Smoothly Undulating Palindromic Primes

World!Of Numbers		HOME plate WON \|
Smoothly Undulating Palindromic Primes (SUPP's)
1 2 3 4 5 6 Near Smoothly Undulating Primes NSUP's (with 6-digit undulators) Near Smoothly Undulating Primes NSUP's (with 22-digit undulators) Palindromic Wing Primes Plateau & Depres. Primes Palindromic Merlon Primes Home Primes Circular Primes SUPP-sorted

101	121	131	141	151	161
171	181	191	313	323	343
353	373	383	717	727	737
747	757	787	797	919	929
949	959	979	989

Smoothly Undulating Palprimes

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 by Andrzej Nowicki

Translated in Dutch "Priemgetallen met een speciale rangschikking van cijfers"

Translated in English "Prime numbers with a special arrangement of digits"

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

Messages

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

(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 !

[ 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:
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."

[ February 14, 2001 ]
Message from Carlos Rivera

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.

[ 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/

The following link includes many details about Landon's proof of the SUPP
and why his proof got lost : Yahoo Message 1942 - 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

[ May 6, 2001 ]
Enters Hans Rosenthal 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!

[ 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) = (10^k –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 ]
Hans Rosenthal 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."

[ October 27, 2002 ]
Hans Rosenthal 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.eu/public/primo/top20.html

[ December 10, 2002 ]
Hans Rosenthal informs :

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"...

[ December 22, 2002 ]
David Broadhurst announced via a message (http://groups.yahoo.com/group/primeform/message/2937) in the
User group for the PrimeForm program that
the following SUPP is prime !

1(41)₃₁₇₁ = (14*10⁶³⁴³–41)/99

[ 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) [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.eu/public/primo/files/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.

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.eu/public/primo/top20.html

“SUPP” + “SUP” Factorization Projects

The starting point of this overview was to mirror Hisanori Mishima's pages regarding SUPP's and SUP's as they seem to be discontinued since 2013 and his website might disappear one day.
At the time I wasn't yet aware that M. Kamada's pages 'abbba.htm' (depression and plateau primes) divided by 11 are often also SUPP's and/or SUP's.
These numbers are only indirectly or implicitly 'smoothly undulating' by considering the odd exponents divided by 11 sometimes multiplied by some 'm' value.
Importantly his lists go up to exponent 300 instead of Mishima's exponent 100 ! Therefore I will add to my mirrored pages from Mishima a link to each Kamada entry were appropriate.
Hereunder you'll find the factor-list of the SUPP's to start with.

SUPP (Smoothly Undulating Prime Palindromes) reference files
1(01)_w = (10*10ⁿ–1)/99	Factorization of Repunits (M. Kamada) with even exponents n divided by 11.
1(21)_w = (12*10ⁿ–21)/99	Factorization of 133...331 (M. Kamada) with odd exponents n divided by 11.
1(31)_w = (13*10ⁿ–31)/99	Factorization of 144...441 (M. Kamada) with odd exponents n divided by 11.
1(41)_w = (14*10ⁿ–41)/99	Factorization of 155...551 (M. Kamada) with odd exponents n divided by 11.
1(51)_w = (15*10ⁿ–51)/99	Factorization of 166...661 (M. Kamada) with odd exponents n divided by 11.
1(61)_w = (16*10ⁿ–61)/99	Factorization of 177...771 (M. Kamada) with odd exponents n divided by 11.
1(71)_w = (17*10ⁿ–71)/99	Factorization of 188...881 (M. Kamada) with odd exponents n divided by 11.
1(81)_w = (18*10ⁿ–81)/99	Factorization of 199...991 (M. Kamada) with odd exponents n divided by 11.
1(91)_w = (19*10ⁿ–91)/99	facsupp191.htm (maintained by Patrick De Geest).	aba(1,9,n)=191...191 (n=1 to 100) (Hisanori Mishima)	Free to factor 4 remaining

3(13)_w = (31*10ⁿ–13)/99	Factorization of 344...443 (M. Kamada) with odd exponents n divided by 11.
3(23)_w = (32*10ⁿ–23)/99	Factorization of 355...553 (M. Kamada) with odd exponents n divided by 11.
3(43)_w = (34*10ⁿ–43)/99	Factorization of 377...773 (M. Kamada) with odd exponents n divided by 11.
3(53)_w = (35*10ⁿ–53)/99	Factorization of 388...883 (M. Kamada) with odd exponents n divided by 11.
3(73)_w = (37*10ⁿ–73)/99	facsupp373.htm (maintained by Patrick De Geest).	aba(3,7,n)=373...373 (n=1 to 100) (Hisanori Mishima)	Free to factor 7 remaining
3(83)_w = (38*10ⁿ–83)/99	facsupp383.htm (maintained by Patrick De Geest).	aba(3,8,n)=383...383 (n=1 to 100) (Hisanori Mishima)	Free to factor 6 remaining

7(17)_w = (71*10ⁿ–17)/99	Factorization of 788...887 (M. Kamada) with odd exponents n divided by 11.
7(27)_w = (72*10ⁿ–27)/99	Factorization of 799...997 (M. Kamada) with odd exponents n divided by 11.
7(37)_w = (73*10ⁿ–37)/99	facsupp737.htm (maintained by Patrick De Geest).	aba(7,3,n)=737...737 (n=1 to 100) (Hisanori Mishima)	Free to factor 6 remaining
7(47)_w = (74*10ⁿ–47)/99	facsupp747.htm (maintained by Patrick De Geest).	aba(7,4,n)=747...747 (n=1 to 100) (Hisanori Mishima)	Free to factor 11 remaining
7(57)_w = (75*10ⁿ–57)/99	facsupp757.htm (maintained by Patrick De Geest).	aba(7,5,n)=757...757 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
7(87)_w = (78*10ⁿ–87)/99	facsupp787.htm (maintained by Patrick De Geest).	aba(7,8,n)=787...787 (n=1 to 100) (Hisanori Mishima)	Free to factor 9 remaining
7(97)_w = (79*10ⁿ–97)/99	facsupp797.htm (maintained by Patrick De Geest).	aba(7,9,n)=797...797 (n=1 to 100) (Hisanori Mishima)	Free to factor 8 remaining

