
1(91)1 (19*103-91)/99 JH Feb 09 2001 PRIME View
1(91)16 (19*1033-91)/99 JH Feb 09 2001 PRIME View
1(91)66 (19*10133-91)/99 JH Feb 09 2001 PRIME View
1(91)984 (19*101969-91)/99 HR Jul 08 2001 PRIME View
1(91)1167 (19*102335-91)/99 HR Sep 04 2001 PRIME View
A062216 ¬
3(13)1 (31*103-13)/99 JH Feb 09 2001 PRIME View
3(13)25 (31*1051-13)/99 JH Feb 09 2001 PRIME View
3(13)41 (31*1083-13)/99 JH Feb 09 2001 PRIME View
3(13)112 (31*10225-13)/99 JH Feb 09 2001 PRIME View
3(13)280 (31*10561-13)/99 JH Feb 09 2001 PRIME View
3(13)5209 (31*1010419-13)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
3(13)9127 (31*1018255-13)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
A062217 ¬
3(23)2 (32*105-23)/99 JH Feb 09 2001 PRIME View
3(23)4 (32*109-23)/99 JH Feb 09 2001 PRIME View
3(23)5 (32*1011-23)/99 JH Feb 09 2001 PRIME View
3(23)1507 (32*103015-23)/99 HR Sep 04 2001 PRIME View
3(23)1703 (32*103407-23)/99 HR Oct 19 2001 PRIME View
3(23)3479 (32*106959-23)/99 HR Jul 08 2003 RECORD
PROVEN
PRIME		 View
3(23)4799 (32*109599-23)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
3(23)5699 (32*1011399-23)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
3(23)8296 (32*1016593-23)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
3(23)12941 (32*1025883-23)/99 HR Oct 19 2001 PROBABLE
PRIME		 View
 ¬
3(43)w (34*10n-43)/99 --- Mon day year - View
A062218 ¬
3(53)1 (35*103-53)/99 JH Feb 09 2001 PRIME View
3(53)2 (35*105-53)/99 JH Feb 09 2001 PRIME View
3(53)11 (35*1023-53)/99 JH Feb 09 2001 PRIME View
3(53)1088 (35*102177-53)/99 HR Jun 17 2001 PRIME View
3(53)1573 (35*103147-53)/99 HR Oct 19 2001 PRIME View
3(53)2078 (35*104157-53)/99 HR Feb 11 2002 PRIME View
3(53)11356 (35*1022713-53)/99 HR Oct 19 2001 PROBABLE
PRIME		 View
3(53)14192 (35*1028385-53)/99 HR Oct 19 2001 PROBABLE
PRIME		 View
A062219 ¬
3(73)1 (37*103-73)/99 JH Feb 09 2001 PRIME View
3(73)10 (37*1021-73)/99 JH Feb 09 2001 PRIME View
3(73)13 (37*1027-73)/99 JH Feb 09 2001 PRIME View
3(73)40 (37*1081-73)/99 JH Feb 09 2001 PRIME View
3(73)157 (37*10315-73)/99 JH Feb 09 2001 PRIME View
3(73)424 (37*10849-73)/99 HR Jun 04 2001 PRIME View
3(73)946 (37*101893-73)/99 HR Jul 08 2001 PRIME View
3(73)1441 (37*102883-73)/99 LN ___ __ 1997 PRIME View
3(73)4795 (37*109591-73)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
3(73)7345 (37*1014691-73)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
A062220 ¬
3(83)1 (38*103-83)/99 JH Feb 09 2001 PRIME View
3(83)4 (38*109-83)/99 JH Feb 09 2001 PRIME View
3(83)7 (38*1015-83)/99 JH Feb 09 2001 PRIME View
3(83)8 (38*1017-83)/99 JH Feb 09 2001 PRIME View
3(83)10 (38*1021-83)/99 JH Feb 09 2001 PRIME View
3(83)28 (38*1057-83)/99 JH Feb 09 2001 PRIME View
3(83)2116 (38*104233-83)/99 HR Apr 02 2002 PRIME View
3(83)2167 (38*104335-83)/99 HR Aug 08 2002 PRIME View
3(83)6610 (38*1013221-83)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
3(83)13223 (38*1026447-83)/99 HR Oct 19 2001 PROBABLE
PRIME		 View
3(83)14948 (38*1029897-83)/99 HR Oct 19 2001 PROBABLE
PRIME		 View
 ¬
7(17)w (71*10n-17)/99 --- Mon day year - View
A062221 ¬    n > 100001 (PDG, September 13, 2004)
7(27)1 (72*103-27)/99 JH Feb 09 2001 PRIME View
7(27)2 (72*105-27)/99 JH Feb 09 2001 PRIME View
7(27)4 (72*109-27)/99 JH Feb 09 2001 PRIME View
7(27)8 (72*1017-27)/99 JH Feb 09 2001 PRIME View
7(27)35 (72*1071-27)/99 JH Feb 09 2001 PRIME View
7(27)49 (72*1099-27)/99 JH Feb 09 2001 PRIME View
7(27)121 (72*10243-27)/99 JH Feb 09 2001 PRIME View
7(27)3797 (72*107595-27)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
7(27)4636 (72*109273-27)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
7(27)26923 (72*1053847-27)/99 PDG Aug 06 2004 RECORD
PROBABLE
PRIME		 View
A062222 ¬
7(37)7 (73*1015-37)/99 JH Feb 09 2001 PRIME View
7(37)19 (73*1039-37)/99 JH Feb 09 2001 PRIME View
