Problems & Puzzles: Puzzles

Puzzle 536. Primes in several bases (again)

This puzzle was suggested by JM Bergot

You are given a string of digits S.

Is it possible that the same string S produce a prime number when S is converted to Base 10 supposing that the string S is in base B, for B=2 to 10?

I (CR) found the following almost "solution":

 S 11111110111101000001 B S converted base 10 supposing S is in base B Prime? 2 1044289 True 3 1742858209 True 4 366487080961 False 5 23841613687501 True 6 731229510940993 True 7 13298697207471649 True 8 164703003365081089 True 9 1519707899947730221 True 10 11111110111101000001 True

Q. Can you find a complete solution?

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Useful?: http://www.easysurf.cc/cnver17.htm#batob10

J. K. Andersen wrote:

http://fatphil.org/maths/trivia/multibase.html  says
S = 110100000010101111110001010011001110001 (447045215857 in decimal) for B=2 to 10 was found by Phil Carmody in 2003 and confirmed minimal by Jim Fougeron: http://tech.groups.yahoo.com/group/primenumbers/message/13356

Furthermore, S = 1000000010000011110100010001000101001010110111001
(282607273285049 in decimal) is minimal for B=2 to 11, found in 2008 by
Andreas Höglund: http://www.mersenneforum.org/showpost.php?p=125758&postcount=71

The sequence of minimal S is in A086884. The decimal value is in A086204.

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Jan van Delden wrote:

I found the following 4 near misses (base 2 - representation), failing at only 1 base

41532851    not prime base 10
572222537  not prime base 3
636174857  not prime base 6
796769213  not prime base 10

Searched until 15.10^8 (base 2).

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On october 17, 2022, Andreas Höglund wrote:

The smallest multibase base 2-12 prime was found by Jim Viebke on mersenneforum:

base 2-12 prime:    10100001011000101000110101011011011101111110100101011
I found the smallest base 2-11 prime back in 2008   Now Jim found 6 more base 2-11 and this base 2-12.

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