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.

41, 313, 3593, 4481, 32633, 42961, 66361, 67073, 165233, 198593, 237161, 266921, 378953, 462073, 465041, 487073, 559001, 594161, 750353, 757633, 815401, 1157033, 1414081, 1416161, 1687393, 2439881, 2793481, 2866121, 2947561, 3344161, 3577913, 3759713, 4295281, 4617073, 4795481, 5654641

Offset: 1

Author

Thomas Ordowski and Altug Alkan, Feb 23 2017

Keywords

• nonn

Comments

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

Examples

			For prime 41 = 5^2 + 4^2 is 5^2 - 4^2 = 3^2 and 5^4 + 4^4 = 881 is prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A002646.

Cf. A002144, A002645.

Programs

Maple

N:= 10^7: # to get all terms <= N Res:= {}:
for w from 1 to floor((2*N)^(1/4)) by 2 do
  for u from 1 to min(w-1, floor((2*N-w^4)^(1/4))) by 2 do
    p:= (u^4 + w^4)/2;
    if isprime(p) and isprime((u^8 + 6*u^4*w^4 + w^8)/8) then
      Res:= Res union {p}
    fi;
od od:
sort(convert(Res,list)); # Robert Israel, Feb 24 2017

Mathematica

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 *)

Formula

a(n) == 1 (mod 8).

a(n) == 1 or 33 (mod 40).