9(19)_w = (91*10ⁿ–19)/99	facsupp919.htm (maintained by Patrick De Geest).	aba(9,1,n)=919...919 (n=1 to 100) (Hisanori Mishima)	Free to factor 7 remaining
9(29)_w = (92*10ⁿ–29)/99	facsupp929.htm (maintained by Patrick De Geest).	aba(9,2,n)=929...929 (n=1 to 100) (Hisanori Mishima)	Free to factor 5 remaining
9(49)_w = (94*10ⁿ–49)/99	facsupp949.htm (maintained by Patrick De Geest).	aba(9,4,n)=949...949 (n=1 to 100) (Hisanori Mishima)	Free to factor 8 remaining
9(59)_w = (95*10ⁿ–59)/99	facsupp959.htm (maintained by Patrick De Geest).	aba(9,5,n)=959...959 (n=1 to 100) (Hisanori Mishima)	Free to factor 6 remaining
9(79)_w = (97*10ⁿ–79)/99	facsupp979.htm (maintained by Patrick De Geest).	aba(9,7,n)=979...979 (n=1 to 100) (Hisanori Mishima)	Free to factor 5 remaining
9(89)_w = (98*10ⁿ–89)/99	facsupp989.htm (maintained by Patrick De Geest).	aba(9,8,n)=989...989 (n=1 to 100) (Hisanori Mishima)	Free to factor 7 remaining

There are of course some SUP's that can never become prime (except the single digits 2, 3, 5, 7), or become SUPP, but are still worth factoring.
Note that 2(02)_w, 3(03)_w, 4(04)_w, 5(05)_w, 6(06)_w, 7(07)_w, 8(08)_w and 9(09)_w can all be reduced to the 1(01)_w case multiplied by d with d = 2, 3, 4, 5, 6, 7, 8, 9.
Hereunder at the left is the table with the remaining list of these SUP's.

Case [k](d₁d₂)_u = [prefix](2-digit undulator). Cases with 6-digit undulators will be investigated at undulsix.htm.
Attentive readers of the factor lists will have noticed with me that some SUP primefactors are themselves near smoothly undulating. 'Near' because of an initial prefix and the 2-digit undulators.
We divide therefore the SUP by a 2 (or a power of 2) or 5 (or a power of 5).
For instance with 2(12)_w / 2² which equals 530303030303... I coined the acronym NSUP's for this lot.
In shorthand (21*10ⁿ–12)/(99*4) results in [53](03)_n. This give rise to a whole new and interesting field where we can try to find evermore primes.
In the above examples we found primes (PRP's) for values n = 3, 5, 7, 21, 77, 103, 143, 521, 1265, 1347, 5017, 9027, 15737, ...
Also watch out for exceptions. E.g. 6(76)_w which equals 2 * 2 * 169191919191919... but [169](19)_n has a covering set {3, 13, 7} and is always composite ! Hence the added 'AC'.

SUP (Smoothly Undulating Composite Palindromes) reference files

2(12)_w = (21*10ⁿ–12)/99	facsup212.htm (maint. by P. De Geest).	aba(2,1,n)=212...212 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
2(32)_w = (23*10ⁿ–32)/99	facsup232.htm (maint. by P. De Geest).	aba(2,3,n)=232...232 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
2(52)_w = (25*10ⁿ–52)/99	facsup252.htm (maint. by P. De Geest).	aba(2,5,n)=252...252 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
2(72)_w = (27*10ⁿ–72)/99	facsup272.htm (maint. by P. De Geest).	aba(2,7,n)=272...272 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
2(92)_w = (29*10ⁿ–92)/99	facsup292.htm (maint. by P. De Geest).	aba(2,9,n)=292...292 (n=1 to 100) (Hisanori Mishima)	Free to factor 7 remaining

4(14)_w = (41*10ⁿ–14)/99	facsup414.htm (maint. by P. De Geest).	aba(4,1,n)=414...414 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
4(34)_w = (43*10ⁿ–34)/99	facsup434.htm (maint. by P. De Geest).	aba(4,3,n)=434...434 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
4(54)_w = (45*10ⁿ–54)/99	facsup454.htm (maint. by P. De Geest).	aba(4,5,n)=454...454 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
4(74)_w = (47*10ⁿ–74)/99	facsup474.htm (maint. by P. De Geest).	aba(4,7,n)=474...474 (n=1 to 100) (Hisanori Mishima)	Free to factor 8 remaining
4(94)_w = (49*10ⁿ–94)/99	facsup494.htm (maint. by P. De Geest).	aba(4,9,n)=494...494 (n=1 to 100) (Hisanori Mishima)	Free to factor 9 remaining

5(15)_w = (51*10ⁿ–15)/99	facsup515.htm (maint. by P. De Geest).	aba(5,1,n)=515...515 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
5(25)_w = (52*10ⁿ–25)/99	facsup525.htm (maint. by P. De Geest).	aba(5,2,n)=525...525 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
5(35)_w = (53*10ⁿ–35)/99	facsup535.htm (maint. by P. De Geest).	aba(5,3,n)=535...535 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
5(45)_w = (54*10ⁿ–45)/99	facsup545.htm (maint. by P. De Geest).	aba(5,4,n)=545...545 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
5(65)_w = (56*10ⁿ–65)/99	facsup565.htm (maint. by P. De Geest).	aba(5,6,n)=565...565 (n=1 to 100) (Hisanori Mishima)	Free to factor 10 remaining
5(75)_w = (57*10ⁿ–75)/99	facsup575.htm (maint. by P. De Geest).	aba(5,7,n)=575...575 (n=1 to 100) (Hisanori Mishima)	Free to factor 8 remaining
5(85)_w = (58*10ⁿ–85)/99	facsup585.htm (maint. by P. De Geest).	aba(5,8,n)=585...585 (n=1 to 100) (Hisanori Mishima)	Free to factor 6 remaining
5(95)_w = (59*10ⁿ–95)/99	facsup595.htm (maint. by P. De Geest).	aba(5,9,n)=595...595 (n=1 to 100) (Hisanori Mishima)	Free to factor 9 remaining

6(16)_w = (61*10ⁿ–16)/99	facsup616.htm (maint. by P. De Geest).	aba(6,1,n)=616...616 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
6(56)_w = (65*10ⁿ–56)/99	facsup656.htm (maint. by P. De Geest).	aba(6,5,n)=656...656 (n=1 to 100) (Hisanori Mishima)	Free to factor 7 remaining
6(76)_w = (67*10ⁿ–76)/99	facsup676.htm (maint. by P. De Geest).	aba(6,7,n)=676...676 (n=1 to 100) (Hisanori Mishima)	Free to factor 6 remaining

