minimal prime

A minimal prime is a prime numberMathworldPlanetmath p that when written in a given base b, no smaller prime q<p can be formed from a substring of the digits of p (the digits need not be consecutive, but they must be in the same order). For example, in base 10, the prime 991 is a minimal prime because all of its possible substrings (9, 9, 1, 99, 91, 91) are either composite or not considered prime. A071062 of Sloane’s OEIS lists the twenty-six base 10 minimal primes.

Clearly, all primes p<b are minimal primes in that base. Such primes are obviously finite, but so are those minimal primes p>b, per Michel Lothaire’s findings. In binary, there are only exactly two minimal primes: 2 and 3, written 10 and 11 respectively. Every larger prime will have 1 as its most significant digit and possibly a 0 somewhere; the 1 and 0 can then be brought together to form 10 (2 in decimal). The exception to this are the Mersenne primesMathworldPlanetmath 2q-1 (or binary repunitsMathworldPlanetmath), but it is even more elegant to prove these are not minimal primes in binary: they contain all smaller Mersenne primes as substrings!


  • 1 M. Lothaire “Combinatorics on words” in Encylopedia of mathematics and its applications 17 New York: Addison-Wesley (1983): 238 - 247
Title minimal prime
Canonical name MinimalPrime
Date of creation 2013-03-22 16:52:23
Last modified on 2013-03-22 16:52:23
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 4
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A41
Classification msc 11A63