A348655 Numbers whose square can be represented in exactly five ways as the sum of a positive square and a positive fourth power.

642916625, 2571666500, 4418701625, 5786249625, 10286666000, 16072915625, 17674806500, 20931496625, 23144998500, 31502914625, 39768314625, 41146664000, 52076246625, 57801168750, 64291662500, 70699226000, 77792911625, 83725986500, 92579994000, 108652909625

Offset: 1

Author

Karl-Heinz Hofmann, Oct 27 2021

Keywords

• nonn

Comments

Numbers z such that there are exactly 5 solutions to z^2 = x^2 + y^4.

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.

Some other terms of the sequence: 20931496625, 23144998500, 31502914625, 41146664000, 52076246625, 64291662500, 77792911625, 83725986500, 92579994000, 108652909625, 126011658500, 144656240625, 164586656000. - Chai Wah Wu, Oct 29 2021

Examples

			5786249625^2 = 5785404300^2 +  9945^4
5786249625^2 = 5608211175^2 + 37740^4
5786249625^2 = 5341153500^2 + 47175^4
5786249625^2 = 4307113575^2 + 62160^4
5786249625^2 = 2036759868^2 + 73593^4
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..3989 (all terms < A346115(7) = 2474052064291275)

Crossrefs

Cf. A111925, A271576, A345645 (in exactly 1 way), A345700 (in exactly 2 ways), A345968 (in exactly 3 ways), A346110 (in exactly 4 ways), A349324 (in exactly 6 ways), A346115 (the least solutions).

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

Extensions

a(8) and beyond from Jon E. Schoenfield, Nov 14 2021