Bhargava factorial


In mathematics, Bhargava's factorial function, or simply Bhargava factorial, is a certain generalization of the factorial function developed by the Fields Medal winning mathematician Manjul Bhargava as part of his thesis in Harvard University in 1996. 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 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 = Z itself, then the integer associated with k, that is k !Z, would turn out to be the ordinary factorial of k.

Motivation for the generalization

The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5×4×3×2×1 = 120. By convention, the value of 0! is defined as 1. This classical factorial function appears prominently in many theorems in number theory. The following are a few of these theorems.
  1. For any positive integers k and l, ! is a multiple of k! l!.
  2. Let f be a primitive integer polynomial, that is, a polynomial in which the coefficients are integers and are relatively prime to each other. If the degree of f is k then the greatest common divisor of the set of values of f for integer values of x is a divisor of k!.
  3. Let a0, a1, a2,..., an be any n + 1 integers. Then the product of their pairwise differences is a multiple of 0! 1!... n!.
  4. Let Z be the set of integers and n any integer. Then the number of polynomial functions from the ring of integers Z to the quotient ring Z/nZ is given by.
Bhargava posed to himself the following problem and obtained an affirmative answer: In the above theorems, can one replace the set of integers by some other set S and define a function depending on S which assigns a value to each non-negative integer k, denoted by k!S, such that the statements obtained from the theorems given earlier by replacing k! by k!S remain true?

The generalisation

Let S be the set of all prime numbers P =.
The first few factorials associated with the set of prime numbers are obtained as follows.

Table of values of vk and k!P
p = 2p = 3p = 5p = 7p = 11...k!P
k = 011111...1×1×1×1×1×... = 1
k = 111111...1×1×1×1×1×... = 1
k = 221111...2×1×1×1×1×... = 2
k = 383111...8×3×1×1×1×... = 24
k = 4163111...16×3×1×1×1×... = 48
k = 51289511...128×9×5×1×1×... = 5760
k = 62569511...256×9×5×1×1×... = 11520

Example: Factorials using the set of natural numbers

Let S be the set of natural numbers Z.
Thus the first few factorials using the natural numbers are

Examples: Some general expressions

The following table contains the general expressions for k!S for some special cases of S.

Sl. No.Set Sk!S
1Set of natural numbersk!
2Set of even integers2k×k!
3Set of integers of the form an + bak×k!
4Set of integers of the form 2n...
5Set of integers of the form qn for some prime q...
6Set of squares of integers!/2

Properties

Let S be an infinite subset of the set Z of integers. For any integer k, let k!S be the Bhargava factorial of k associated with the set S. Manjul Bhargava proved the following results which are generalisations of corresponding results for ordinary factorials.
  1. For any positive integers k and l, !S is a multiple of k!S × l!S.
  2. Let f be a primitive integer polynomial, that is, a polynomial in which the coefficients are integers and are relatively prime to each other. If the degree of f is k then the greatest common divisor of the set of values of f for values of x in the set S is a divisor of k!S.
  3. Let a0, a1, a2,..., an be any n + 1 integers in the set S. Then the product of their pairwise differences is a multiple of 0!S 1!S... n!S.
  4. Let Z be the set of integers and n any integer. Then the number of polynomial functions from S to the quotient ring Z/nZ is given by.