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Search: id:A040028
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| A040028 |
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Primes p such that x^3 = 2 has a solution mod p. |
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+0
23
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| 2, 3, 5, 11, 17, 23, 29, 31, 41, 43, 47, 53, 59, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 149, 157, 167, 173, 179, 191, 197, 223, 227, 229, 233, 239, 251, 257, 263, 269, 277, 281, 283, 293, 307, 311, 317, 347, 353, 359, 383, 389, 397, 401, 419, 431, 433
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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This is the union of {3}, A003627 (primes congruent to 2 mod 3) and A014572 (primes of the form x^2+27y^2). By Thm. 4.15 of [Cox], p is of the form x^2+27y^2 if and only if p is congruent to 1 mod 3 and 2 is a cubic residue mod p. If p is not congruent to 1 mod 3, then every number is a cubic residue mod p, including 2. - Andrew V. Sutherland (drew(AT)math.mit.edu), Apr 26 2008
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REFERENCES
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David A. Cox, "Primes of the Form x^2+ny^2", 1998, John Wiley & Sons.
Kenneth Ireland and Michael Rosen, "A Classical Introduction to Modern Number Theory", second ed., 1990, Springer-Verlag.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
Index entries for related sequences
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MATHEMATICA
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f[p_] := Block[{k = 2}, While[k < p && Mod[k^3, p] != 2, k++ ]; If[k == p, 0, 1]]; Select[ Prime[ Range[100]], f[ # ] == 1 &] (from Robert G. Wilson v Jul 26)
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PROGRAM
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(MAGMA) [ p: p in PrimesUpTo(433) | exists(t){x : x in ResidueClassRing(p) | x^3 eq 2} ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Dec 02 2008]
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CROSSREFS
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Cf. A001132. Number of primes p < 10^n for which 2 is a cubic residue (mod p) is in A097142.
Cf. A003627, A014572.
Sequence in context: A079545 A154755 A040095 this_sequence A049589 A049583 A049596
Adjacent sequences: A040025 A040026 A040027 this_sequence A040029 A040030 A040031
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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