A346115 Least number k such that k^2 can be expressed in exactly n ways as x^2 + y^4 with {x, y} >= 1.

5, 65, 1625, 469625, 642916625, 15697403475, 2474052064291275

Offset: 1

Author

Karl-Heinz Hofmann, Jul 05 2021

Keywords

• nonn
• hard
• more

Comments

a(5) <= 642916625. - Hugo Pfoertner, Jul 07 2021

From Karl-Heinz Hofmann, Sep 02 2021: (Start)

a(6) <= 15697403475.

2 * 10^10 < a(7) <= 2474052064291275.

These two conjectured values arise from the "green group". Up to term a(5) the least solutions are in the "blue group". Follow the links below to get more information about the different colored groups.

Terms cannot be a square (see the comment from Altug Alkan in A111925).

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

If the terms additionally have 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 y too. The special prime factor 2 has the same behavior, i.e., if the term is even, x and y must be even too. (End)

Examples

			a(1)=A345645(1); a(2)=A345700(1); a(3)=A345968(1); a(4)=A346110(1).
		

Links

• Karl-Heinz Hofmann, All valid {z,x1,y1,x2,y2,x3,y3,x4,y4} sets up to 10^9
• Karl-Heinz Hofmann, A 3D Animation of the 4 solutions and the least 5 solution up to 10^9

Crossrefs

Cf. A000290, A000583, A111925.

Cf. A271576 (1 or more solutions), A345645 (1 solution), A345700 (2 solutions), A345968 (3 solutions), A346110 (4 solutions), A348655 (5 solutions), A349324 (6 solutions).

Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).

Extensions

a(5) confirmed by Martin Ehrenstein, Jul 09 2021

a(6) confirmed by Karl-Heinz Hofmann, Oct 15 2021

a(7) confirmed by Jon E. Schoenfield, Nov 15 2021