A262340 - OEIS

%I #10 Aug 21 2017 02:40:40

%S 5,41,137,281,617,857,977,1097,1217,1321,1657,1697,2137,4217,4457,

%T 4937,5297,6257,6337,7537,7577,7817,7937,9137,10009,10169,10289,10337,

%U 10457,10529,11369,11497,11681,11897,12809,13177,13721,14489,15329,16889,17417

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

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

%C 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.)

%H Charles R Greathouse IV, <a href="/A262340/b262340.txt">Table of n, a(n) for n = 1..10000</a>

%H Art of Problem Solving, <a href="http://www.artofproblemsolving.com/wiki/index.php/Fermat&#39;s_Two_Squares_Theorem">Fermat's Two Squares Theorem</a>

%H MathWorld, <a href="http://mathworld.wolfram.com/Fermats4nPlus1Theorem.html">Fermat's 4n+1 Theorem</a>

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

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

%o (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

%Y Cf. A028916, A185086.

%K nonn

%O 1,1

%A _Jonathan Sondow_, Oct 03 2015