Enzo Creti searched primes of the form $$270270270\cdots 270270271$$
The general expression for those numbers is $$\frac{270\cdot (1000^{n}-1)}{999}+1$$ when $\ "270"\ $ appears $\ n-1\ $ times. The even exponents $\ n>2\ $ giving a prime we found yet are $\ 12,740,788,7964\ $
Those even exponents have the following properties :
- They are divisible by $\ 4\ $ , but not by $\ 8\ $
- If we subtract $\ 1\ $ , we get a prime number. (even an emirp, that is a prime that is also prime when the digits in base $\ 10\ $ are written down in reverse order)
Moreover, the odd exponents that are multiples of $\ 3\ $ ($\ 3,9,129\ $) have the form $\ 2^n + 1\ $
Is this pure coincidence or can it be shown that only such even exponents (apart from $\ 2\ $) can make the expression prime ?