OFFSET
1,6
COMMENTS
In 1998 J. Friedlander and H. Iwaniec proved that there are infinitely many primes of the form x^4 + y^2 with x and y integers.
Conjecture: (i) a(n) > 0 for all n > 1. In other words, for each odd prime p, there is a prime q < p of the form x^4 + y^2 with y + 1 prime such that q is a quadratic nonresidue modulo p.
(ii) For any odd prime p not among 3, 5, 13, 37, 277, there is a prime q < p of the form x^4 + y^2 with y + 1 prime such that q is a quadratic residue modulo p.
Part (i) of the conjecture verified for all odd primes p < 2*10^9.
We even conjecture further that for any prime p > 5 there is a prime q < p of the form x^4 + y^2 with y + 1 prime such that q is a primitive root modulo p.
See also A344173 for a similar conjecture.
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.
EXAMPLE
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.
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 10 2021
STATUS
approved