relating the sums of the p-th powers of the first n positive integers for p = 0, 1, 2, ..., k.

That same year, Pascal had a religious experience, and he mostly gave up work in mathematics.

Cycloid

In 1658, Pascal, while suffering from a toothache, began considering several problems concerning the cycloid. His toothache disappeared, and he took this as a heavenly sign to proceed with his research. Eight days later he had completed his essay and, to publicize the results, proposed a contest.

Pascal proposed three

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