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A095978 Number of solutions to y^2=x^3+x (mod p) as p runs through the primes. 5
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Feb 26 2016, after Robert Israel*)

CROSSREFS

Cf. A000040, A002144, A002972, A267859.

Sequence in context: A080088 A098715 A167886 * A156763 A169653 A129012

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

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Last modified December 5 13:38 EST 2016. Contains 278768 sequences.