A349664 a(n) is the number of solutions for n^4 = z^2 - x^2 with {z,x} >= 1.

0, 1, 2, 3, 2, 7, 2, 5, 4, 7, 2, 17, 2, 7, 12, 7, 2, 13, 2, 17, 12, 7, 2, 27, 4, 7, 6, 17, 2, 37, 2, 9, 12, 7, 12, 31, 2, 7, 12, 27, 2, 37, 2, 17, 22, 7, 2, 37, 4, 13, 12, 17, 2, 19, 12, 27, 12, 7, 2, 87, 2, 7, 22, 11, 12, 37, 2, 17, 12, 37, 2, 49, 2, 7, 22

Offset: 1

Author

Karl-Heinz Hofmann, Dec 13 2021

Keywords

• nonn

Comments

If n is an odd prime^i, the number of solutions is 2*i.

If n = 2^i, the number of solutions is 2*i-1.

These two facts are not generally valid in reverse for terms > 6.

If a(n) = 2, n is an odd prime. This is generally valid in reverse.

For more information about these facts see the link.

The calculation of the terms is done with an algorithm of Jon E. Schoenfield, which is described in A349324.

Conditions to be satisfied for a valid, countable solution:

- z cannot be a square.

- z must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).

- If z has prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and n too.

- If z is even, x and n must be even too.

- The lower limit of the ratio x/n is sqrt(2).

- high limits of z and x:

| n is odd | n is even

---------+------------------+------------------

z limit | (n^4 + 1)/2 | (n^4 + 4)/4

x limit | (n^4 + 1)/2 - 1 | (n^4 + 4)/4 - 2

Examples

			a(6) = 7 (solutions): 6^4 = 1296 = 325^2 - 323^2 = 164^2 - 160^2 = 111^2 - 105^2 = 85^2 - 77^2 = 60^2 - 48^2 = 45^2 - 27^2 = 39^2 - 15^2.
		

Links

• Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
• Karl-Heinz Hofmann, What the terms can tell about n.

Crossrefs

Cf. A000290, A000583, A271576, A349663, A002144, A002145, A346115.

Cf. A345645, A345700, A345968, A346110, A348655, A349324.

Programs

Mathematica

a[n_] := Length[Solve[n^4 == z^2 - x^2 && x >= 1 && z >= 1, {x, z}, Integers]]; Array[a, 75] (* Amiram Eldar, Dec 14 2021 *)

PARI

a(n) = numdiv(if(n%2, n^4, n^4/4))\2; \\ Jinyuan Wang, Dec 19 2021