A334802 Positive integers of the form x^4 - y^4 that have exactly 4 divisors.

15, 65, 671, 3439, 12209, 102719, 113521, 178991, 246559, 515201, 1124111, 1342879, 2964961, 3940399, 9951391, 21254449, 27220159, 34209169, 45259649, 48986321, 70710641, 92110289, 93084991, 125620111, 131687681, 144402721, 201792079, 211782751, 276694241

Offset: 1

Author

C. Kenneth Fan, May 12 2020

Keywords

• nonn

Comments

If a(n) = pq, where p > q are both prime, then p is the hypotenuse and q is a leg of a primitive Pythagorean triple. (x^4-y^4 = (x^2+y^2)(x+y)(x-y), hence x-y=1 and x^2+y^2 and x+y are both prime. Note that x^2+y^2 can never be (x+y)^2 so a(n) is never the cube of a prime.)

Examples

			2^4 - 1^4 = 15 = 3*5 and (3, 4, 5) is a Pythagorean triple, so 15 is a term.
6^4 - 5^4 = 671 = 11*61 and (11, 60, 61) is a Pythagorean triple, so 671 is a term.
		

Links

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

Crossrefs

Cf. A068501.

Intersection of A030513 and A147857.

Programs

Maple

f:= proc(y) if isprime(2*y+1) and isprime(2*y^2 + 2*y+1) then (2*y+1)*(2*y^2+2*y+1) fi end proc:
map(f, [$1..1000]); # Robert Israel, Jun 16 2020

Mathematica

Select[(#^4 - (#-1)^4) & /@ Range[420], DivisorSigma[0, #] == 4 &] (* Giovanni Resta, May 12 2020 *)

Formula

a(n) = (b(n)+1)^4 - b(n)^4 with b(n)=A068501(n).

a(n) = A048161(n)*A067756(n).