7(37)283 (73*10567-37)/99 JH Feb 09 2001 PRIME View
7(37)1264 (73*102529-37)/99 HR Sep 04 2001 PRIME View
7(37)7168 (73*1014337-37)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
A062223 ¬
7(47)2 (74*105-47)/99 JH Feb 09 2001 PRIME View
7(47)8 (74*1017-47)/99 JH Feb 09 2001 PRIME View
7(47)1034 (74*102069-47)/99 HR Aug 09 2001 PRIME View
7(47)3407 (74*106815-47)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
7(47)10208 (74*1020417-47)/99 HR Oct 19 2001 PROBABLE
PRIME		 View
7(47)12872 (74*1025745-47)/99 HR Oct 19 2001 PROBABLE
PRIME		 View
A062224 ¬
7(57)1 (75*103-57)/99 JH Feb 09 2001 PRIME View
7(57)8 (75*1017-57)/99 JH Feb 09 2001 PRIME View
7(57)38 (75*1077-57)/99 JH Feb 09 2001 PRIME View
7(57)71 (75*10143-57)/99 JH Feb 09 2001 PRIME View
7(57)74 (75*10149-57)/99 JH Feb 09 2001 PRIME View
7(57)256 (75*10513-57)/99 JH Feb 09 2001 PRIME View
7(57)539 (75*101079-57)/99 HR Aug 09 2001 PRIME View
7(57)707 (75*101415-57)/99 HR Aug 09 2001 PRIME View
7(57)3124 (75*106249-57)/99 HR Aug 21 2003 PRIME View
7(57)6632 (75*1013265-57)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
7(57)7289 (75*1014579-57)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
7(57)7646 (75*1015293-57)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
A062225 ¬
7(87)1 (78*103-87)/99 JH Feb 09 2001 PRIME View
7(87)2 (78*105-87)/99 JH Feb 09 2001 PRIME View
7(87)10 (78*1021-87)/99 JH Feb 09 2001 PRIME View
7(87)13 (78*1027-87)/99 JH Feb 09 2001 PRIME View
7(87)47 (78*1095-87)/99 JH Feb 09 2001 PRIME View
7(87)1037 (78*102075-87)/99 HR Aug 09 2001 PRIME View
7(87)1082 (78*102165-87)/99 HR Aug 09 2001 PRIME View
7(87)1523 (78*103047-87)/99 HR Oct 19 2001 PRIME View
7(87)1751 (78*103503-87)/99 HR Feb 11 2002 PRIME View
7(87)8395 (78*1016791-87)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
7(87)17441 (78*1034883-87)/99 HR Oct 19 2001 PROBABLE
PRIME		 View
A062226 ¬
7(97)1 (79*103-97)/99 JH Feb 09 2001 PRIME View
7(97)178 (79*10357-97)/99 JH Feb 09 2001 PRIME View
7(97)268 (79*10537-97)/99 JH Feb 09 2001 PRIME View
7(97)838 (79*101677-97)/99 HR Aug 09 2001 PRIME View
7(97)1528 (79*103057-97)/99 HR Oct 19 2001 PRIME View
A062227 ¬
9(19)1 (91*103-19)/99 JH Feb 09 2001 PRIME View
9(19)4 (91*109-19)/99 JH Feb 09 2001 PRIME View
9(19)5 (91*1011-19)/99 JH Feb 09 2001 PRIME View
9(19)8 (91*1017-19)/99 JH Feb 09 2001 PRIME View
9(19)11 (91*1023-19)/99 JH Feb 09 2001 PRIME View
9(19)12614 (91*1025229-19)/99 HR Jun 15 2001 PROBABLE
PRIME		 View
A062228 ¬
9(29)1 (92*103-29)/99 JH Feb 09 2001 PRIME View
9(29)4 (92*109-29)/99 JH Feb 09 2001 PRIME View
9(29)97 (92*10195-29)/99 JH Feb 09 2001 PRIME View
9(29)257 (92*10515-29)/99 JH Feb 09 2001 PRIME View
9(29)428 (92*10857-29)/99 HR Jun 04 2001 PRIME View
9(29)5696 (92*1011393-29)/99 HR Jun 04 2001 PROBABLE
PRIME		 View
A062229 ¬
9(49)2 (94*105-49)/99 JH Feb 09 2001 PRIME View
9(49)8 (94*1017-49)/99 JH Feb 09 2001 PRIME View
9(49)32 (94*1065-49)/99 JH Feb 09 2001 PRIME View
9(49)71 (94*10143-49)/99 JH Feb 09 2001 PRIME View
9(49)275 (94*10551-49)/99 JH Feb 09 2001 PRIME View
A062230 ¬
9(59)2 (95*105-59)/99 JH Feb 09 2001 PRIME View
9(59)8 (95*1017-59)/99 JH Feb 09 2001 PRIME View
9(59)104 (95*10209-59)/99 JH Feb 09 2001 PRIME View
9(59)647 (95*101295-59)/99 HR Aug 09 2001 PRIME View
A062231 ¬
9(79)4 (97*109-79)/99 JH Feb 09 2001 PRIME View
9(79)13 (97*1027-79)/99 JH Feb 09 2001 PRIME View
9(79)22 (97*1045-79)/99 JH Feb 09 2001 PRIME View
9(79)118 (97*10237-79)/99 JH Feb 09 2001 PRIME View
A062232 ¬
9(89)4 (98*109-89)/99 JH Feb 09 2001 PRIME View
9(89)80 (98*10161-89)/99 JH Feb 09 2001 PRIME View
9(89)109 (98*10219-89)/99 JH Feb 09 2001 PRIME View
9(89)2429 (98*104859-89)/99 HR Oct 19 2001 PRIME View
9(89)10994 (98*1021989-89)/99 HR Oct 19 2001 PROBABLE
PRIME		 View



