Search: id:A040028
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%I A040028
%S A040028 2,3,5,11,17,23,29,31,41,43,47,53,59,71,83,89,101,107,109,113,127,131,
%T A040028 137,149,157,167,173,179,191,197,223,227,229,233,239,251,257,263,269,
%U A040028 277,281,283,293,307,311,317,347,353,359,383,389,397,401,419,431,433
%N A040028 Primes p such that x^3 = 2 has a solution mod p.
%C A040028 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
%D A040028 David A. Cox, "Primes of the Form x^2+ny^2", 1998, John Wiley & Sons.
%D A040028 Kenneth Ireland and Michael Rosen, "A Classical Introduction to Modern 
               Number Theory", second ed., 1990, Springer-Verlag.
%H A040028 T. D. Noe, <a href="b040028.txt">Table of n, a(n) for n=1..1000</a>
%H A040028 <a href="Sindx_Pri.html#smp">Index entries for related sequences</a>
%t A040028 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)
%o A040028 (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]
%Y A040028 Cf. A001132. Number of primes p < 10^n for which 2 is a cubic residue 
               (mod p) is in A097142.
%Y A040028 Cf. A003627, A014572.
%Y A040028 Sequence in context: A079545 A154755 A040095 this_sequence A049589 A049583 
               A049596
%Y A040028 Adjacent sequences: A040025 A040026 A040027 this_sequence A040029 A040030 
               A040031
%K A040028 nonn
%O A040028 1,1
%A A040028 N. J. A. Sloane (njas(AT)research.att.com).

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