Following attempts at an analytic proof of the Pentagonal Number Theorem, we report on the discovery of a general principle leading to the unexpected cancellation of oscillating sums, of which $\sum_{n^2\leq x}(-1)^ne^{\sqrt{x-n^2}}$ is an example (to get the idea of the result). After stating the motivation, and our theorem, we apply it to prove a number of results on integer partitions, the distribution of prime numbers, and the Prouhet-Tarry-Escott Problem... (read more)

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