7(67)_w = (76*10ⁿ–67)/99	facsup767.htm (maint. by P. De Geest).	aba(7,6,n)=767...767 (n=1 to 100) (Hisanori Mishima)	Free to factor 6 remaining

8(18)_w = (81*10ⁿ–18)/99	facsup818.htm (maint. by P. De Geest).	aba(8,1,n)=818...818 (n=1 to 100) (Hisanori Mishima)	All factored (n ⩽ 100)
8(38)_w = (83*10ⁿ–38)/99	facsup838.htm (maint. by P. De Geest).	aba(8,3,n)=838...838 (n=1 to 100) (Hisanori Mishima)	Free to factor 7 remaining
8(58)_w = (85*10ⁿ–58)/99	facsup858.htm (maint. by P. De Geest).	aba(8,5,n)=858...858 (n=1 to 100) (Hisanori Mishima)	Free to factor 6 remaining
8(78)_w = (87*10ⁿ–78)/99	facsup878.htm (maint. by P. De Geest).	aba(8,7,n)=878...878 (n=1 to 100) (Hisanori Mishima)	Free to factor 3 remaining
8(98)_w = (89*10ⁿ–98)/99	facsup898.htm (maint. by P. De Geest).	aba(8,9,n)=898...898 (n=1 to 100) (Hisanori Mishima)	Free to factor 8 remaining

[k](d₁d₂)_u = NSUP's (Near Smoothly Undulating Primes)

2(12)_w / 2² (2110ⁿ–12)/(994) = [53](03)_u	Link to table
2(32)_w / 2⁵ (2310ⁿ–32)/(9932) = [726](01)_u	Link to table
2(52)_w / 2² (2510ⁿ–52)/(994) = [63](13)_u	Link to table
2(72)_w / 2³ (2710ⁿ–72)/(998) = [34](09)_u	Link to table
2(92)_w / 2² (2910ⁿ–92)/(994) = [73](23)_u	Link to table

4(14)_w / 2 (4110ⁿ–14)/(992) = [2](07)_u	Link to table
4(34)_w / 2 (4310ⁿ–34)/(992) = [2](17)_u	Link to table
4(54)_w / 2 (4510ⁿ–54)/(992) = [2](27)_u	Link to table
4(74)_w / 2 (4710ⁿ–74)/(992) = [2](37)_u	Link to table
4(94)_w / 2 (4910ⁿ–94)/(992) = [2](47)_u	Link to table

5(15)_w / 5 (5110ⁿ–15)/(995) = [1](03)_u	Link to table
5(25)_w / 5² (5210ⁿ–25)/(9925) = [21](01)_n	Link to table
5(35)_w / 5 (5310ⁿ–35)/(995) = [1](07)_u	Link to table
5(45)_w / 5 (5410ⁿ–45)/(995) = [1](09)_u	Link to table
5(65)_w / 5 (5610ⁿ–65)/(995) = [1](13)_u	Link to table
5(75)_w / 5² (5710ⁿ–75)/(9925) = [23](03)_u	Link to table
5(85)_w / 5 (5810ⁿ–85)/(995) = [1](17)_u	Link to table
5(95)_w / 5 (5910ⁿ–95)/(995) = [1](19)_u	Link to table

6(16)_w / 2⁴ (6110ⁿ–16)/(9916) = [3851](01)_u	Link to table
6(56)_w / 2³ (6510ⁿ–56)/(998) = [82](07)_u	Link to table
6(76)_w / 2² (6710ⁿ–76)/(994) = [169](19)_u = AC

Undulator(s) more than 2 digits_w^u

8(18)_w / 2 (8110ⁿ–18)/(992) = [4](09)_u	Link to table
8(38)_w / 2 (8310ⁿ–38)/(992) = [4](19)_u	Link to table
8(58)_w / 2 (8510ⁿ–58)/(992) = [4](29)_u	Link to table
8(78)_w / 2 (8710ⁿ–78)/(992) = [4](39)_u	Link to table
8(98)_w / 2 (8910ⁿ–98)/(992) = [4](49)_u	Link to table

Following condition must be imposed that gcd(A,B) = 1 (in Factorization of ABA...ABA), i.e. A and B are coprime, since if A and B have a common factor > 1, then we can divide this factor from the number,
e.g. factor 69696...69696 is equivalent to factor 23232...23232.

