A095978 Number of solutions to y^2=x^3+x (mod p) as p runs through the primes.

2, 3, 3, 7, 11, 19, 15, 19, 23, 19, 31, 35, 31, 43, 47, 67, 59, 51, 67, 71, 79, 79, 83, 79, 79, 99, 103, 107, 115, 127, 127, 131, 159, 139, 163, 151, 179, 163, 167, 147, 179, 163, 191, 207, 195, 199, 211, 223, 227, 259, 207, 239, 271, 251, 255, 263

Offset

1,1

Comments

The only rational solution of y^2 = x^3 + x is (y, x) = (0, 0). See the Silverman reference, Theorem 44.1 with a proof on pp. 388 - 390 (in the 4th edition, 2014, Theorem 1, pp. 354 - 356). - Wolfdieter Lang, Feb 08 2016

This is also the number of solutions to y^2 = x^3 - 4*x (mod p) as p runs through the primes. - Seiichi Manyama, Sep 16 2016

References

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, Theorem 45.1 on p. 399. In the 4th edition, 2014, Theorem 1 on p. 365.

Links

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

Formula

a(1) = 2; if prime(n) == 3 (mod 4) then a(n) = prime(n); if prime(n) = A002144(m) then if A002972(m) == 1 (mod 4) then a(n) = prime(n) - 2*A002972(m), otherwise a(n) = prime(n) + 2*A002972(m).

Example

n = 21: prime(21) = A000040(21) = 73 = A002144(9)  == 1 (mod 4), A002972(9) = 3 == 3 (mod 4) (not 1 (mod 4)), a(n) = 73 + 2*3 = 79.
n = 22: prime(22) = A000040(22) = 79 == 3 (mod 4), a(n) = prime(22) = 79.

Maple

a:= proc(n)
  local p, xy, x;
  p:= ithprime(n);
  if p mod 4 = 3 then return p fi;
  xy:= [Re, Im](GaussInt:-GIfactors(p)[2][1][1]);
  x:= op(select(type, xy, odd));
  if x mod 4 = 1 then p - 2*x else p + 2*x fi
end proc:
a(1):= 2:
map(a, [$1..100]); # Robert Israel, Feb 09 2016

Mathematica

a[n_] := Module[{p, xy, x}, p = Prime[n]; If[Mod[p, 4]==3, Return[p]]; xy = {Re[#], Im[#]}& @ FactorInteger[p, GaussianIntegers -> True][[2, 1]]; x = SelectFirst[xy, OddQ]; If[Mod[x, 4]==1, p - 2*x, p + 2*x]]; a[1] = 2; Array[a, 100] (* Jean-François Alcover, Feb 26 2016, after Robert Israel*)

Crossrefs

Cf. A000040, A002144, A002972, A267859.

Sequence in context: A080088 A098715 A167886 * A370361 A156763 A169653

Adjacent sequences: A095975 A095976 A095977 * A095979 A095980 A095981

Keyword

nonn

Author

Lekraj Beedassy, Jul 16 2004

Extensions

Edited. Update of reference, formula corrected, examples given, and a(21) - a(56) from Wolfdieter Lang, Feb 06 2016

Status

approved