Problems & Puzzles: Puzzles Puzzle 34. Prime Triplets in arithmetic progression In 1988 Dubner searched for triplets in arithmetic progression with the first term equal to 3, like (3, 5, 7), (3, 7, 11), (3, 11, 19), (3, 23, 43), etc. In particular he searched for triplets of the form (3, a + 1, 2*a –1). Here are three of his biggest triplets:
Can you find a higher triplet? (Ref. 1, p. 198, Table 25) Addendum: At the middle of december I wrote to H. Dubner asking if he has gotten any larger triplets. This is his answer: "Carlos, I have not done any more work on prime triplets in arithmetic progression starting with 3. Table 25 is still accurate. Since that work was done in 1988 it would be relatively easy to find much larger triplets with the faster computers that are now available, but I currently have no plans to do anything in this area. I remember having a very good time doing that project." Solution Vasiliy Danilov
(8/2/99) has advised to these pages that Warut
Roonguthai has obtained a higher triplet in
arithmetic progression, and published it here: http://www.utm.edu/~primes/PrimesL/msg00385.html (3, 475977645*2^44639+1, 475977645*2^446401) with 1, 13447 & 13447 digits respectively. Warut Roonguthai found (18/06/99) the second prime in the triplet using Proth.exe, while the third one was previously found by Kimmo Herranen in 1998 using Proth.exe also. The method used by Roonguthai was the same than before and described in the PrimeL link shown above. J. K. Andersen wrote (June 2010)
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