OFFSET
1,9
COMMENTS
Conjecture: a(n) > 0 for all n > 6.
This is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013. It also implies that there are infinitely many primes p with p^2 - 2 also prime.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 22 2023
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.
EXAMPLE
a(10) = 1 since 10 = 2*2 + 3 + 3 with 2, 3, 2^2 - 2 = 2, 3^2 - 2 = 7 all prime.
a(19) = 2 since 19 = 3 + 3 + 13 = 5 + 7 + 7 with 3, 13, 5, 7, 3^2 - 2 = 7, 13^2 - 2 = 167, 5^2 - 2 = 23, 7^2 - 2 = 47 all prime.
MATHEMATICA
pp[n_]:=PrimeQ[n^2-2]
pq[n_]:=PrimeQ[n]&&pp[n]
a[n_]:=Sum[If[pp[Prime[i]]&&pp[Prime[j]]&&pq[n-(2-Mod[n, 2])Prime[i]-Prime[j]], 1, 0], {i, 1, PrimePi[n/(4-Mod[n, 2])]}, {j, i, PrimePi[(n-(2-Mod[n, 2])Prime[i])/2]}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 21 2013
STATUS
approved