On the representations of an integer as a sum of two or four triangular numbers

摘要

Review: The author employs Ramanujan's 1ψ1 summation to give a short, elegant proof of the following theorem on triangular numbers k(k+1)/2: The number of representations of a positive integer n as a sum of 4 triangular numbers equals ∑d∣(2n+1)d. The reviewer (and evidently also the author) does not know who first discovered this theorem, but it is essentially in Ramanujan's notebooks. Ramanujan also discovered a corresponding theorem for representations of n by a sum of 8 triangular numbers