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:
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.
The entries 1 & 12 are the two AP22 known solutions (1 by Moran, Pritchard and Thyssen in 1993; 12 by Markus Frind in 2003).
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:
There we can read the following message from J. Wroblewski:
(*) 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 +
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: