A344174 Number of primes p of the form x^4 + y^2 with y + 1 prime such that p is smaller than prime(n) and also a quadratic nonresidue modulo prime(n).
0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 3, 1, 2, 2, 3, 2, 1, 2, 2, 2, 1, 3, 1, 3, 4, 4, 4, 3, 4, 2, 4, 3, 1, 3, 5, 4, 5, 3, 3, 3, 3, 3, 5, 4, 2, 4, 2, 3, 3, 3, 3, 5, 4, 5, 4, 8, 3, 4, 4, 2, 6, 5, 4, 6, 5, 8, 3, 4, 3, 5, 3, 3, 6, 4, 6, 4, 4, 4, 3, 5, 4, 7, 6, 3, 6, 5, 4, 7, 4, 5, 2, 4, 6, 2, 4, 6, 7
Offset: 1
Keywords
Examples
a(2) = 1, and the prime 1^4 + (2-1)^2 = 2 is a quadratic nonresidue modulo prime(2) = 3. a(4) = 1, and the prime 1^4 + (3-1)^2 = 5 is a quadratic nonresidue modulo prime(4) = 7. a(24) = 1, and the prime 1^4 + (7-1)^2 = 37 is a quadratic nonresidue modulo prime(24) = 89. a(36) = 1, and the prime 1^4 + (11-1)^2 = 101 is a quadratic nonresidue modulo prime(36) = 151. a(204) = 5, and the primes 3^4 + (11-1)^2 = 181, 3^4 + (17-1)^2 = 337, 5^4 + (5-1)^2 = 641, 5^4 + (17-1)^2 = 881 and 5^4 + (23-1)^2 = 1109 are all quadratic nonresidues modulo prime(204) = 1249.
Links
- J. Friedlander and H. Iwaniec, The polynomial x^2 + y^4 captures its primes, arXiv:math/9811185 [math.NT], 1998; Ann. of Math. 148 (1998), 945-1040.
Programs
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Mathematica
tab={0};Do[p:=p=Prime[n];tt={};Do[If[PrimeQ[b+1]&&PrimeQ[a^4+b^2]&&JacobiSymbol[a^4+b^2,p]==-1,tt=Append[tt,a^4+b^2]],{a,1,(p-1)^(1/4)},{b,1,(p-1-a^4)^(1/2)}];tab=Append[tab,Length[Union[tt]]],{n,2,100}];Print[tab]
Comments