In number theory, the Kempner function is defined for a given positive integer divides the , but does

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 without trials.

The Kempner function is also sometimes called the Smarandache function following Florentin Smarandache's rediscovery of the function

Properties

is always at is prime if and only is as small as possible are the factorials: for

is the smallest possible degree of a monic polynomial with integer coefficients, whose values over the integers are all divisible means that there is a cubic polynomial whose values are all zero modulo 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 coincides with the largest prime factor (in the sense that the asymptotic density of the set of exceptions is zero).

Computational complexity

is the maximum, over the prime powers., its Kempner function may be found in polynomial time until finding the first one whose factorial contains enough multiples The same algorithm 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.

. 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 is composite, the greatest common divisor will necessarily be a nontrivial divisor 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.

See also

• Ruler function, a simpler integer function with an inconsistent growth rate

References and notes