Bhargava Factorial
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Bhargava's factorial function, or simply Bhargava factorial, is a certain generalization of the
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
function developed by the
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winning mathematician
Manjul Bhargava Manjul Bhargava (born 8 August 1974) is a Canadian-American mathematician. He is the Brandon Fradd, Class of 1983, Professor of Mathematics at Princeton University, the Stieltjes Professor of Number Theory at Leiden University, and also holds A ...
as part of his thesis in
Harvard University Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...
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 \mathbb 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'' = \mathbb itself, then the integer associated with ''k'', that is ''k'' !\mathbb , would turn out to be the ordinary factorial of ''k''.


Motivation for the generalization

The
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
of a
non-negative integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
''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 Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
. The following are a few of these theorems. # For any positive integers ''m'' and ''n'', (''m'' + ''n'')! is a multiple of ''m''! ''n''!. # Let ''f''(''x'') be a primitive integer polynomial, that is, a polynomial in which the coefficients are integers and are
relatively prime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
to each other. If the degree of ''f''(''x'') is ''k'' then the
greatest common divisor In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest co ...
of the set of values of ''f''(''x'') for integer values of ''x'' is a
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
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 \mathbb be the set of integers and ''n'' any integer. Then the number of
polynomial function In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative int ...
s from the
ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often de ...
\mathbb to the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
\mathbb/n\mathbb is given by \prod_^ \frac. 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 \mathbb, or a subset of some
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) 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 an arbitrary infinite subset of the set ''Z'' of integers. * Choose a prime number ''p''. * Construct an ordered sequence of numbers chosen from ''S'' as follows (such a sequence is called a ''p''-ordering of ''S''): :# ''a''0 is any arbitrary element of ''S''. :# ''a''1 is any arbitrary element of ''S'' such that the highest power of ''p'' that divides ''a''1 − ''a''0 is minimum. :# ''a''2 is any arbitrary element of ''S'' such that the highest power of ''p'' that divides (''a''2 − ''a''0)(''a''2 − ''a''1) is minimum. :# ''a''3 is any arbitrary element of ''S'' such that the highest power of ''p'' that divides (''a''3 − ''a''0)(''a''3 − ''a''1)(''a''3 − ''a''2) is minimum. :# ... and so on. * Construct a ''p''-ordering of ''S'' for each prime number ''p''. (For a given prime number ''p'', the ''p''-ordering of ''S'' is not unique.) * For each non-negative integer ''k'', let ''v''''k''(''S'', ''p'') be the highest power of ''p'' that divides (''a''''k'' − ''a''0)(''a''''k'' − ''a''1)(''a''''k'' − ''a''2) ... (''a''''k'' − ''a''''k'' − 1). The sequence is called the associated ''p''-sequence of ''S''. This is independent of any particular choice of ''p''-ordering of ''S''. (We assume that ''v''0(''S'', ''p'') = 1 always.) * The factorial of the integer ''k'', associated with the infinite set ''S'', is defined as k!_=\prod_p v_k(S,p), where the product is taken over all prime numbers ''p''.


