Bhargava factorial

In mathematics, Bhargava's factorial function, or simply the Bhargava factorial, is a generalization of the factorial function developed by the Fields Medal winning mathematician Manjul Bhargava as part of his bachelor's thesis at Harvard University in 1996. Bhargava subsequently published a revised version in The American Mathematical Monthly in 2000. The Bhargava factorial has the property that many number-theoretic results involving the ordinary factorials remain true even when the factorials are replaced by the Bhargava factorials. Using an arbitrary infinite subset S of the set ${\displaystyle \mathbb {Z} }$ of integers, Bhargava associated a positive integer with every positive integer k, which he denoted by k !_S, with the property that if one takes S = ${\displaystyle \mathbb {Z} }$ itself, then the integer associated with k, that is k !_{${\displaystyle \mathbb {Z} }$} , would turn out to be the ordinary factorial of k.