Problems & Puzzles: Puzzles

Puzzle 1105 A variation related to Puzzle 1103 Emmanuel Vantieghem sent the following variation related to Puzzle 1103 I suggest the following variation of puzzle 1103 :   Successive primes P= p+10^(2^t), for t=0, 1, 2, ...   It's a fact that sets with more than five primes will appear. My actual champion is  p = 40680037 : p + 10^2 ^t: is prime for  t = 0, 1, 2, 3, 4, 5, 6  and  7. I'm sure there are primes  p  which will produce longer series !   Q. Find a prime better than p=40680037

---

During the week 2-8 October, 2022, contributions came from Giorgos Kalogeropoulos, Emmanuel Vantieghem, Oscar Volpatti, Jean-Marc Rebert

***

Giorgos wrote:

The least prime which produces a longer series is p = 347322708637

p + 10^2 ^t: is prime for  t = 0, 1, 2, 3, 4, 5, 6, 7  and  8

***

Emmanuel wrote:

I continued my search for a better  p.

I put my result in the next table where I listed  p(n) = the smallest prime  p  such that  p + 10^(2^t)  is prime for  t = 0, 1, ..., n :

n     p(n)
0     3
1     3
2     7
3     7
4    7489
5    185089
6    228403
7    40680037
8    347322708637
9      > 5*10^11 (but I guess it will be > 10^13).

***

Oscar wrote

I checked all primes below 1.2*10^13, finding several seeds which produce 9 successes:
 

347322708637,

1605611274637,

8859341434201,

9507146662999,

9621844324231,

10177552956643,

10497972271639.

***

Jean-Marc wrote:

Q. Find a prime better than p=40680037.

 

p = 2            : p + 10^2^t is prime for t = 0 and is a 3-digit composite number for t = 1.

p = 3            : p + 10^2^t is prime for t = 0, 1 and is a 5-digit composite number for t = 2.

p = 7            : p + 10^2^t is prime for t = 0, 1, 2, 3 and is a 17-digit composite number for t = 4.

p = 3037         : p + 10^2^t is prime for t = 0, 1, 2, 3, 4 and is a 33-digit composite number for t = 5.

p = 185089       : p + 10^2^t is prime for t = 0, 1, 2, 3, 4, 5 and is a 65-digit composite number for t = 6.

p = 920707       : p + 10^2^t is prime for t = 0, 1, 2, 3, 4, 5, 6 and is a 129-digit composite number for t = 7.

p = 40680037     : p + 10^2^t is prime for t = 0, 1, 2, 3, 4, 5, 6, 7 and is a 257-digit composite number for t = 8.

p = 129582646033 : p + 10^2^t is prime for t = 0, 1, 2, 3, 4, 5, 6, 7, 8 and is a 513-digit composite number for t = 9. *

 

 

***