Baçam Sets and Baçam Primes

Hypothetically, one shouldn’t be sure of the base a typically written number is in unless explicitly stated. For example, 3623 could be any base seven or higher. Unfortunately (or fortunately), most people automatically assume any number to be in base ten. But what if we automatically assume the most likely base? What is the most likely base?

As base numbers go up, the likelihood of that base being the one used decreases. The number “10220001112110210” can be much more safely assumed to be in base three than in base of some number that can be written as the sum of two fourth powers two different ways. Let’s for now interpret any written number as being in the base of one larger than its largest digit and see what happens. Let’s call this base the minimal applicable base, which I will abbreviate “baçam,” to also make fun of the letter c while at it. A number’s baçam set can now therefore be defined as the set of integers greater than 1 b such that when the number is expressed in base b the largest digit in the representation is b-1. For example, twelve’s baçam set is (with bases written in base ten) {2 (1100), 4 (30), 13 (C)}, and fifteen’s baçam set is {2 (1111), 3 (120), 4 (33), 8 (17), 16 (E)}.

As one might notice, some numbers have notably larger baçam sets than others. To find a number with a very large baçam set, take a number that is one less than a highly abundant number, since a number n has any factor of n+1 greater than 1 in its baçam set; this fact can be easily verified via modular arithmetic. One can also notice that other than n+1, 2 is always in a number n’s baçam set.

If a number n has only 2 and n+1 in its baçam set, it reminds us of the property that defined primes; thus we can call numbers with a baçam set of size 2 baçam primes. Notice that excepting the special case of 3, a necessary (but far not sufficient) condition for n to be a baçam prime is for n+1 to be a prime. There are two numbers, 2 and 3, which are both primes and baçam primes, and 3 is the only odd baçam prime. The first eight baçam primes, in base ten representation, are 2, 3, 4, 10, 36, 40, 82, and 256; in base two, they are 10, 11, 100, 1010, 100100, 101000, 1010010, 100000000. Using Python, it appears that there are no more baçam primes under $2^{16}+1$, so is 256 the largest baçam prime?