The “SUPP” Table

The reference table for Smoothly Undulating Palindromic Primes
This collection is complete for probable primes up to 100,000 (ref. RC) 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 PDG = Patrick De Geest RC = Ray Chandler
SUPP	Formula Accolades = prime exp Blue exp = # of digits	Who	When	Status	Program Output Logs
¬
1(01)₁	(10*10^{3}–01)/99 IMPORTANT NOTE	JH	Feb 09 2001	PRIME	View
A062209 ¬ A056803 ¬
1(21)₃	(12*10^{7}–21)/99	JH	Feb 09 2001	PRIME	View
1(21)₅	(12*10^{11}–21)/99	JH	Feb 09 2001	PRIME	View
1(21)₂₁	(12*10^{43}–21)/99	JH	Feb 09 2001	PRIME	View
1(21)₆₉	(12*10^{139}–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*10^{1399}–21)/99	HR	Jun 17 2001	PRIME	View
1(21)₇₉₈	(12*10^{1597}–21)/99	HR	Jun 17 2001	PRIME	View
1(21)₉₈₉	(12*10^{1979}–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(21)₂₃₂₄₉	(12*10^{46499}–21)/99	RC	Oct 12 2010	PROBABLE PRIME	View
¬
1(31)₁	(13*10^{3}–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*10^{11}–41)/99	JH	Feb 09 2001	PRIME	View
1(41)₁₃₈	(14*10^{277}–41)/99	JH	Feb 09 2001	PRIME	View
1(41)₂₃₉	(14*10^{479}–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*10^{6343}–41)/99	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*10^{3}–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*10^{89}–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
1(51)₂₂₃₂₆	(15*10⁴⁴⁶⁵³–51)/99	RC	Oct 04 2010	PROBABLE PRIME	View
A062212 ¬
1(61)₃	(16*10^{7}–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*10^{109}–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*10^{229}–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*10^{31}–71)/99	JH	Feb 09 2001	PRIME	View
1(71)₁₈	(17*10^{37}–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*10^{3}–81)/99	JH	Feb 09 2001	PRIME	View
1(81)₂	(18*10^{5}–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*10^{163}–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*10^{30259}–81)/99	HR	Oct 19 2001	PROBABLE PRIME	View
1(81)₁₅₅₃₁	(18*10^{31063}–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
1(81)₃₃₃₂₈	(18*10⁶⁶⁶⁵⁷–81)/99	RC	Jan 31 2011	PROBABLE PRIME	View
A062215 ¬
1(91)₁	(19*10^{3}–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*10^{3}–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*10^{83}–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
3(13)₂₁₉₃₄	(31*10⁴³⁸⁶⁹–13)/99	RC	Sep 30 2010	PROBABLE PRIME	View
A062217 ¬
3(23)₂	(32*10^{5}–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*10^{11}–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*10^{3407}–23)/99	HR	Oct 19 2001	PRIME	View
3(23)₃₄₇₉	(32*10^{6959}–23)/99	HR	Jul 08 2003	PRIME	View
3(23)₄₇₉₉	(32*10⁹⁵⁹⁹–23)/99	HR	Jun 04 2001	RECORD PROVEN PRIME	View
3(23)₅₆₉₉	(32*10^{11399}–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*10^{3}–53)/99	JH	Feb 09 2001	PRIME	View
3(53)₂	(35*10^{5}–53)/99	JH	Feb 09 2001	PRIME	View
3(53)₁₁	(35*10^{23}–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*10^{4157}–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*10^{3}–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*10^{3}–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*10^{17}–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
3(83)₄₅₉₉₈	(38*10^{91997}–83)/99	RC	Jul 29 2011	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*10^{3}–27)/99	JH	Feb 09 2001	PRIME	View
7(27)₂	(72*10^{5}–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*10^{17}–27)/99	JH	Feb 09 2001	PRIME	View
7(27)₃₅	(72*10^{71}–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	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*10^{5}–47)/99	JH	Feb 09 2001	PRIME	View
7(47)₈	(74*10^{17}–47)/99	JH	Feb 09 2001	PRIME	View
7(47)₁₀₃₄	(74*10^{2069}–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*10^{3}–57)/99	JH	Feb 09 2001	PRIME	View
7(57)₈	(75*10^{17}–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*10^{149}–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
7(57)₂₀₈₂₈	(75*10⁴¹⁶⁵⁷–57)/99	RC	Sep 16 2010	PROBABLE PRIME	View
7(57)₃₆₄₇₀	(75*10⁷²⁹⁴¹–57)/99	RC	Mar 24 2011	PROBABLE PRIME	View
A062225 ¬
7(87)₁	(78*10^{3}–87)/99	JH	Feb 09 2001	PRIME	View
7(87)₂	(78*10^{5}–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*10^{34883}–87)/99	HR	Oct 19 2001	PROBABLE PRIME	View
A062226 ¬
7(97)₁	(79*10^{3}–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
7(97)₂₅₈₃₁	(79*10⁵¹⁶⁶³–97)/99	RC	Nov 3 2010	PROBABLE PRIME	View
7(97)₃₃₂₂₃	(79*10⁶⁶⁴⁴⁷–97)/99	RC	Jan 29 2011	PROBABLE PRIME	View
A062227 ¬
9(19)₁	(91*10^{3}–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*10^{11}–19)/99	JH	Feb 09 2001	PRIME	View
9(19)₈	(91*10^{17}–19)/99	JH	Feb 09 2001	PRIME	View
9(19)₁₁	(91*10^{23}–19)/99	JH	Feb 09 2001	PRIME	View
9(19)₁₂₆₁₄	(91*10^{25229}–19)/99	HR	Jun 15 2001	PROBABLE PRIME	View
A062228 ¬
9(29)₁	(92*10^{3}–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*10^{857}–29)/99	HR	Jun 04 2001	PRIME	View
9(29)₅₆₉₆	(92*10^{11393}–29)/99	HR	Jun 04 2001	PROBABLE PRIME	View
A062229 ¬
9(49)₂	(94*10^{5}–49)/99	JH	Feb 09 2001	PRIME	View
9(49)₈	(94*10^{17}–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
9(49)₄₆₄₉₀	(94*10⁹²⁹⁸¹–49)/99	RC	Jul 30 2011	RECORD PROBABLE PRIME	View
A062230 ¬
9(59)₂	(95*10^{5}–59)/99	JH	Feb 09 2001	PRIME	View
9(59)₈	(95*10^{17}–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
9(89)₂₆₄₆₅	(98*10⁵²⁹³¹–89)/99	RC	Nov 9 2010	PROBABLE PRIME	View
9(89)₄₄₂₉₇	(98*10⁸⁸⁵⁹⁵–89)/99	RC	Jul 23 2011	PROBABLE PRIME	View

