OFFSET
1,5
COMMENTS
Conjecture: a(n) > 0 except for n = 1, 16, 292.
This implies not only Goldbach's conjecture for even numbers, but also Ming-Zhi Zhang's conjecture (cf. A036468) that any odd number greater than one can be written as x + y (x, y > 0) with x^2 + y^2 prime.
We have verified the conjecture for n up to 10^7.
Conjecture verified for n up tp 10^9. - Mauro Fiorentini, Sep 21 2023
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.
EXAMPLE
a(7) = 1 since 2*7 = 11 + 3, and (11-1)^2 + 3^2 = 109 is prime.
a(19) = 1 since 2*19 = 7 + 31, and (7-1)^2 + 31^2 = 997 is prime.
MATHEMATICA
a[n_]:=Sum[If[PrimeQ[2n-Prime[i]]&&PrimeQ[(Prime[i]-1)^2+(2n-Prime[i])^2], 1, 0], {i, 1, PrimePi[2n-2]}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 12 2013
STATUS
approved