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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-François 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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