The “NSUP” Table

The reference table for Near Smoothly Undulating Primes Cases with 2-digit undulators derived from the composite set of SUP's
This collection is complete for probable primes up to 30,000 digits.		PDG = Patrick De Geest
NSUP	Formula Accolades = prime exp	Who	When	Status	Prime Certificat
¬
2(12)₁/2² = [53](03)₀	(2110^{3}–12)/(994)	PDG	Aug 01 2022	PRP	View
2(12)₃/2² = [53](03)₂	(2110^{7}–12)/(994)	PDG	Aug 01 2022	PRP	View
2(12)₁₀/2² = [53](03)₉	(2110²¹–12)/(994)	PDG	Aug 01 2022	PRP	View
2(12)₃₈/2² = [53](03)₃₇	(2110⁷⁷–12)/(994)	PDG	Aug 01 2022	PRP	View
2(12)₅₁/2² = [53](03)₅₀	(2110^{103}–12)/(994)	PDG	Aug 01 2022	PRP	View
2(12)₇₁/2² = [53](03)₇₀	(2110¹⁴³–12)/(994)	PDG	Aug 01 2022	PRP	View
2(12)₂₆₀/2² = [53](03)₂₅₉	(2110^{521}–12)/(994)	PDG	Aug 01 2022	PRP	View
2(12)₆₃₂/2² = [53](03)₆₃₁	(2110¹²⁶⁵–12)/(994)	PDG	Aug 01 2022	PRP	View
2(12)₆₇₃/2² = [53](03)₆₇₂	(2110¹³⁴⁷–12)/(994)	PDG	Aug 01 2022	PRP	View
2(12)₂₅₀₈/2² = [53](03)₂₅₀₇	(2110⁵⁰¹⁷–12)/(994)	PDG	Aug 01 2022	PRP	View
2(12)₄₅₁₃/2² = [53](03)₄₅₁₂	(2110⁹⁰²⁷–12)/(994)	PDG	Aug 01 2022	PRP	View
2(12)₇₈₆₈/2² = [53](03)₇₈₆₇	(2110^{15737}–12)/(994)	PDG	Aug 01 2022	PRP	View
¬
2(32)₉₁/2⁵ = [726](01)₈₉	(2310¹⁸³–32)/(9932)	PDG	Aug 01 2022	PRP	View
2(32)₇₂₁/2⁵ = [726](01)₇₁₉	(2310¹⁴⁴³–32)/(9932)	PDG	Aug 01 2022	PRP	View
2(32)₃₀₀₀/2⁵ = [726](01)₂₉₉₈	(2310⁶⁰⁰¹–32)/(9932)	PDG	Aug 01 2022	PRP	View
¬
2(52)₅/2² = [63](13)₄	(2510^{11}–52)/(994)	PDG	Aug 01 2022	PRP	View
2(52)₁₅/2² = [63](13)₁₄	(2510^{31}–52)/(994)	PDG	Aug 01 2022	PRP	View
2(52)₄₂/2² = [63](13)₄₁	(2510⁸⁵–52)/(994)	PDG	Aug 01 2022	PRP	View
2(52)₂₀₄/2² = [63](13)₂₀₃	(2510^{409}–52)/(994)	PDG	Aug 01 2022	PRP	View
2(52)₇₀₂/2² = [63](13)₇₀₁	(2510¹⁴⁰⁵–52)/(994)	PDG	Aug 01 2022	PRP	View
¬
2(72)₃/2³ = [34](09)₂	(2710^{7}–72)/(998)	PDG	Aug 01 2022	PRP	View
2(72)₁₂/2³ = [34](09)₁₁	(2710²⁵–72)/(998)	PDG	Aug 01 2022	PRP	View
2(72)₅₅/2³ = [34](09)₅₄	(2710¹¹¹–72)/(998)	PDG	Aug 01 2022	PRP	View
2(72)₂₉₄₁/2³ = [34](09)₂₉₄₀	(2710⁵⁸⁸³–72)/(998)	PDG	Aug 01 2022	PRP	View
2(72)₃₈₅₃/2³ = [34](09)₃₈₅₂	(2710⁷⁷⁰⁷–72)/(998)	PDG	Aug 01 2022	PRP	View
2(72)₈₄₈₇/2³ = [34](09)₈₄₈₆	(2710¹⁶⁹⁷⁵–72)/(998)	PDG	Aug 01 2022	PRP	View
¬	n ⩾ 67231 (PDG, August 8, 2022)
2(92)₁/2² = [73](23)₀	(2910^{3}–92)/(994)	PDG	Aug 01 2022	PRP	View
2(92)₃/2² = [73](23)₂	(2910^{7}–92)/(994)	PDG	Aug 01 2022	PRP	View
2(92)₂₈/2² = [73](23)₂₇	(2910⁵⁷–92)/(994)	PDG	Aug 01 2022	PRP	View
2(92)₂₂₆/2² = [73](23)₂₂₅	(2910⁴⁵³–92)/(994)	PDG	Aug 01 2022	PRP	View
¬
4(14)₀/2 = [2](07)₀	(4110¹–14)/(992)	PDG	Aug 01 2022	PRP	View
4(14)₂/2 = [2](07)₂	(4110^{5}–14)/(992)	PDG	Aug 01 2022	PRP	View
4(14)₁₂₉/2 = [2](07)₁₂₉	(4110²⁵⁹–14)/(992)	PDG	Aug 01 2022	PRP	View
4(14)₂₄₉/2 = [2](07)₂₄₉	(4110^{499}–14)/(992)	PDG	Aug 01 2022	PRP	View
4(14)₃₁₅/2 = [2](07)₃₁₅	(4110^{631}–14)/(992)	PDG	Aug 01 2022	PRP	View
4(14)₅₅₇/2 = [2](07)₅₅₇	(4110¹¹¹⁵–14)/(992)	PDG	Aug 01 2022	PRP	View
4(14)₆₁₅/2 = [2](07)₆₁₅	(4110^{1231}–14)/(992)	PDG	Aug 01 2022	PRP	View
4(14)₉₆₅/2 = [2](07)₉₆₅	(4110^{1931}–14)/(992)	PDG	Aug 01 2022	PRP	View
4(14)₄₆₀₅/2 = [2](07)₄₆₀₅	(4110⁹²¹¹–14)/(992)	PDG	Aug 01 2022	PRP	View
¬
4(34)₀/2 = [2](17)₀	(4310¹–34)/(992)	PDG	Aug 01 2022	PRP	View
4(34)₃/2 = [2](17)₃	(4310^{7}–34)/(992)	PDG	Aug 01 2022	PRP	View
4(34)₆/2 = [2](17)₆	(4310^{13}–34)/(992)	PDG	Aug 01 2022	PRP	View
4(34)₄₅/2 = [2](17)₄₅	(4310⁹¹–34)/(992)	PDG	Aug 01 2022	PRP	View
4(34)₄₈/2 = [2](17)₄₈	(4310^{97}–34)/(992)	PDG	Aug 01 2022	PRP	View
