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A065904
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Integers i > 1 for which there is one prime p such that i is a solution mod p of x^4 = 2.
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5
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2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 19, 20, 21, 22, 23, 24, 26, 29, 31, 32, 37, 38, 39, 40, 41, 42, 43, 44, 49, 50, 52, 53, 54, 59, 60, 61, 62, 64, 65, 70, 72, 73, 74, 75, 77, 79, 80, 82, 83, 85, 87, 89, 93, 94, 95, 96, 97, 99, 100, 101, 103, 108, 109, 111, 116, 119, 121
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OFFSET
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1,1
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COMMENTS
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Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for i > 1: There is a prime p such that i is a solution mod p of x^4 = 2 iff i^4 - 2 has a prime factor > i; i is a solution mod p of x^4 = 2 iff p is a prime factor of i^4 - 2 and p > i. i^4 - 2 has at most three prime factors > i. For i such that i^4 - 2 has no resp. two resp. three prime factors > i; cf. A065903 resp. A065905 resp. A065906.
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LINKS
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FORMULA
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a(n) = n-th integer i such that i^4 - 2 has one prime factor > i.
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EXAMPLE
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a(3) = 4, since 4 is (after 2 and 3) the third integer i for which there is one prime p > i (viz. 127) such that i is a solution mod p of x^4 = 2, or equivalently, 4^4 - 2 = 254 = 2*127 has one prime factor > 4 (cf. A065902).
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MAPLE
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filter:= n -> nops(select(`>`, numtheory:-factorset(n^4-2), n))=1:
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MATHEMATICA
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okQ[n_] := Length[Select[FactorInteger[n^4 - 2][[All, 1]], # > n&]] == 1;
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PROG
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(PARI): a065904(m) = local(c, n, f, a, s, j); c = 0; n = 2; while(c<m, f = factor(n^4-2); a = matsize(f)[1]; s = []; for(j = 1, a, if(f[j, 1]>n, s = concat(s, f[j, 1]))); if(matsize(s)[2] == 1, print1(n, ", "); c++); n++) a065904(70)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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