Sources Revealed
brown line

Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
Various undulating numbers, primes and palindromic primes are categorised as follows :
%N Undulating squares. under A016073
%N Undulating primes (digits alternate). under A032758
%N Undulating numbers (of form abababab... in base 10). under A033619
%N Undulating palindromic primes of form [AB]nA with
       alternating prime and nonprime digits. under A039944
%N Non-trivial undulants; base 10 numbers >100 which are of the form
       aba, abab, ababa..., where a!=b. under A046075
%N Indices of binary undulants; numbers n such that 2^n contains the
       alternating sequence of digits 010... or 101... under A046076
%N a(d-2) is the smallest member of A046076 containing an undulating
       sequence of 010... or 101... of maximal length d=3, 4, ... under A046077
%N Palindromic primes with just two distinct digits. under A056730
%N Numbers of (2n+1)-digit palindromic primes that undulate. under A057332
%N Numbers of n-digit primes that undulate. under A057333
%N Palindromic primes with just two distinct prime digits. under A058375
%N Primes in which digits alternately rise and fall (or vice versa);
       sometimes called undulating primes. under A059168
%N Strictly undulating primes (digits alternate and differ by 1). under A059170
%N Undulating palindromic primes: numbers that are prime, palindromic
       in base 10, and the digits alternate: ababab... with a != b. under A059758
%N Smoothly undulating palindromic primes of the general form
       (ab*10^m-ba)/99 exist for digitlengths a(n). under A077799
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.


Prime Curios! - site maintained by G. L. Honaker Jr. and Chris Caldwell

7 (9th and 25th entry)
101
131
151
181
191
264
313
353
373
383
727
757
787
797
919
929
12121 smoothly undulating composite
69696 smoothly undulating composite
72727
78787
94949
1212121
696969696 smoothly undulating composite
919191919
906343609
12121212121
151515151515151
74747474747474747
35353535353535353535353
13131...13131 (25-digits)
17171...17171 (31-digits)
19191...19191 (33-digits)
17171...17171 (37-digits)
73737...73737 (39-digits)
12121...12121 (43-digits)
18181...18181 (77-digits)
12121...12121 (139-digits)
16161...16161 (229-digits)
72727...72727 (243-digits)
37373...37373 (2883-digits)
32323...32323 (3407-digits)
35353...35353 (4157-digits)
98989...98989 (4859-digits)
17171...17171 (4885-digits)
75757...75757 (6249-digits)
14141...14141 (6343-digits)
32323...32323 (6959-digits)


Clifford A. Pickover's book "Keys to Infinity".

contains a chapter about these undulating primes (Chapter 20, pages 159 to 161),
titled "The Undulation of the Monks". The last alinea is in fact an appeal to the public:
.
I am interested in hearing from readers who have searched for undulating primes
with larger periods of undulation, such as found in the prime number 5,995,995,995
(which does not finish its last cycle of undulation).
.
Well, I am interested as well !

All of Hans Rosenthal's probable primes above 10000 digits are also
submitted to the PRP TOP records table maintained by Henri & Renaud Lifchitz.
See : http://www.primenumbers.net/prptop/prptop.php

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(All rights reserved) - Last modified : October 24, 2004.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com