A262340 Primes of the form p^2 + b^4 where p is a prime.

5, 41, 137, 281, 617, 857, 977, 1097, 1217, 1321, 1657, 1697, 2137, 4217, 4457, 4937, 5297, 6257, 6337, 7537, 7577, 7817, 7937, 9137, 10009, 10169, 10289, 10337, 10457, 10529, 11369, 11497, 11681, 11897, 12809, 13177, 13721, 14489, 15329, 16889, 17417

Offset: 1

Author

Jonathan Sondow, Oct 03 2015

Keywords

• nonn

Comments

It is not known whether there are infinitely many primes of such form.

Same as the intersection of A185086 (primes of the form p^2 + k^2 where p is a prime) with A028916 (primes of the form a^2 + b^4). (Proof: Clearly, p^2 + b^4 is in A185086 and in A028916. Conversely, if a(n) = p^2 + k^2 = a^2 + b^4, then by the uniqueness part of Fermat's two squares (or 4n+1) theorem, (p,k) = (a,b^2) or (p,k) = (b^2,a). But the latter is impossible since p is prime, so a(n) = p^2 + b^4.)

Examples

			5 = 2^2 + 1^4, so a(1) = 5.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Art of Problem Solving, Fermat's Two Squares Theorem
• MathWorld, Fermat's 4n+1 Theorem

Crossrefs

Cf. A028916, A185086.

Programs

Mathematica

nn = 14; Union[ Flatten[ Table[ Select[ Prime[n]^2 + Range[nn]^4, PrimeQ[#] && # < nn^4 &], {n,PrimePi[nn^2]}]]]

PARI

list(lim)=my(v=List(),p2,t); forprime(p=2,sqrtint(lim\=1), p2=p^2; forstep(x=1+p%2,sqrtnint(lim-p2,4),2, if(isprime(t=p2+x^4), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 21 2017