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I know that there are many binary operations to show that something is true
for example we can show if number is power of two or something else is there some theory or special binary method to show if number is prime?
Detecting if a number is prime is not very easy!
Read this article about the PRIMES is in P breakthrough: http://www.ams.org/notices/200305/fea-bornemann.pdf to give you an idea of how tough a problem this actually is.
This news article might be an easier read: http://members.cox.net/mathmistakes/primes.htm
In short, if you find a simple ‘binary method’ you will be famous!