I've been pondering about primes for years. I noticed something and was wondering if anyone had some insight or research papers which have already covered this.
Today I'd like to talk about prime factorization.
It should be apparent to anyone who has studied prime factorization that any number over 10 (or 7, coincidentally - Is this because 5+2=7?), cannot be prime if the final digits of this number in base 10 are 2,4,6,8,0, or 5. You can consider 0 as a 10, and you might notice something. All of these numbers are multiples of the prime factors of 10.
The prime factors of 10 are 2 and 5. The final digits which cannot be in a prime number, no matter how large it is, are all multiples of 2 (2,4,6,8,10) or multiples of 5 (5,10).
This, to me, has a sort of symmetry with the Goldbach theorems.
I am wondering if a number base which is the product of two primes will always have this property. It seems like it inherently would be so, but I'd like some help in understanding why.
For example, given a number in base 21 (3*7) or base 14 (2*7), is it always the case that all numbers ending in multiples of those prime factors will never themselves be prime numbers?