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:

 First prime Second prime Third prime Digits 3 415587*10^800+1 831174*10^800 - 1 806 3 235398*10^1000+1 470796*10^1000 – 1 1006 3 87114*10^1100+1 174228*10^1100 – 1 1106

Can you find a higher triplet?

(Ref. 1, p. 198, Table 25)

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/Primes-L/msg00385.html
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This 20/06/99 Warut Roonguthai communicated by e-mail another and larger triplet:

(3, 475977645*2^44639+1, 475977645*2^44640-1) 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 Prime-L link shown above.

J. K. Andersen wrote (June 2010)

David Broadhurst thought of the same challenge as this puzzle:
http://tech.groups.yahoo.com/group/primeform/message/10358

I told him about puzzle 34 and he found a new record in
http://tech.groups.yahoo.com/group/primeform/message/10360 :
(3, 26866689*2^44999+1, 26866689*2^45000-1)

I used The Prime Pages database similarly to Warut Roonguthai and
soon found two more records:
(3, 8311875*2^73162+1, 8311875*2^73163-1)
(3, 955*2^103848+1, 955*2^103849-1)

The 22032-digit 8311875*2^73163-1 was found by Tim Nikkel in 2000:
http://primes.utm.edu/primes/page.php?id=8097

The 31265-digit 955*2^103848+1 was found by Anton Oleynick in 2001:
http://primes.utm.edu/primes/page.php?id=4170

They both used Proth.exe. I found the other primes with PrimeForm/GW.

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