4(34)₂₉₁/2 = [2](17)₂₉₁	(4310⁵⁸³–34)/(992)	PDG	Aug 01 2022	PRP	View
4(34)₂₃₈₈/2 = [2](17)₂₃₈₈	(4310⁴⁷⁷⁷–34)/(992)	PDG	Aug 01 2022	PRP	View
¬
4(54)₀/2 = [2](27)₀	(4510¹–54)/(992)	PDG	Aug 01 2022	PRP	View
4(54)₁/2 = [2](27)₁	(4510^{3}–54)/(992)	PDG	Aug 01 2022	PRP	View
4(54)₂/2 = [2](27)₂	(4510^{5}–54)/(992)	PDG	Aug 01 2022	PRP	View
4(54)₃/2 = [2](27)₃	(4510^{7}–54)/(992)	PDG	Aug 01 2022	PRP	View
4(54)₈/2 = [2](27)₈	(4510^{17}–54)/(992)	PDG	Aug 01 2022	PRP	View
4(54)₄₈₆/2 = [2](27)₄₈₆	(4510⁹⁷³–54)/(992)	PDG	Aug 01 2022	PRP	View
4(54)₄₉₇/2 = [2](27)₄₉₇	(4510⁹⁹⁵–54)/(992)	PDG	Aug 01 2022	PRP	View
4(54)₇₀₃/2 = [2](27)₇₀₃	(4510¹⁴⁰⁷–54)/(992)	PDG	Aug 01 2022	PRP	View
4(54)₁₄₅₁₄/2 = [2](27)₁₄₅₁₄	(4510²⁹⁰²⁹–54)/(992)	PDG	Aug 01 2022	PRP	View
¬
4(74)₀/2 = [2](37)₀	(4710¹–74)/(992)	PDG	Aug 01 2022	PRP	View
4(74)₅₇/2 = [2](37)₅₇	(4710¹¹⁵–74)/(992)	PDG	Aug 01 2022	PRP	View
¬
4(94)₀/2 = [2](47)₀	(4910¹–94)/(992)	PDG	Aug 01 2022	PRP	View
4(94)₃₃₆/2 = [2](47)₃₃₆	(4910^{673}–94)/(992)	PDG	Aug 01 2022	PRP	View
4(94)₃₉₆/2 = [2](47)₃₉₆	(4910⁷⁹³–94)/(992)	PDG	Aug 01 2022	PRP	View
¬
5(15)₀/5 = [1](03)₀	(5110¹–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₁/5 = [1](03)₁	(5110^{3}–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₂/5 = [1](03)₂	(5110^{5}–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₄/5 = [1](03)₄	(5110⁹–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₉/5 = [1](03)₉	(5110^{19}–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₂₂/5 = [1](03)₂₂	(5110⁴⁵–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₂₈/5 = [1](03)₂₈	(5110⁵⁷–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₃₉/5 = [1](03)₃₉	(5110^{79}–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₉₆/5 = [1](03)₉₆	(5110^{193}–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₁₃₈/5 = [1](03)₁₃₈	(5110^{277}–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₁₅₃₂/5 = [1](03)₁₅₃₂	(5110³⁰⁶⁵–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₁₅₅₃/5 = [1](03)₁₅₅₃	(5110³¹⁰⁷–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₃₀₂₂/5 = [1](03)₃₀₂₂	(5110⁶⁰⁴⁵–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₃₃₂₅/5 = [1](03)₃₃₂₅	(5110⁶⁶⁵¹–15)/(995)	PDG	Aug 01 2022	PRP	View
5(15)₉₈₈₈/5 = [1](03)₉₈₈₈	(5110^{19777}–15)/(995)	PDG	Aug 01 2022	PRP	View
¬
5(25)₃/5² = [21](01)₂	(5210^{7}–25)/(9925)	PDG	Aug 01 2022	PRP	View
5(25)₁₈/5² = [21](01)₁₇	(5210^{37}–25)/(9925)	PDG	Aug 01 2022	PRP	View
5(25)₁₇₀/5² = [21](01)₁₆₉	(5210³⁴¹–25)/(9925)	PDG	Aug 01 2022	PRP	View
5(25)₂₂₇/5² = [21](01)₂₂₆	(5210⁴⁵⁵–25)/(9925)	PDG	Aug 01 2022	PRP	View
5(25)₃₀₈₆/5² = [21](01)₃₀₈₅	(5210^{6173}–25)/(9925)	PDG	Aug 01 2022	PRP	View
5(25)₅₈₄₀/5² = [21](01)₅₈₃₉	(5210^{11681}–25)/(9925)	PDG	Aug 01 2022	PRP	View
¬
5(35)₁/5 = [1](07)₁	(5310^{3}–35)/(995)	PDG	Aug 01 2022	PRP	View
5(35)₄/5 = [1](07)₄	(5310⁹–35)/(995)	PDG	Aug 01 2022	PRP	View
5(35)₆/5 = [1](07)₆	(5310^{13}–35)/(995)	PDG	Aug 01 2022	PRP	View
5(35)₃₄/5 = [1](07)₃₄	(5310⁶⁹–35)/(995)	PDG	Aug 01 2022	PRP	View
5(35)₁₅₆₃/5 = [1](07)₁₅₆₃	(5310³¹²⁷–35)/(995)	PDG	Aug 01 2022	PRP	View
¬
5(45)₀/5 = [1](09)₀	(5410¹–45)/(995)	PDG	Aug 01 2022	PRP	View
5(45)₁/5 = [1](09)₁	(5410^{3}–45)/(995)	PDG	Aug 01 2022	PRP	View
5(45)₂/5 = [1](09)₂	(5410^{5}–45)/(995)	PDG	Aug 01 2022	PRP	View
5(45)₃/5 = [1](09)₃	(5410^{7}–45)/(995)	PDG	Aug 01 2022	PRP	View
5(45)₅/5 = [1](09)₅	(5410^{11}–45)/(995)	PDG	Aug 01 2022	PRP	View
5(45)₁₂/5 = [1](09)₁₂	(5410²⁵–45)/(995)	PDG	Aug 01 2022	PRP	View
