Numbers and words

[simont]

This morning I tried to return a phone call. I managed to dial the wrong number three times at widely separated intervals, and (I later worked out) on all three occasions I transposed the same pair of adjacent digits. And despite carefully cross-checking between the number shown on the display of my phone, the number on the email (my employer's

Comments

[gerald_duck]

Drat. Now you've got me wondering something not entirely related.

My first thought was that maybe your subconscious was rearranging digits to make the number prime - which would be a feat of Ramanujan proportions. Though clearly both the real number and your transposition ending in 5 militated against this possibility.

But now I'm pondering collections of digits that form a prime number however they are arranged. For example, (1,1,3) where 113 131 and 311 are all prime. It feels displeasing to have repeated digits in the collection, but with that constraint it soon becomes clear no examples of more than two digits exist in base ten. Without the constraint, repunit primes like (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) become especially degenerate examples.

Considering other bases, binary is especially unedifying since Mersenne primes are the only examples. Being a curious sort, I wondered if larger collections of distinct digits worked in other bases. Having written a quick program, I find that all permutations of the digits (1,3,9) and (3,6,A) in base 11 are prime; plenty more in higher bases.

I have a proof that no collections of four digits exist in any base such that all permutations of those digits are prime. But this comment is too small to contain it. And I don't for a moment believe that's actually true, but it's certainly true up to base 100 and a brute force search is O(n4).

[simont]

I note that the Wikipedia page you cite links to a page describing just those things. It appears to be quite hard to think of natural questions in recreational maths that haven't been investigated yet :-)

[gerald_duck]

Humph!

In my defence, the Wikipedia article on prime numbers does not link to that page. Guess where I went looking. )-8

At least my conclusions seem to have been correct. And, especially, at least I didn't waste time looking for any more base-ten permutable primes after I'd found the three-digit ones.

That Wikipedia article doesn't touch on gerald_duck's hopefully-far-from-last theorem, though.