cp's OEIS Frontend

A282867 Primes of the form x^2 + y^2 with x > y such that x^2 - y^2 is a square and x^4 + y^4 is a prime.

%I A282867 #24 Jul 23 2024 15:54:10
%S A282867 41,313,3593,4481,32633,42961,66361,67073,165233,198593,237161,266921,
%T A282867 378953,462073,465041,487073,559001,594161,750353,757633,815401,
%U A282867 1157033,1414081,1416161,1687393,2439881,2793481,2866121,2947561,3344161,3577913,3759713,4295281,4617073,4795481,5654641
%N A282867 Primes of the form x^2 + y^2 with x > y such that x^2 - y^2 is a square and x^4 + y^4 is a prime.
%C A282867 Primes of the form (u^4 + v^4)/2 with u and v odd and (u^8 + 6*u^4*v^4 + v^8)/8 prime. - _Robert Israel_, Feb 24 2017
%H A282867 Robert Israel, <a href="/A282867/b282867.txt">Table of n, a(n) for n = 1..10000</a>
%F A282867 a(n) == 1 (mod 8).
%F A282867 a(n) == 1 or 33 (mod 40).
%e A282867 For prime 41 = 5^2 + 4^2 is 5^2 - 4^2 = 3^2 and 5^4 + 4^4 = 881 is prime.
%p A282867 N:= 10^7: # to get all terms <= N Res:= {}:
%p A282867 for w from 1 to floor((2*N)^(1/4)) by 2 do
%p A282867   for u from 1 to min(w-1, floor((2*N-w^4)^(1/4))) by 2 do
%p A282867     p:= (u^4 + w^4)/2;
%p A282867     if isprime(p) and isprime((u^8 + 6*u^4*w^4 + w^8)/8) then
%p A282867       Res:= Res union {p}
%p A282867     fi;
%p A282867 od od:
%p A282867 sort(convert(Res,list)); # _Robert Israel_, Feb 24 2017
%t A282867 Select[Total[#^2]&/@Select[Subsets[Range[3000],{2}],IntegerQ[Sqrt[#[[2]]^2-#[[1]]^2]] && PrimeQ[ Total[#^4]]&],PrimeQ]//Union (* _Harvey P. Dale_, Jul 23 2024 *)
%Y A282867 Subsequence of A002646.
%Y A282867 Cf. A002144, A002645.
%K A282867 nonn
%O A282867 1,1
%A A282867 _Thomas Ordowski_ and _Altug Alkan_, Feb 23 2017