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

Author

Lekraj Beedassy, Jul 16 2004

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

Cf. A000040, A002144, A002972, A267859.

Programs

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*)

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).

Extensions

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