Problems & Puzzles: Puzzles

Puzzle 268.  23 primes in A.P.

"Caminante, no hay camino, sólo estelas en la mar" (A. Machado)

May is a good month for me. This month (and this particular week) we arrive to six (6) years publishing these prime puzzles pages (thanks so much, friends!)

Interesting news have arrived also to the prime lovers:

1) Almost for sure the 41st Mersenne known prime number has been discovered (non-officially yet, 22/5/04) by the GIMPS group

2) In the theoretical field an old conjecture has been proved (by Ben Green and Terence Tao): the prime numbers do contain arithmetic progressions of length k for all k.

But what about effective arithmetic progressions of k prime numbers?

As for sure you know, the record (k=22) was established by A. Moran, P. Pritchard and A. Thyssen.

When and how (hard) was this record established?

This record was established in 1993. 'More than sixty computers were used in this search' (Ribenboim, p. 286).

How hard is now to beat this record?

Well, probably this worth more than many words and boring calculations: last year (April 19, 2003 to be exact) Markus Frind found a second (larger) example of the same length (k=22) using an AMD 1800XP during only 10 days.

So, perhaps to beat this old computational record is nowadays at hands of a single-solitaire records-hunter.

Q. Can you get one more k=22 primes in A.P. and/or the first known k=23 primes in A.P.(*)?

News form the last minutes!!!

A few hours after I released this puzzle Markus Frind  wrote to me to let me know that he, Paul Underwood and Paul Jobling are working together hunting the AP23, but in the meanwhile they have gotten other 15 AP22 solutions, kindly submitted in order other readers consider them.

1         11,410,337,850,553 + k*20660*23#
2         11,430,832,489,277 + k*65971*23#
3         57,564,516,803,819 + k*172009*23#
4         61,168,090,488,293 + k*123663*23#
5         82,343,600,010,653 + k*133371*23#
6         94,244,389,707,469 + k*50879*23#
7       102,458,805,947,077 + k*106082*23#
8       128,248,121,159,371 + k*132852*23#
9       168,219,776,115,997 + k*99615*23#
10       219,555,913,683,761 + k*80504*23#
11       239,043,302,770,157 + k*17205*23#
12       376,859,931,192,959 + k*83146*23#
13       379,825,257,999,359 + k*95753*23#
14       388,518,661,431,227 + k*121347*23#
15       391,070,459,886,979 + k*53110*23#
16       497,003,857,949,969 + k*7826*23#
17       515,629,258,109,087 + k*31432*23#

The entries 1 & 12 are the two AP22 known solutions (1 by Moran, Pritchard and Thyssen in 1993; 12 by Markus Frind in 2003).

"Most likely there are hundreds of solutions in between what we have already found. Our multiplier m (k*m*23#) is between 1 and ~300,000 depending on the version of the program...From the way we have our search set up (randomized), we would expect to find a AP23 for every 4.5 AP22's we find. as you can see Poisson  hasn't been our friend..."


Don't you see something strange in the Frind-Underwood-Jobling results How is that, that they have not find any AP23 result after 15 AP22 if the expected results are one AP23 for each 5 AP22? Is this flaw a result of the 'randomized search'? Is it that the prediction is not true? Is it that other hidden factor is the cause of the flaw...?


On March 2007, Paul Underwood wrote:

Although not an AP23, an AP24 was luckily found a couple of months back:  with use of 75 computers.

There we can read the following message from J. Wroblewski:

Today, January 18, 2007, at 3:06 am one of the computers I am using

468395662504823 + 205619*23#*n, n=0..23

I looked at the file at 4:23 and verified the result a few minutes
later. I believe this is record AP23 with last term 1523454717745013
and this is the first AP24.





(*) minimal difference for the k=23 primes in A.P. must be 223092870=2*3*5*7*11*13*17*19*23, according to an old M. Cantor's theorem (Ribenboim, p. 284)


Markus Frind wrote (11/6/04):

Here is another AP22..... 121,840,659,021,089 + K*150945*23#

 Finally, the 25/07/04 Markus Frind wrote the good new:

On July 24th 2004 myself (Markus Frind), Paul Jobling and Paul Underwood discovered the worlds first 23 primes in arithmetic progression. 56,211,383,760,397 +K*44,546,738,095,860 for K =0 to 22. More information can be found at


My comments:

56,211,383,760,397 +K*44,546,738,095,860 for K =0 to 22 is equal to 56,211,383,760,397 +K*199678*23# for K =0 to 22.

The center of this 25 A. P. is located at the 12th prime, in this case at the prime 56,211,383,760,397 +11*199678*23# = 56,211,383,760,397 +11*199678*223092870 = 56,211,383,760,397 +11*44,546,738,095,860 = 546,225,502,814,857.

Sub-question: is this the earliest example of 23 primes in A.P., that is to say, are there another 23 primes in A.P. such that the center is a prime smaller than 546,225,502,814,857?


The same Markus Frind wrote on March 31, 2006:

I found the following AP23's that solve your sub problem.

403185216600637 +9523*23#
478447998087407+ 23548*23#
100308707032367 + 156722·23





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