In number theory, the Kempner function S ( n ) {\displaystyle S(n)} is defined for a given positive integer n {\displaystyle n} to be the smallest number s {\displaystyle s} such that n {\displaystyle n} divides the factorial s ! {\displaystyle s!} . For example, the number 8 {\displaystyle 8} does not divide 1 ! {\displaystyle 1!} , 2 ! {\displaystyle 2!} , or 3 ! {\displaystyle 3!} , but does divide 4 ! {\displaystyle 4!} , so S ( 8 ) = 4 {\displaystyle S(8)=4} . This function has a highly irregular growth rate: it grows linearly on the prime numbers but only grows sublogarithmically at the factorial numbers.
History This function was first considered by François Édouard Anatole Lucas in 1883, followed by Joseph Jean Baptiste Neuberg in 1887. In 1918, A. J. Kempner gave an algorithm for computing S ( n ) {\displaystyle S(n)} without trials. The Kempner function is also sometimes called the Smarandache function following Florentin Smarandache's rediscovery of the function in 1980.
Properties Since n {\displaystyle n} divides n ! {\displaystyle n!} , S ( n ) {\displaystyle S(n)} is always at most n {\displaystyle n} . A number n > 4 {\displaystyle n>4} is prime if and only if S ( n ) = n {\displaystyle S(n)=n} . That is, the numbers n {\displaystyle n} for which S ( n ) {\displaystyle S(n)} is as large as possible relative to n {\displaystyle n} are the primes. In the other direction, the numbers for which S ( n ) {\displaystyle S(n)} is as small as possible are the factorials: S ( k ! ) = k {\displaystyle S(k!)=k} , for all k ≥ 1 {\displaystyle k\geq 1} .
S ( n ) {\displaystyle S(n)} is the smallest possible degree of a monic polynomial with integer coefficients, whose values over the integers are all divisible by n {\displaystyle n} . For instance, the fact that S ( 6 ) = 3 {\displaystyle S(6)=3} means that there is a cubic polynomial whose values are all zero modulo 6, for instance the polynomial
x ( x − 1 ) ( x − 2 ) = x 3 − 3 x 2 + 2 x , {\displaystyle x(x-1)(x-2)=x^{3}-3x^{2}+2x,}
but that all quadratic or linear polynomials (with leading coefficient one) are nonzero modulo 6 at some integers. In one of the advanced problems in The American Mathematical Monthly, set in 1991 and solved in 1994, Paul Erdős pointed out that the function S ( n ) {\displaystyle S(n)} coincides with the largest prime factor of n {\displaystyle n} for "almost all" n {\displaystyle n} (in the sense that the asymptotic density of the set of exceptions is zero).
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