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A346645
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Mersenne prime exponents p which are twin primes, so p-2 and/or p+2 is prime.
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0
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3, 5, 7, 13, 17, 19, 31, 61, 107, 521, 1279, 4423, 110503, 132049, 20996011, 24036583
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OFFSET
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1,1
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COMMENTS
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When p is 5, both p-2 and p+2 are prime. No other number can have this property.
74207281 is a term, but its position in the sequence is not known, although it is highly likely to be the next term. As of July 26, 2021, the GIMPS project has tested all exponents below 104044933 at least once for Mersenne prime numbers and found no exponents > 82589933. However, only exponents below 56420383 have been verified.
As of July 26, 2021, the sequence of Mersenne prime exponents with a twin prime above the exponent (A087858) contains 5 terms, whereas the sequence of Mersenne prime exponents with a twin prime below the exponent (A297674) contains 12 terms, which is 2.4x as many as the former sequence. This is an empirical observation. It is uncertain whether either sequence will increase in length, but given that it is conjectured that there are infinitely many twin primes, and also conjectured there are an infinite number of Mersenne primes, both sequences may increase in length, in which case this sequence might too.
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LINKS
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EXAMPLE
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p 2^p-1 p-2 p+2
-- ---------- --- ----
3 7 1 5*
5 31 3* 7*
7 127 5* 9
13 8191 11* 15
17 131071 15 19*
19 524287 17* 21
31 2147483647 29* 33
Integers in columns 1 and 2 are always prime.
Primes in columns 3 and 4 are marked by an asterisk.
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MATHEMATICA
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twinPrimeQ[n_]:=PrimeQ[n] && (PrimeQ[n-2] || PrimeQ[n+2])
Table[MersennePrimeExponent[a], {a, 47}] //Select[twinPrimeQ]
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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