binary relation among prime numbers
binary, primes, series
Solution
Binary is just writing numbers as a sum of powers of two. It's not significantly different from decimal in a mathematical sense. So no, there will not be any theorems in binary that don't have some parallel in decimal.
In decimal, no number ending in an even numeral or 5 can be prime, except 2 and 5. In binary, no number ending in `0` can be prime, except `10` (which is 2).
EDIT: See this answer I wrote a couple years ago for an example of how to quickly generate primes using binary arithmetic optimizations, not advanced math. It's just a sieve of Erastosthenes, but thousands-of-years-old math, predating even the decimal system, is still amenable to SSE vectorization.
Problem
Do we have any theory stating a relation between primes in binary system. I mean, in decimal system we have a pattern stating that "a number which is divided by 1 and itself is a prime". This was learned in my school when i was kid. But modern computation is performed on bits, in sense they are 1's and 0's. But we calculate the prime nature based on our school knowledge. It works fine when the numbers are small. But questions calculating largest prime in integers, this logic doesn't make sense. So if there exists any theory(may be already existing) stating a relation among primes in binary represention, then we can save lot of computing power. For ex, starting with a binary representation of prime, changing or adding bits yields next prime number saves lot of computational power. This might not make sense. But these were my thoughts from last night. Please correct me if I am wrong or it doesn't make a sense at all.