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Search: id:A059772
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| A059772 |
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Smallest prime p such that n is a solution mod p of x^2 = 2, or 0 if no such prime exists. |
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+0
4
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| 0, 7, 7, 23, 17, 47, 31, 79, 0, 17, 71, 167, 97, 223, 127, 41, 23, 359, 199, 439, 241, 31, 41, 89, 337, 727, 0, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 0, 47, 73, 881, 1847, 967, 0, 151, 2207, 1151, 2399, 1249, 113, 193, 401, 0, 3023, 1567, 191, 0, 71
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OFFSET
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2,2
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COMMENT
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Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for n > 1: There is a prime p such that n is a solution mod p of x^2 = 2 iff n^2-2 has a prime factor > n; n is a solution mod p of x^2 = 2 iff p is a prime factor of n^2-2 and p > n. n^2-2 has at most one prime factor > n, consequently such a factor is the only prime p such that n is a solution mod p of x^2 = 2. For n such that n^2-2 has no prime factor > n (the zeros in the sequence), cf. A060515.
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FORMULA
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If n^2-2 has a (unique) prime factor p > n, then a(n) = p, else a(n) = 0.
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EXAMPLE
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a(11) = 17, since 11 is a solution mod 17 of x^2 = 2 and 11 is not a solution mod p of x^2 = 2 for primes p < 17. Although 11^2 = 2 mod 7, prime 7 is excluded because 7 < 11 and 11 = 4 mod 7.
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CROSSREFS
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Cf. A038873, A059770, A059771, A060515.
Sequence in context: A146466 A146141 A062368 this_sequence A114369 A027844 A111217
Adjacent sequences: A059769 A059770 A059771 this_sequence A059773 A059774 A059775
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KEYWORD
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nonn
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AUTHOR
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Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Feb 21 2001
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EXTENSIONS
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Offset corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 21 2009
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