Integrable Highest Weight Modules over Affine Superalgebras and Number Theory

The problem of representing an integer as a sum of squares of integers has had a long history. One of the first after antiquity was A. Girard who in 1632 conjectured that an odd prime p can be represented as a sum of two squares iff p ≡ 1 mod 4, and P. Fermat in 1641 gave an “irrefutable proof” of this conjecture. The subsequent work on this problem culminated in papers by A.M. Legendre (1798) and C.F. Gauss (1801) who found explicit formulas for the number of representations of an integer as a sum of two squares. C.G. Bachet in 1621 conjectured that any positive integer can be represented as a sum of four squares of integers, and it took efforts of many mathematicians for about 150 years before J.-L. Lagrange gave a proof of this conjecture in 1770.

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