5(45)₇₁₆/5 = [1](09)₇₁₆	(5410^{1433}–45)/(995)	PDG	Aug 01 2022	PRP	View
5(45)₂₈₆₇/5 = [1](09)₂₈₆₇	(5410⁵⁷³⁵–45)/(995)	PDG	Aug 01 2022	PRP	View
5(45)₅₇₃₈/5 = [1](09)₅₇₃₈	(5410¹¹⁴⁷⁷–45)/(995)	PDG	Aug 01 2022	PRP	View
¬
5(65)₀/5 = [1](13)₀	(5610¹–65)/(995)	PDG	Aug 01 2022	PRP	View
5(65)₁/5 = [1](13)₁	(5610^{3}–65)/(995)	PDG	Aug 01 2022	PRP	View
5(65)₆/5 = [1](13)₆	(5610^{13}–65)/(995)	PDG	Aug 01 2022	PRP	View
5(65)₁₀/5 = [1](13)₁₀	(5610²¹–65)/(995)	PDG	Aug 01 2022	PRP	View
5(65)₈₁₀/5 = [1](13)₈₁₀	(5610^{1621}–65)/(995)	PDG	Aug 01 2022	PRP	View
¬
5(75)₁/5² = [23](03)₀	(5710^{3}–75)/(9925)	PDG	Aug 01 2022	PRP	View
5(75)₃/5² = [23](03)₂	(5710^{7}–75)/(9925)	PDG	Aug 01 2022	PRP	View
5(75)₇/5² = [23](03)₆	(5710¹⁵–75)/(9925)	PDG	Aug 01 2022	PRP	View
5(75)₂₁/5² = [23](03)₂₀	(5710^{43}–75)/(9925)	PDG	Aug 01 2022	PRP	View
5(75)₄₀/5² = [23](03)₃₉	(5710⁸¹–75)/(9925)	PDG	Aug 01 2022	PRP	View
5(75)₆₀/5² = [23](03)₅₉	(5710¹²¹–75)/(9925)	PDG	Aug 01 2022	PRP	View
5(75)₇₃/5² = [23](03)₇₂	(5710¹⁴⁷–75)/(9925)	PDG	Aug 01 2022	PRP	View
5(75)₅₇₁/5² = [23](03)₅₇₀	(5710¹¹⁴³–75)/(9925)	PDG	Aug 01 2022	PRP	View
5(75)₇₈₃/5² = [23](03)₇₈₂	(5710^{1567}–75)/(9925)	PDG	Aug 01 2022	PRP	View
5(75)₂₉₈₀/5² = [23](03)₂₉₇₉	(5710⁵⁹⁶¹–75)/(9925)	PDG	Aug 01 2022	PRP	View
¬
5(85)₀/5 = [1](17)₀	(5810¹–85)/(995)	PDG	Aug 01 2022	PRP	View
5(85)₂/5 = [1](17)₂	(5810^{5}–85)/(995)	PDG	Aug 01 2022	PRP	View
5(85)₆/5 = [1](17)₆	(5810^{13}–85)/(995)	PDG	Aug 01 2022	PRP	View
5(85)₁₂/5 = [1](17)₁₂	(5810²⁵–85)/(995)	PDG	Aug 01 2022	PRP	View
5(85)₃₃/5 = [1](17)₃₃	(5810^{67}–85)/(995)	PDG	Aug 01 2022	PRP	View
5(85)₉₀/5 = [1](17)₉₀	(5810^{181}–85)/(995)	PDG	Aug 01 2022	PRP	View
5(85)₁₂₆₂/5 = [1](17)₁₂₆₂	(5810²⁵²⁵–85)/(995)	PDG	Aug 01 2022	PRP	View
5(85)₆₈₇₂/5 = [1](17)₆₈₇₂	(5810¹³⁷⁴⁵–85)/(995)	PDG	Aug 01 2022	PRP	View
5(85)₁₀₃₆₅/5 = [1](17)₁₀₃₆₅	(5810^{20731}–85)/(995)	PDG	Aug 01 2022	PRP	View
5(85)₁₃₆₆₅/5 = [1](17)₁₃₆₆₅	(5810²⁷³³¹–85)/(995)	PDG	Aug 01 2022	PRP	View
¬
5(95)₀/5 = [1](19)₀	(5910¹–95)/(995)	PDG	Aug 01 2022	PRP	View
5(95)₂₄/5 = [1](19)₂₄	(5910⁴⁹–95)/(995)	PDG	Aug 01 2022	PRP	View
5(95)₃₈₁/5 = [1](19)₃₈₁	(5910⁷⁶³–95)/(995)	PDG	Aug 01 2022	PRP	View
5(95)₉₇₄₁/5 = [1](19)₉₇₄₁	(5910^{19483}–95)/(995)	PDG	Aug 01 2022	PRP	View
¬
6(16)₂/2⁴ = [3851](01)₀	(6110^{5}–16)/(9916)	PDG	Aug 01 2022	PRP	View
6(16)₅/2⁴ = [3851](01)₃	(6110^{11}–16)/(9916)	PDG	Aug 01 2022	PRP	View
6(16)₆₂/2⁴ = [3851](01)₆₀	(6110¹²⁵–16)/(9916)	PDG	Aug 01 2022	PRP	View
6(16)₉₂₆/2⁴ = [3851](01)₉₂₄	(6110¹⁸⁵³–16)/(9916)	PDG	Aug 01 2022	PRP	View
¬
6(56)₄/2³ = [82](07)₃	(6510⁹–56)/(998)	PDG	Aug 01 2022	PRP	View
6(56)₁₀/2³ = [82](07)₉	(6510²¹–56)/(998)	PDG	Aug 01 2022	PRP	View
6(56)₁₃/2³ = [82](07)₁₂	(6510²⁷–56)/(998)	PDG	Aug 01 2022	PRP	View
6(56)₃₇/2³ = [82](07)₃₆	(6510⁷⁵–56)/(998)	PDG	Aug 01 2022	PRP	View
6(56)₃₀₄₇/2³ = [82](07)₃₀₄₆	(6510⁶⁰⁹⁵–56)/(998)	PDG	Aug 01 2022	PRP	View
¬
8(18)₁/2 = [4](09)₁	(8110^{3}–18)/(992)	PDG	Aug 01 2022	PRP	View
8(18)₇/2 = [4](09)₇	(8110¹⁵–18)/(992)	PDG	Aug 01 2022	PRP	View
8(18)₁₄/2 = [4](09)₁₄	(8110^{29}–18)/(992)	PDG	Aug 01 2022	PRP	View
8(18)₅₈/2 = [4](09)₅₈	(8110¹¹⁷–18)/(992)	PDG	Aug 01 2022	PRP	View
8(18)₁₄₃/2 = [4](09)₁₄₃	(8110²⁸⁷–18)/(992)	PDG	Aug 01 2022	PRP	View
8(18)₃₈₃/2 = [4](09)₃₈₃	(8110⁷⁶⁷–18)/(992)	PDG	Aug 01 2022	PRP	View
8(18)₄₈₈/2 = [4](09)₄₈₈	(8110^{977}–18)/(992)	PDG	Aug 01 2022	PRP	View
8(18)₄₉₉/2 = [4](09)₄₉₉	(8110⁹⁹⁹–18)/(992)	PDG	Aug 01 2022	PRP	View
