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December 9

If the Mersenne number 2^p-1 is prime, then must it be the smallest Mersenne prime == 1 mod p?

If the Mersenne number 2^p-1 is prime, then must it be the smallest Mersenne prime == 1 mod p? (i.e. there is no prime q < p such that 2^q-1 is also a Mersenne prime == 1 mod p) If p is prime (no matter 2^p-1 is prime or not), 2^p-1 is always == 1 mod p. However, there are primes p such that there is a prime q < p such that 2^q-1 is also a Mersenne prime == 1 mod p:

2^19-1 is Mersenne prime == 1 mod 73 (p=73, q=19)
2^31-1 is Mersenne prime == 1 mod 151 (p=151, q=31)
2^61-1 is Mersenne prime == 1 mod 151 (p=151, q=61)
2^17-1 is Mersenne prime == 1 mod 257 (p=257, q=17)
2^31-1 is Mersenne prime == 1 mod 331 (p=331, q=31)
2^61-1 is Mersenne prime == 1 mod 331 (p=331, q=61)
2^127-1 is Mersenne prime == 1 mod 337 (p=337, q=127)
2^89-1 is Mersenne prime == 1 mod 353 (p=353, q=89)
2^89-1 is Mersenne prime == 1 mod 397 (p=397, q=89)

but for these primes p, 2^p-1 is not prime, and my question is: Is there a prime p such that 2^p-1 is a prime and there is a prime q < p such that 2^q-1 is also a Mersenne prime == 1 mod p?

If 2^11-1 is prime, then this is true, since 2^11-1 is == 1 mod 31 and 2^31-1 is prime, but 2^11-1 is not prime
If 2^23-1 or 2^67-1 is prime, then this is true, since 2^23-1 and 2^67-1 are == 1 mod 89 and 2^89-1 is prime, but 2^23-1 and 2^67-1 are not primes
If 2^29-1 or 2^43-1 or 2^71-1 or 2^113-1 is prime, then this is true, since 2^29-1 and 2^43-1 and 2^71-1 and 2^113-1 are == 1 mod 127 and 2^127-1 is prime, but 2^29-1 and 2^43-1 and 2^71-1 and 2^113-1 are not primes
If 2^191-1 or 2^571-1 or 2^761-1 or 2^1901-1 is prime, then this is true, since 2^191-1 and 2^571-1 and 2^761-1 and 2^1901-1 are == 1 mod 2281 and 2^2281-1 is prime, but 2^191-1 and 2^571-1 and 2^761-1 and 2^1901-1 are not primes
If 2^1609-1 is prime, then this is true, since 2^1609-1 is == 1 mod 3217 and 2^3217-1 is prime, but 2^1609-1 is not prime

Another question: For any prime p, is there always a Mersenne prime == 1 mod p? 220.132.216.52 (talk) 19:03, 9 December 2024 (UTC)[reply]

Neither question is easy. For the first, relations

2^{q-1}\equiv 1{\pmod {p}}

would imply that the integer 2 is not a primitive root mod p, and that its order divides

q-1

for the prime q. This is a sufficiently infrequent occurrence that it seems likely that all Mersenne numbers could be ruled out statistically, but not enough is known about their distribution. For the second, it is not even known if there are infinitely many Mersenne primes. Tito Omburo (talk) 19:23, 9 December 2024 (UTC)[reply]

I found that: 2^9689-1 is the smallest Mersenne prime == 1 mod 29, 2^44497-1 is the smallest Mersenne prime == 1 mod 37, 2^756839-1 is the smallest Mersenne prime == 1 mod 47, 2^57885161-1 is the smallest Mersenne prime == 1 mod 59, 2^4423-1 is the smallest Mersenne prime == 1 mod 67, 2^9941-1 is the smallest Mersenne prime == 1 mod 71, 2^3217-1 is the smallest Mersenne prime == 1 mod 97, 2^21701-1 is the smallest Mersenne prime == 1 mod 101, and none of the 52 known Mersenne primes are == 1 mod these primes p < 1024: 79, 83, 103, 173, 193, 197, 199, 227, 239, 277, 307, 313, 317, 349, 359, 367, 373, 383, 389, 409, 419, 431, 443, 461, 463, 467, 479, 487, 503, 509, 523, 547, 563, 587, 599, 613, 647, 653, 659, 661, 677, 709, 727, 733, 739, 743, 751, 757, 769, 773, 797, 809, 821, 823, 827, 829, 839, 853, 857, 859, 863, 887, 907, 911, 919, 929, 937, 941, 947, 971, 977, 983, 991, 1013, 1019, 1021 220.132.216.52 (talk) 20:51, 9 December 2024 (UTC)[reply]

Also,

2^19937-1 is Mersenne prime == 1 mod 2^16+1
2^521-1 is Mersenne prime == 1 mod 2^13-1
2^3021377-1 is Mersenne prime == 1 mod 2^17-1
2^2281-1 is Mersenne prime == 1 mod 2^19-1
2^21701-1 is Mersenne prime == 1 mod 2^31-1
2^19937-1 is Mersenne prime == 1 mod 2^89-1
2^86243-1 is Mersenne prime == 1 mod 2^107-1

but none of these primes p has 2^p-1 is known to be prime, the status of 2^(2^89-1)-1 and 2^(2^107-1)-1 are still unknown (see double Mersenne number), but if at least one of them is prime, then will disprove this conjecture (none of the 52 known Mersenne primes are == 1 mod 2^61-1 or 2^127-1), I think that this conjecture may be as hard as the New Mersenne conjecture. 220.132.216.52 (talk) 20:55, 9 December 2024 (UTC)[reply]

Also, for the primes p < 10000, there is a prime q < p such that 2^q-1 is also a Mersenne prime == 1 mod p only for p = 73, 151, 257, 331, 337, 353, 397, 683, 1321, 1613, 2113, 2731, 4289, 4561, 5113, 5419, 6361, 8191, 9649 (this sequence is not in OEIS), however, none of these primes p have 2^p-1 prime. 220.132.216.52 (talk) 02:23, 10 December 2024 (UTC)[reply]