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A230261 Number of ways to write 2*n - 1 = p + q with p, p + 6 and q^4 + 1 all prime, where q is a positive integer. 3

%I #15 Oct 15 2013 03:11:10

%S 0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,2,4,3,3,4,4,3,3,4,1,5,4,3,5,5,5,4,6,4,

%T 5,5,3,3,5,4,4,2,6,8,5,4,6,7,5,5,7,6,5,7,4,6,6,3,6,5,7,6,4,6,7,6,2,7,

%U 6,2,5,5,3,7,7,5,7,9,6,7,4,6,6,4,3,9,7,4,9,9,6,5,10,8,5,9,6,7,8,4

%N Number of ways to write 2*n - 1 = p + q with p, p + 6 and q^4 + 1 all prime, where q is a positive integer.

%C Conjecture: (i) a(n) > 0 for all n > 3. Also, any odd number greater than 6 can be written as p + q (q > 0) with p, p + 6 and q^2 + 1 all prime.

%C (ii) Any integer n > 1 can be written as x + y (x, y > 0) with x^4 + 1 and y^2 + y + 1 both prime.

%C (iii) Each integer n > 2 can be expressed as x + y (x, y > 0) with 4*x^2 + 3 and 4*y^2 -3 both prime.

%C Either of parts (i) and (ii) implies that there are infinitely many primes of the form x^4 + 1.

%H Zhi-Wei Sun, <a href="/A230261/b230261.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588.

%e a(6) = 2 since 2*6-1 = 5 + 6 = 7 + 4, and 5, 5+6 = 11, 7, 7+6 = 13, 6^4+1 = 1297 and 4^4+1 = 257 are all prime.

%e a(25) = 1 since 2*25-1 = 47 + 2, and 47, 47+6 = 53, 2^4+1 = 17 are all prime.

%t a[n_]:=Sum[If[PrimeQ[Prime[i]+6]&&PrimeQ[(2n-1-Prime[i])^4+1],1,0],{i,1,PrimePi[2n-2]}]

%t Table[a[n],{n,1,100}]

%o (PARI) a(n)=my(s,p=5,q=7);forprime(r=11,2*n+4,if(r-p==6&&isprime((2*n-1-p)^4+1),s++); if(r-q==6&&isprime((2*n-1-q)^4+1),s++); p=q;q=r);s \\ _Charles R Greathouse IV_, Oct 14 2013

%Y Cf. A000068, A023201, A037896, A219864, A227908, A227909, A230241, A230252, A230254.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Oct 14 2013

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