An integer is prime if it has only two distinct divisors: 1 and itself. It is composite if it has more than that. For example, 17 is divisible only by 1 and itself: it is therefore prime, whereas 18, which has six divisors (1, 2, 3, 6, 9 and 18), is composite. Note, however, that 0 and 1 have a special status: they are neither prime nor composite.
Asserting that a number is prime cannot, a priori, be told just from looking at it — that is, from its written form: substantial calculations are often needed for large numbers. This is generally no longer true, however, for showing that a number is composite. If its usual written form (that is, in base 10) has certain properties, we can easily tell that the number is composite: for example, if its last digit is 0, 2, 4, 6 or 8, the number is divisible by 2; if its last digit is 0 or 5, it is divisible by 5; if the sum of its digits is divisible by 3, it is divisible by 3, and so on. Other divisibility tests, likewise based on the way the number is written, also exist (see Calcul mental, Tangente hors-série 95, 2025).
So what happens to a prime number when its digits undergo transformations? Let's first look at what happens when we permute them.
Facing permutations
Take, for example, the number 379: it is prime, and the numbers obtained by permuting its digits — 397, 739, 793, 937 and 973 — are prime, with the exception of 793, which is divisible by 13, and 973, which is divisible by 7. Can we find a prime number such that every permutation of its digits gives a prime number? The answer is yes. For example, 199 is prime, and the numbers obtained by permuting its digits, 919 and 991, are also prime. Such a number is called a permutable prime. The terminology absolute prime or absolutely prime number is also used. These numbers were introduced in 1951 by the German mathematician Hans-Egon Richert (1924–1993).
