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Kempner function

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Graph of the Kempner function

In number theory, the Kempner function [1] is defined for a given positive integer to be the smallest number such that divides the factorial . For example, the number does not divide , , or , but does divide , so .

This function has the property that it has a highly inconsistent growth rate: it grows linearly on the prime numbers but only grows sublogarithmically at the factorial numbers.

History

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This function was first considered by François Édouard Anatole Lucas in 1883,[2] followed by Joseph Jean Baptiste Neuberg in 1887.[3] In 1918, A. J. Kempner gave the first correct algorithm for computing .[4]

The Kempner function is also sometimes called the Smarandache function following Florentin Smarandache's rediscovery of the function in 1980.[5]

Properties

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Since divides , is always at most . A number greater than 4 is a prime number if and only if .[6] That is, the numbers for which is as large as possible relative to are the primes. In the other direction, the numbers for which is as small as possible are the factorials: , for all .

is the smallest possible degree of a monic polynomial with integer coefficients, whose values over the integers are all divisible by .[1] For instance, the fact that means that there is a cubic polynomial whose values are all zero modulo 6, for instance the polynomial 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 coincides with the largest prime factor of for "almost all" (in the sense that the asymptotic density of the set of exceptions is zero).[7]

Computational complexity

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The Kempner function of an arbitrary number is the maximum, over the prime powers dividing , of .[4] When is itself a prime power , its Kempner function may be found in polynomial time by sequentially scanning the multiples of until finding the first one whose factorial contains enough multiples of . The same algorithm can be extended to any whose prime factorization is already known, by applying it separately to each prime power in the factorization and choosing the one that leads to the largest value.

For a number of the form , where is prime and is less than , the Kempner function of is .[4] It follows from this that computing the Kempner function of a semiprime (a product of two primes) is computationally equivalent to finding its prime factorization, believed to be a difficult problem. More generally, whenever is a composite number, the greatest common divisor of and will necessarily be a nontrivial divisor of , allowing to be factored by repeated evaluations of the Kempner function. Therefore, computing the Kempner function can in general be no easier than factoring composite numbers.

References and notes

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  1. ^ a b Called the Kempner numbers in the Online Encyclopedia of Integer Sequences: see Sloane, N. J. A. (ed.). "Sequence A002034 (Kempner numbers: smallest number m such that n divides m!)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^ Lucas, E. (1883). "Question Nr. 288". Mathesis. 3: 232.
  3. ^ Neuberg, J. (1887). "Solutions de questions proposées, Question Nr. 288". Mathesis. 7: 68–69.
  4. ^ a b c Kempner, A. J. (1918). "Miscellanea". The American Mathematical Monthly. 25 (5): 201–210. doi:10.2307/2972639. JSTOR 2972639.
  5. ^ Hungerbühler, Norbert; Specker, Ernst (2006). "A generalization of the Smarandache function to several variables". Integers. 6: A23, 11. MR 2264838.
  6. ^ R. Muller (1990). "Editorial" (PDF). Smarandache Function Journal. 1: 1. ISBN 84-252-1918-3.
  7. ^ Erdős, Paul; Kastanas, Ilias (1994). "The smallest factorial that is a multiple of n (solution to problem 6674)" (PDF). The American Mathematical Monthly. 101: 179. doi:10.2307/2324376. JSTOR 2324376..

This article incorporates material from Smarandache function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.