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.
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.
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.
[ 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.
Some nontrivial combinations can never produce primes...
By Julien Peter Benney (email)
Thus in both last cases only for w of the form 3n+2
is there any chance of a prime !
[ February 9, 2001 ]
Jeff Heleen wrote :
" 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.
[ February 12, 2001 ]
Jeff Heleen wrote :
" I have found the following website:
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."
[ February 14, 2001 ]
Message from Carlos Rivera
There are several extra terms :
Carlos argues 'As a matter of fact the real SUPP 's
(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)
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.
[ April 2001 ]
Start of above date 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/
[ May 6, 2001 ]
Enters Hans Rosenthal with new and more impressive data !
Here is a probable prime of length 10419 for your SUPP page:
Hans added that many more of the _abababa_ type will follow within not too long.
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).
( 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!
[ June 26, 2001 ]
Carlos Rivera writes 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 !
[ September 4, 2001 ]
Hans Rosenthal broke Landon Curt Noll's old record
by prime proving the following SUPP of 3015 digits !
3(23)1507 = (32*103015-23)/99
[ October 19, 2001 ]9(89)2429 = (98*104859-89)/99
Hans Rosenthal sent in a list of five new records.
The largest one he prime proved is the following SUPP of 4859 digits !
" 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."
[ October 27, 2002 ]1(71)2442 = (17*104885-71)/99
Hans Rosenthal sent a new SUPP record of 4885 digits !
( Announced at Walter Schneider's site at Undulants - now a broken link )
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
[ December 10, 2002 ]ps. This might be an interesting link for the SUPP reference page:
It's very likely that François Morain will independently verify the Primo
Hans Rosenthal informs :
certificate of the (former) "largest known SUPP"...
[ December 22, 2002 ]1(41)3171 = (14*106343-41)/99
David Broadhurst announced via a message (#2937) in the
User group for the PrimeForm program that
the following SUPP is prime !
[ December 23, 2002 ]
Reaction from Hans Rosenthal ¬
" 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."
[ July 13, 2003 ]
Hans Rosenthal 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) 
having 6959 decimal digits, with the program Primo , 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.
SUPP Factorization Projects