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A173255 Smaller member p of a twin prime pair (p, p+2) such that the sum p+(p+2) is a fifth power: 2*(p+1) = k^5 for some integer k. 8

%I #12 Sep 08 2022 08:45:50

%S 4076863487,641194278911,16260080320511,174339220049999,

%T 420586798122287,388931440807883087,1715002302605720111,

%U 2051821692518399999,4617724356355049999,5873208011345484287,58698987193722272687,76578949263222449999,180701862444484649999,562030251929933709311

%N Smaller member p of a twin prime pair (p, p+2) such that the sum p+(p+2) is a fifth power: 2*(p+1) = k^5 for some integer k.

%C Since k^5 = 2*(p+1) is even, k is also even.

%C The lesser of twin primes p (except for 3) are congruent to -1 modulo 3 (see third comment in A001359), the greater of twin primes p+2 (except for 5) are congruent to 1 modulo 3. Therefore p+1 is a multiple of 3. Since k^5 = 2*(p+1) is a multiple of 3, k is also a multiple of 3. Hence k is divisible by 2 and by 3, i.e. a multiple of 6.

%C The lesser of twin primes except for 3 (A001359) are congruent to 1, 7 or 9 modulo 10; this applies also to the terms of the present sequence, a subsequence of A001359.

%H Amiram Eldar, <a href="/A173255/b173255.txt">Table of n, a(n) for n = 1..10000</a>

%e p = 4076863487 and p+2 form a twin prime pair, their sum 8153726976 = 96^5 is a fifth power. Hence 4076863487 is in the sequence.

%e p = 641194278911 and p+2 form a twin prime pair, their sum 1282388557824 = 264^5 is a fifth power. Hence 641194278911 is in the sequence.

%e p = 388931440807883087 and p+2 form a twin prime pair, their sum 777862881615766176 = 3786^5 is a fifth power. Hence 388931440807883087 is in the sequence.

%e 3786 is the smallest value of k that gives a prime when divided by 6, it corresponds to a(6): 3786 = 6*631 and 631 is prime. The next value of k that gives a prime when divided by 6 is 10326 and corresponds to a(11): 10326 = 6*1721 and 1721 is prime.

%e If p is a term and k^5 the corresponding fifth power, then a fifth-power multiple c^5*k^5 does not necessarily correspond to a term q. The fifth power 96^5 corresponds to a(1), but q = 2^5*96^5/2-1 = 130459631615 = 5*7607*3429989 is not prime, much less is (q, q+2) a twin prime pair.

%e If p is a term and k^5 the corresponding fifth power, and if k^5 is the product c^5*d^5 of two fifth powers where d is even, then d^5 does not necessarily correspond to a term q. The fifth power 3786^5 = 3^5*1262^5 corresponds to a(6), but q = 1262^5/2-1 = 1600540908674415 = 3*5*577*55171*3351883 is not prime, much less is (q, q+2) a twin prime pair.

%t Select[Range[2, 10^5, 2]^5/2 - 1, And@@PrimeQ[# + {0, 2}] &] (* _Amiram Eldar_, Dec 24 2019 *)

%o (Magma) /* gives triples <p, k^5, k> */ [ <p, k^5, k>: k in [2..10500 by 2] | IsPrime(p) and IsPrime(p+2) where p is (k^5 div 2)-1 ];

%Y Cf. A001359, A006512, A014574, A061308, A069496, A119859, A172271, A172494

%K nonn

%O 1,1

%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Feb 14 2010

%E Edited, non-specific references and keywords base, hard removed, MAGMA program added and listed terms verified by the Associate Editors of the OEIS, Feb 26 2010

%E More terms from _Amiram Eldar_, Dec 24 2019

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