8(18)₁₂₀₃/2 = [4](09)₁₂₀₃	(8110²⁴⁰⁷–18)/(992)	PDG	Aug 01 2022	PRP	View
8(18)₈₇₅₄/2 = [4](09)₈₇₅₄	(8110^{17509}–18)/(992)	PDG	Aug 01 2022	PRP	View
8(18)₁₁₇₀₈/2 = [4](09)₁₁₇₀₈	(8110^{23417}–18)/(992)	PDG	Aug 01 2022	PRP	View
¬
8(38)₁/2 = [4](19)₁	(8310^{3}–38)/(992)	PDG	Aug 01 2022	PRP	View
8(38)₃/2 = [4](19)₃	(8310^{7}–38)/(992)	PDG	Aug 01 2022	PRP	View
8(38)₄₃/2 = [4](19)₄₃	(8310⁸⁷–38)/(992)	PDG	Aug 01 2022	PRP	View
8(38)₈₇/2 = [4](19)₈₇	(8310¹⁷⁵–38)/(992)	PDG	Aug 01 2022	PRP	View
8(38)₈₁₁/2 = [4](19)₈₁₁	(8310¹⁶²³–38)/(992)	PDG	Aug 01 2022	PRP	View
8(38)₉₇₉/2 = [4](19)₉₇₉	(8310¹⁹⁵⁹–38)/(992)	PDG	Aug 01 2022	PRP	View
8(38)₁₃₃₇₂/2 = [4](19)₁₃₃₇₂	(8310²⁶⁷⁴⁵–38)/(992)	PDG	Aug 01 2022	PRP	View
¬
8(58)₂/2 = [4](29)₂	(8510^{5}–58)/(992)	PDG	Aug 01 2022	PRP	View
8(58)₆/2 = [4](29)₆	(8510^{13}–58)/(992)	PDG	Aug 01 2022	PRP	View
8(58)₉₀/2 = [4](29)₉₀	(8510^{181}–58)/(992)	PDG	Aug 01 2022	PRP	View
8(58)₉₈/2 = [4](29)₉₈	(8510^{197}–58)/(992)	PDG	Aug 01 2022	PRP	View
8(58)₁₄₁/2 = [4](29)₁₄₁	(8510^{283}–58)/(992)	PDG	Aug 01 2022	PRP	View
8(58)₄₄₃/2 = [4](29)₄₄₃	(8510^{887}–58)/(992)	PDG	Aug 01 2022	PRP	View
8(58)₅₆₀/2 = [4](29)₅₆₀	(8510¹¹²¹–58)/(992)	PDG	Aug 01 2022	PRP	View
8(58)₆₈₉/2 = [4](29)₆₈₉	(8510¹³⁷⁹–58)/(992)	PDG	Aug 01 2022	PRP	View
8(58)₁₁₃₉₃/2 = [4](29)₁₁₃₉₃	(8510^{22787}–58)/(992)	PDG	Aug 01 2022	PRP	View
¬
8(78)₁/2 = [4](39)₁	(8710^{3}–78)/(992)	PDG	Aug 01 2022	PRP	View
8(78)₁₀/2 = [4](39)₁₀	(8710²¹–78)/(992)	PDG	Aug 01 2022	PRP	View
8(78)₁₆/2 = [4](39)₁₆	(8710³³–78)/(992)	PDG	Aug 01 2022	PRP	View
8(78)₃₇/2 = [4](39)₃₇	(8710⁷⁵–78)/(992)	PDG	Aug 01 2022	PRP	View
8(78)₁₃₂/2 = [4](39)₁₃₂	(8710²⁶⁵–78)/(992)	PDG	Aug 01 2022	PRP	View
8(78)₁₅₀/2 = [4](39)₁₅₀	(8710³⁰¹–78)/(992)	PDG	Aug 01 2022	PRP	View
8(78)₂₇₁₆/2 = [4](39)₂₇₁₆	(8710⁵⁴³³–78)/(992)	PDG	Aug 01 2022	PRP	View
8(78)₉₉₂₂/2 = [4](39)₉₉₂₂	(8710¹⁹⁸⁴⁵–78)/(992)	PDG	Aug 01 2022	PRP	View
8(78)₁₁₃₇₀/2 = [4](39)₁₁₃₇₀	(8710^{22741}–78)/(992)	PDG	Aug 01 2022	PRP	View
¬
8(98)₁/2 = [4](49)₁	(8910^{3}–98)/(992)	PDG	Aug 01 2022	PRP	View
8(98)₂₁/2 = [4](49)₂₁	(8910^{43}–98)/(992)	PDG	Aug 01 2022	PRP	View
8(98)₂₈/2 = [4](49)₂₈	(8910⁵⁷–98)/(992)	PDG	Aug 01 2022	PRP	View
8(98)₄₃/2 = [4](49)₄₃	(8910⁸⁷–98)/(992)	PDG	Aug 01 2022	PRP	View
8(98)₁₆₉/2 = [4](49)₁₆₉	(8910³³⁹–98)/(992)	PDG	Aug 01 2022	PRP	View
8(98)₂₀₁/2 = [4](49)₂₀₁	(8910⁴⁰³–98)/(992)	PDG	Aug 01 2022	PRP	View
8(98)₂₁₀/2 = [4](49)₂₁₀	(8910^{421}–98)/(992)	PDG	Aug 01 2022	PRP	View
8(98)₂₉₈/2 = [4](49)₂₉₈	(8910⁵⁹⁷–98)/(992)	PDG	Aug 01 2022	PRP	View
8(98)₁₉₄₈/2 = [4](49)₁₉₄₈	(8910³⁸⁹⁷–98)/(992)	PDG	Aug 01 2022	PRP	View
8(98)₃₃₅₁/2 = [4](49)₃₃₅₁	(8910^{6703}–98)/(992)	PDG	Aug 01 2022	PRP	View
8(98)₁₃₂₁₃/2 = [4](49)₁₃₂₁₃	(8910²⁶⁴²⁷–98)/(992)	PDG	Aug 01 2022	PRP	View

Sources Revealed

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

12121 smoothly undulating composite

69696 smoothly undulating composite

696969696 smoothly undulating composite

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

[ TOP OF PAGE]

Patrick De Geest - Belgium

- Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

101	121	131	141	151	161
171	181	191	313	323	343
353	373	383	717	727	737
747	757	787	797	919	929
949	959	979	989

101	121	131	141	151	161
171	181	191	313	323	343
353	373	383	717	727	737
747	757	787	797	919	929
949	959	979	989

101	121	131	141	151	161
171	181	191	313	323	343
353	373	383	717	727	737
747	757	787	797	919	929
949	959	979	989