Show that$\mathbf{P}\mathbf{R}\mathbf{I}\mathbf{M}\mathbf{E}\mathbf{S}\mathbf{}\mathbf{=}\mathbf{\left\{}\mathbf{m}\mathbf{\right|}$is a prime number in binary} ?. (Hint: For$\mathit{p}\mathbf{>}\mathbf{1}$, the multiplicative group$\mathbf{Z}\mathbf{*}\mathbf{p}\mathbf{}\mathbf{=}\mathbf{\left\{}\mathbf{x}\mathbf{\right|}\mathbf{x}$ is relatively prime toand } is both cyclic and of order$\mathit{p}\mathbf{-}\mathbf{1}$if$\mathit{p}$is prime. You may use this fact without justifying it. The stronger statement$\mathit{P}$is now known to be true, but it is more difficult to prove.)

Now, consider two situations:

Consider a situation where all prime numbers are greater than 1.

Here, a situation is considered where x is a relative prime number the value of lies between $0$and $1$.

Both conditions are cyclic as the both situations can be combined and can lie between $1$and$P$ .

Order of these conditions is $\$P-1\$$if $\$P\$$is prime as the range lies between$1$ and$\$P\$$ then the order between$P-1$. This fact is alone sufficient to prove the statement is a prime number in binary and second considered situation are quite enough itself to justify the statement.

It can be proved by the fact of belonging of prime numbers to$co-RP$ and consider prime numbers belong to $co-RP$.

Thus, required statement will be true as well, because belongingness of can be proved only if belongs to $co-RP$and $co-RP$as well.

Thus it is quite obvious that PRIMES is a prime number in binary.