A230561 Smallest number that is the sum of two positive n-th powers in >= n ways.

2, 50, 87539319

Offset: 1

Author

Jonathan Sondow, Oct 23 2013

Keywords

• hard
• more
• nonn
• bref

Comments

Guy, 2004: "Euler knew that 635318657 = 133^4 + 134^4 = 59^4 + 158^4, and Leech showed this to be the smallest example. No one knows of three such equal sums." Thus no one knows whether a(4) exists, which requires four such equal sums.

a(4) > 10^21 (if it exists). There is no number <= 10^21 that is the sum of two positive 4th powers in >= three ways. - Donovan Johnson, Jan 07 2014

Examples

			2 = 1^1 + 1^1.
50 = 1^2 + 7^2 = 5^2 + 5^2.
87539319 = 167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1.

Crossrefs

Cf. A048610, A011541 for a(2), a(3).

Cf. also A016078, A230477.

Formula

a(n) >= A016078(n) for n > 1, with equality at least for n = 2, and inequality at least for n = 3.