A294738 Numbers that are the sum of 5 nonzero squares in exactly 4 ways.

62, 70, 71, 72, 75, 76, 82, 84, 89, 97, 108, 129, 132

Offset: 1

Author

Robert Price, Nov 07 2017

Keywords

• nonn
• fini
• full

Comments

This sequence is likely finite and complete as the next term, if it exists, is > 50000.

From a proof by David A. Corneth on Nov 08 2017 in A294736: This sequence is complete, see the von Eitzen Link and Price's computation that the next term must be > 50000. Proof. The link mentions "for positive integer n, if n > 5408 then the number of ways to write n as a sum of 5 squares is at least Floor(Sqrt(n - 101) / 8)". So for n > 5408, there are more than eight ways to write n as a sum of 5 squares. For n <= 5408, it has been verified if n is in the sequence by inspection. Hence the sequence is complete.

References

• E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

Links

• H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
• D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
• Eric Weisstein's World of Mathematics, Square Number.
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A025429, A025357, A294675, A294736, A294737.