Example: Factorials using set of prime numbers

Let ''S'' be the set of all prime numbers ''P'' = . :* Choose ''p'' = 2 and form a ''p''-ordering of ''P''. ::* Choose ''a''0 = 19 arbitrarily from ''P''. ::* To choose ''a''1: :::* The highest power of ''p'' that divides 2 − ''a''0 = −17 is 20 = 1. Also, for any ''a'' ≠ 2 in P, ''a'' − ''a''0 is divisible by 2. Hence, the highest power of ''p'' that divides (''a''1 − ''a''0) is minimum when ''a''1 = 2 and the minimum power is 1. Thus ''a''1 is chosen as 2 and ''v''1(''P'', 2) = 1. ::* To choose ''a''2: :::* It can be seen that for each element ''a'' in ''P'', the product ''x'' = (''a'' − ''a''0)(''a'' − ''a''1) = (''a'' − 19)(''a'' − 2) is divisible by 2. Also, when ''a'' = 5, ''x'' is divisible 2 and it is not divisible by any higher power of 2. So, ''a''2 may be chosen as 5. We have ''v''2(''P'', 2) = 2. ::* To choose ''a''3: :::* It can be seen that for each element ''a'' in ''P'', the product ''x'' = (''a'' − ''a''0)(''a'' − ''a''1)(''a'' − ''a''2) = (''a'' − 19)(''a'' − 2)(''a'' − 5) is divisible by 23 = 8. Also, when ''a'' = 17, ''x'' is divisible by 8 and it is not divisible by any higher power of 2. Choose ''a''3 = 17. Also we have ''v''3(''P'',2) = 8. ::* To choose ''a''4: :::* It can be seen that for each element ''a'' in ''P'', the product ''x'' = (''a'' − ''a''0)(''a'' − ''a''1)(''a'' − ''a''2)(''a'' − ''a''3) = (''a'' − 19)(''a'' − 2)(''a'' − 5)(''a'' − 17) is divisible by 24 = 16. Also, when ''a'' = 23, ''x'' is divisible 16 and it is not divisible by any higher power of 2. Choose ''a''4 = 23. Also we have ''v''4(''P'',2) = 16. ::* To choose ''a''5: :::* It can be seen that for each element ''a'' in ''P'', the product ''x'' = (''a'' − ''a''0)(''a'' − ''a''1)(''a'' − ''a''2)(''a'' − ''a''3)(''a'' − ''a''4) = (''a'' − 19)(''a'' − 2)(''a'' − 5)(''a'' − 17)(''a'' − 23) is divisible by 27 = 128. Also, when ''a'' = 31, ''x'' is divisible 128 and it is not divisible by any higher power of 2. Choose ''a''5 = 31. Also we have ''v''5(''P'',2) = 128. ::* The process is continued. Thus a 2-ordering of P is and the associated 2-sequence is , assuming that ''v''0(''P'', 2) = 1. :* For ''p'' = 3, one possible ''p''-ordering of ''P'' is the sequence and the associated ''p''-sequence of ''P'' is . :* For ''p'' = 5, one possible ''p''-ordering of ''P'' is the sequence and the associated ''p''-sequence is . :* It can be shown that for ''p'' ≥ 7, the first few elements of the associated ''p''-sequences are . The first few factorials associated with the set of prime numbers are obtained as follows .
Table of values of ''v''''k''(''P'', p) and ''k''!''P''


Example: Factorials using the set of natural numbers

Let ''S'' be the set of natural numbers \mathbb. :* For ''p'' = 2, the associated ''p''-sequence is . :* For ''p'' = 3, the associated ''p''-sequence is . :* For ''p'' = 5, the associated ''p''-sequence is . :* For ''p'' = 7, the associated ''p''-sequence is . :* ... and so on. Thus the first few factorials using the natural numbers are :* 0!\mathbb = 1×1×1×1×1×... = 1. :* 1!\mathbb = 1×1×1×1×1×... = 1. :* 2!\mathbb = 2×1×1×1×1×... = 2. :* 3!\mathbb = 2×3×1×1×1×... = 6. :* 4!\mathbb = 8×3×1×1×1×... = 24. :* 5!\mathbb = 8×3×5×1×1×... = 120. :* 6!\mathbb = 16×9×5×1×1×... = 720.


Examples: Some general expressions

The following table contains the general expressions for ''k''!''S'' for some special cases of ''S''. {, class="wikitable" style="margin:1em auto;" , - ! Sl. No. !! Set ''S'' !! ''k''!''S'' , - , 1 , , Set of natural numbers , , ''k''! , - , 2 , , Set of even integers , , 2''k''×''k''! , - , 3 , , Set of integers of the form ''an'' + ''b'' , , ''a''''k''×''k''! , - , 4 , , Set of integers of the form 2''n'' , , (2''k'' − 1)(2''k'' − 2) ... (2''k'' − 2''k'' − 1) , - , 5 , , Set of integers of the form ''q''''n'' for some prime ''q'', , (''q''''k'' − 1)(''q''''k'' − ''q'') ... (''q''''k'' − ''q''''k'' − 1) , - , 6 , , Set of squares of integers , , (2''k'')!/2


References

Combinatorics Factorial and binomial topics