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

1,6

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

Table of n, a(n) for n=1..100.

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

Cf. A000040, A000290, A000583, A028916, A281792, A344173.

Sequence in context: A338912 A086598 A211261 * A336431 A074746 A133188

Adjacent sequences: A344171 A344172 A344173 * A344175 A344176 A344177

Keyword

nonn

Author

Zhi-Wei Sun, May 10 2021

Status

approved