Absolute Primes
A. Slinko
For a long time primes have attracted the attention of mathematicians, especially those primes that possess some sort of symmetry. The mysterious and inconceivable repunits An = 111 . . . 1(n) , whose decimal representation contains only units, form an important class of them1 . For a repunit An to be prime, the number n of digits in its decimal representation must be also prime. But this condition is far from being sufficient: for instance, A3 = 111 = 3 · 37 and A5 = 11111 = 41 · 271. Some repunits are nonetheless prime: A2 , A19 , A23 , A317 and possibly A1031 , give the examples. The question of primeness of repunits was discussed by M. Gardner [1] and later in [2-4]. It is completely unclear whether the number of prime repunits is finite or infinite. The prime repunits are examples of integers which are prime and remain prime after an arbitrary permutation of their decimal digits. Integers with this property are called either permutable primes according to H.-E. Richert [5], who introduced them some 40 years ago, or absolute primes according to T.N. Bhagava and P.H. Doyle [6], and A.W.Johnson [7]. The intent of this note is to give a short proof, which does not require significant number crunching, of all known facts referring to absolute primes different from repunits. Analyzing the table of primes which are less than 103 , we find 21 absolute primes different form the repunit 11: 2, 3, 5, 7, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991.
By a subscript in brackets we will indicate the number of digits in the decimal representation of the integer.
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The first observation is easy to get: Lemma 1: A multidigit absolute prime contains in its decimal representations only the four digits 1, 3, 7, 9. Proof: If any of the digits 0, 2, 4, 5, 6, 8 appear in the representation of an integer, then by shifting this digit to the units place we get a multiple of 2 or a multiple of 5. Now we can confine the area of