Problems & Puzzles: Puzzles

 

 

Problems & Puzzles: Puzzles

Puzzle 1271 91997


On 3/26/2026 Sariyar sent the fillowing nice Curio & Challenge to the always interesting pages of my friend G. L. Honaker, Jr.

91997 is the largest prime number n such that after its digits respectively concatenated to the end of the preceding numbers beginning from n-1, all new numbers are prime, i.e., 919969, 919951, 919949, 919939, 919927 are all primes?


 
Q. Is there a larger example.

 


From June 12 to 18, 2026, contributions came from Michael Branicky, Simon Cavegn:

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Michael wrote:

I found no larger examples in a search up to 10^12 (that is, 12 digits and less).

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Simon wrote:

Could not find a larger solution up to 18234000000000.

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Ashaz Jameel wrote on 27 June 2026:

I believe that there are only a finite number of primes of this form (if any other examples even exist). The prime number must only contain the numbers {1,3,7,9} (as any other number would be divisible by 2 or 5).For the first part of the preceding numbers (e.g. 91996, 91995...) they differ by 10 ≡ 1 mod 3 => one in every 3 numbers can't end in {3,9} as then they would be a multiple of 3. Following this logic, one in every 7 numbers can't end in a 7 for the same reason. Therefore, one in every 3*7=21 numbers can't end in a {3,7,9} => it ends in a 1. But considering mod 11, every 11 numbers a given number is congruent to -1 mod 11 => it can't end in a 1. Combining this, we get that one in every 3*7*11=231 numbers can't end in a {1,3,7,9} => it cannot end in any number. Therefore, the upper limit to primes of this form is 10**231. Whilst computationally we haven't gotten near this limit, I bet that you could probably reduce this down further.

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