minimal prime

A minimal prime is a prime number $p$ that when written in a given base $b$, no smaller prime 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 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 primes $2^q-1$ (or binary repunits), but it is even more elegant to prove these are not minimal primes in binary: they contain all smaller Mersenne primes as substrings!