A002225 a(n) is the smallest prime p such that each of the first n primes has three cube roots mod p. (Formerly M5224 N2274)

31, 307, 643, 5113, 21787, 39199, 360007, 360007, 4775569, 10318249, 10318249, 65139031, 387453811, 913900417, 2278522747, 2741702809, 25147657981, 118748663779, 156776294593, 747206701687, 1151810360731, 1151810360731, 1151810360731

Offset

1,1

Comments

a(n) is the smallest prime p == 1 (mod 3) such that each of the first n primes is a cubic residue mod p. - Robert Israel, Aug 02 2016

References

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).

A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XVI.

Links

Table of n, a(n) for n=1..23.

A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots, Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968 [Annotated scans of selected pages]

Example

For n = 2, the first two primes 2 and 3 each have three cube roots mod 307: 2 has cube roots 52, 270, 292 and 3 has cube roots 192, 194, 228. - Robert Israel, Aug 02 2016

Maple

Primes:= [2]: pp:= 7:
for n from 1 to 12 do
  for p from pp by 6 while
    not(isprime(p) and andmap(t -> t &^ ((p-1)/3) mod p = 1, Primes))
  do od:
  A[n]:= p;
  pp:= p;
  Primes:= [op(Primes), nextprime(Primes[-1])];
od:
seq(A[i], i=1..12); # Robert Israel, Aug 02 2016

Mathematica

(* This naive program being very slow, limit is set to 8 terms *) lim=8; np[] := While[p=NextPrime[p]; Mod[p, 3]!=1]; crQ[n_, p_] := Reduce[ 0<x<p && Mod[x^3, p]==n, x, Integers]=!=False; p=2; pp={p}; A002225={}; While[Length[A002225] < lim, If[And @@ (crQ[#, p]& /@ pp), AppendTo[pp, NextPrime[ Last[pp]]]; Print[p]; AppendTo[A002225, p], np[] ] ]; A002225 (* Jean-François Alcover, Sep 09 2011 *)

Crossrefs

Smallest prime p such that each of the first n primes has q q-th roots mod p: A147972 (q=2), this sequence (q=3), A002226 (q=5), A002227 (q=7), A002228 (q=11), A060363 (q=13), A060364 (q=17).

Cf. A002223, A002224.

Subset of A014752. Except for a(1), subset of A014753. Except for a(1) and a(2), subset of A040044.

Sequence in context: A141912 A147297 A295975 * A124335 A368953 A368946

Adjacent sequences: A002222 A002223 A002224 * A002226 A002227 A002228

Keyword

nonn,nice,more

Author

N. J. A. Sloane

Extensions

More terms from Don Reble, Oct 09 2001

Name corrected by Robert Israel, Aug 02 2016

a(18)-a(23) from Sergey Paramonov, Apr 11 2024

Status

approved