|
In mathematics, Bhargava's factorial function, or simply Bhargava factorial, is a certain generalization of the 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 we take ''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.〔 #For any positive integers ''k'' and ''l'', (''k'' + ''l'')! is a multiple of ''k''! ''l''!. #Let ''f''(''x'') 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''(''x'') is ''k'' then the greatest common divisor of the set of values of ''f''(''x'') for integer values of ''x'' is a divisor of ''k''!. #Let ''a''0, ''a''1, ''a''2, . . . , ''a''''n'' be any ''n'' + 1 integers. Then the product of their pairwise differences is a multiple of 0! 1! ... ''n''!. #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'' (a subset of ''Z'', or a subset of some ring) and define a function depending on ''S'' which assigns a value to each non-negative inter ''k'', denoted by ''k''!''S'', such that the statements obtained from the theorems given earlier by replacing ''k''! by ''k''!''S'' remain true? 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bhargava factorial」の詳細全文を読む スポンサード リンク
|