OFFSET
1,9
COMMENTS
Conjecture: a(n)>0 for all n>8.
This conjecture is stronger than Lemoine's conjecture. It has been verified for n up to 10^8.
Conjecture verified for 2n-1 up to 10^9. - Mauro Fiorentini, Jul 20 2023
Zhi-Wei Sun also made the following general conjecture: For any positive integer n, the set E(n) of positive odd integers not of the form p+2q with p, q, p^2+4(2^n-1)q^2 all prime, is finite. In particular, if we let M(n) denote the maximal element of E(n), then M(1)=3449, M(2)=1711, E(3)={1,3,5,7,31,73}, E(4)={1,3,5,7,9,11,13,15},
M(5)=6227, M(6)=1051, M(7)=2239, M(8)=2599, M(9)=7723,
M(10)=781, M(11)=1163, M(12)=587, M(13)=11443,
M(14)=2279, M(15)=157, M(16)=587, M(17)=32041,
M(18)=1051, M(19)=2083, M(20)=4681.
Conjecture verified for 2n-1 up to 10^9 for n <= 4 and up to 10^6 for n <= 20. - Mauro Fiorentini, Jul 20 2023
Zhi-Wei Sun also guessed that for any positive even integer d not congruent to 2 modulo 6 there exists a prime p(d) such that for any prime p>p(d) there is a prime q<p with p^2+dq^2 prime. In particular, we may take
p(4)=p(6)=3, p(10)=5, p(12)=3, p(16)=2, p(18)=3,
p(22)=11, p(24)=17, and p(28)=p(30)=7.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588 [math.NT], 2012-2017.
Wikipedia, Lemoine's conjecture
EXAMPLE
a(10)=1 since the only primes p and q with p^2+60q^2 prime and p+2q=19 are p=13 and q=3.
MATHEMATICA
a[n_]:=a[n]=Sum[If[PrimeQ[q]==True&&PrimeQ[2n-1-2q]==True&&PrimeQ[(2n-1-2q)^2+60q^2]==True, 1, 0], {q, 1, n-1}]
Do[Print[n, " ", a[n]], {n, 1, 20000}]
PROG
(PARI) A218825(n)={my(c=0, n21=n*2-1); forprime(q=2, n-1, isprime(n21-2*q) || next; isprime(q^2*60+(n21-2*q)^2) && c++); c} \\ M. F. Hasler, Nov 07 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 07 2012
STATUS
approved