Problems & Puzzles: Puzzles

Puzzle 1222 A second puzle about twin primes As a follow up to Puzzle 1221, now we ask for primes of the type: p!+q, other than 11 & 127, where p & q are twin primes. Q. Send all the examples alike you can find

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From May 24 to 30, 2025, contributions came from Michael Branicky, Simon Cavegn, Paul Cleary,

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Michael wrote:

The more general OEIS A084749 (Numbers m such that m! + p is a prime, where p is the smallest prime > m) has the data:
        0, 1, 2, 3, 4, 5, 6, 7, 10, 33, 44, 48, 52, 64, 73, 92, 119, 182, 487, 603, 987, 4884, 6822, 8070, 11079

The highest prime term is 487 and the highest twin prime term is 5.

Thus, any further twin-prime examples would have p > 11079 (which was found by yours truly).

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I checked all twin primes <= 65003, finding no more pairs.

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Simon wrote:

No more results found.
Searched up to the twin: 69827! + 69829 (has 307921 digits)


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Paul wrote:

I wasn't able to find another example using twin primes, searching up to the prime 20543.  There were however a few more solutions if we allowed consecutive primes, these occurred at:-

2 and 3, prime 5

7 and 11, with  prime 5051

73 and 79 with a prime of length 1100  and the last one

17659 and 17669 with a prime of length 67331

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