A059914 -id:A059914 - cp's OEIS frontend

A014752 Primes of the form x^2 + 27y^2.

31, 43, 109, 127, 157, 223, 229, 277, 283, 307, 397, 433, 439, 457, 499, 601, 643, 691, 727, 733, 739, 811, 919, 997, 1021, 1051, 1069, 1093, 1327, 1399, 1423, 1459, 1471, 1579, 1597, 1627, 1657, 1699, 1723, 1753, 1777, 1789, 1801, 1831, 1933, 1999, 2017

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Primes p == 1 (mod 3) such that 2 is a cubic residue mod p.
Primes p == 1 (mod 6) such that 2 and -2 are both cubes (one implies the other) mod p. - Warren D. Smith
Subsequence of A040028, complement of A045309 relative to A040028. For p in this sequence, x^3 == 2 (mod p) has three solutions in integers from 0 to p-1, whose sum is p (A059899) or 2*p (A059914). The solutions are given in A060122, A060123 and A060124. - Klaus Brockhaus, Mar 02 2001
Primes p = 3m+1 such that 2^m == 1 (mod p). Subsequence of A016108 which also includes composites satisfying this congruence. - Alzhekeyev Ascar M, Feb 22 2012

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1982, Prop. 9.6.2, p. 119.

N. J. A. Sloane and T. D. Noe, Table of n, a(n) for n = 1..17753 (The first 1000 terms were computed by T. D. Noe)
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Bishnu Paudel and Chris Pinner, The integer group determinants for the abelian groups of order 18, arXiv:2412.10638 [math.NT], 2024. See p. 3.
Zak Seidov, Corresponding values of x and y
Bram van Asch, On the structure of the ring Z[2^(1/3)], Internat. J. Pure Appl. Math., 16 (No. 2, 2004), 243-251. See Prop. 7.

Cf. A040028, A045309, A059899, A059914, A060122, A060123, A060124, A014753.

[p: p in PrimesUpTo(2500) | NormEquation(27, p) eq true]; // Vincenzo Librandi, Jul 24 2016

With[{nn=50},Take[Select[Union[First[#]^2+27Last[#]^2&/@Tuples[Range[ nn], 2]],PrimeQ],nn]] (* Harvey P. Dale, Jul 28 2014 *)
nn = 1398781;re = Sort[Reap[Do[Do[If[PrimeQ[p = x^2 + 27*y^2], Sow[{p, x, y}]], {x, Sqrt[nn - 27*y^2]}], {y, Sqrt[nn/27]}]][[2, 1]]]; (* For all 17753 values of a(n), x(n) and y(n). - Zak Seidov, May 20 2016 *)

{ fc(a,b,c,M) = my(p,t1,t2,n); t1 = listcreate();
for(n=1,M, p = prime(n);
t2 = qfbsolve(Qfb(a,b,c),p); if(t2 == 0,, listput(t1,p)));
print(t1);
}
fc(1,0,27,1000);
\\ N. J. A. Sloane, Jun 06 2014

list(lim)=my(v=List()); forprimestep(p=31,lim,6, if(Mod(2,p)^(p\3)==1, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Apr 06 2022

a(n) ~ 6n log n by the Landau prime ideal theorem. - Charles R Greathouse IV, Apr 06 2022

Definition provided by T. D. Noe, May 08 2005
Entry revised by Michael Somos and N. J. A. Sloane, Jul 28 2006
Defective Mma program replaced with PARI program, b-file recomputed and extended by N. J. A. Sloane, Jun 06 2014

A001133 Primes p such that the multiplicative order of 2 modulo p is (p-1)/3.

Original entry on oeis.org

43, 109, 157, 229, 277, 283, 307, 499, 643, 691, 733, 739, 811, 997, 1021, 1051, 1069, 1093, 1459, 1579, 1597, 1627, 1699, 1723, 1789, 1933, 2179, 2203, 2251, 2341, 2347, 2749, 2917, 3163, 3181, 3229, 3259, 3373, 4027, 4339, 4549, 4597, 4651, 4909, 5101, 5197, 5323, 5413, 5437, 5653, 6037

Offset: 1

N. J. A. Sloane

nonn

M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 59.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A040028, A014752, A059914.

Cf. also A001134, A001135, A001136, A115586, A115591.

[ p: p in PrimesUpTo(4597) | r eq 1 and Order(R!2) eq q where q,r is Quotrem(p,3) where R is ResidueClassRing(p) ]; // Klaus Brockhaus, Dec 02 2008

Reap[For[p = 2, p < 10^4, p = NextPrime[p], If[MultiplicativeOrder[2, p] == (p-1)/3, Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 10 2015 *)

forprime(p=3,10^4,if(znorder(Mod(2,p))==(p-1)/3,print1(p,", "))); \\ Joerg Arndt, May 17 2013

More terms and better definition from Don Reble, Mar 11 2006

A059899 Primes p such that x^3 = 2 has more than one solution mod p and the sum of the (three) solutions is p.

Original entry on oeis.org

31, 229, 397, 439, 457, 499, 601, 643, 691, 727, 739, 811, 919, 997, 1021, 1051, 1093, 1327, 1459, 1657, 1699, 1753, 1933, 1999, 2113, 2179, 2203, 2251, 2281, 2341, 2347, 2383, 2671, 2731, 2767, 2791, 2833, 2953, 2971, 3061, 3229, 3259, 3331, 3373, 3391

Offset: 1

Klaus Brockhaus, Mar 02 2001

nonn

Subsequence of A040028 and of A014752, complement of A059914 relative to A014752. Solutions mod p are represented by integers from 0 to p-1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A040028, A014752, A059914.

filter:= proc(p) local S;
  if not isprime(p) then return false fi;
  S:= map(t -> rhs(t[1]), [msolve(x^3=2,p)]);
  nops(S) = 3 and convert(S,`+`) = p
end proc:
select(filter, [seq(i,i=7..5000, 6)]); # Robert Israel, Aug 13 2024

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A014752 Primes of the form x^2 + 27y^2.

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A001133 Primes p such that the multiplicative order of 2 modulo p is (p-1)/3.

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Magma

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PARI

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A059899 Primes p such that x^3 = 2 has more than one solution mod p and the sum of the (three) solutions is p.

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Author

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Maple