A002223 Smallest prime p of form p = 8k-1 such that first n primes (p_1=2, ..., p_n) are quadratic residues mod p. (Formerly M4382 N1843)

7, 23, 71, 311, 479, 1559, 5711, 10559, 18191, 31391, 366791, 366791, 366791, 4080359, 12537719, 30706079, 36415991, 82636319, 120293879, 120293879, 131486759, 131486759, 2929911599, 2929911599, 7979490791, 33857579279, 33857579279, 89206899239, 121560956039

Offset

1,1

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. XV.

Links

Jinyuan Wang, Table of n, a(n) for n = 1..57

N. D. Bronson and D. A. Buell, Congruential sieves on FPGA computers, pp. 547-551, in Mathematics of Computation 1943-1993 (Vancouver, 1993), Proc. Symp. Appl. Math., Vol. 48, Amer. Math. Soc. 1994.

D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451. [Annotated scanned copy].

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

12^2 = 2 mod 71, 28^2 = 3 mod 71, 17^2 = 5 mod 71.

Mathematica

np[] := While[p = NextPrime[p]; Mod[p, 8] != 7]; p = 2; A002223 = {}; pp = {2}; np[]; While[ Length[A002223] < 26, If[Union[ JacobiSymbol[#, p] &[pp]] === {1}, AppendTo[pp, NextPrime[Last[pp]]]; Print[p]; AppendTo[A002223, p], np[]]]; A002223 (* Jean-François Alcover, Sep 09 2011 *)

Crossrefs

Cf. A001986, A001992 (the nonsquare versions).

Cf. A002224, A002225, A045535.

Sequence in context: A045535 A001984 A147972 * A034563 A242496 A356684

Adjacent sequences: A002220 A002221 A002222 * A002224 A002225 A002226

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

The Bronson-Buell reference gives terms through prime(n) = 227.

More terms from Don Reble, Sep 19 2001

Status

approved