Number is product of two primes

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Number is product of two primes

Product of primes #

One way to prove that some number $\varN$ is the product of two primes $\varN = \varp \varq$ is by showing that:

$\varN$ is square-free
$\varN$ only has two divisors

If we only showed that $\varN$ is square-free, then it could be of the form $\varN = \varp \varq \varr$. On the other hand, if we only proved that $\varN$ has two divisors, it could be of the form $\varN = \varp^2\varq$.

When the primes are congruent with $3 \; \mathsf{mod}\; 4$, we can show that $\varN$ is the product of two primes much more efficiently with the Paillier-Blum modulus proof.

Number is square-free: Proves that a number is square-free.
Two prime divisors: Proves that a number has two prime divisors.
Product of two primes: Proves that a number is the product of two distinct primes: in parallel, run the square-freeness proof together with the two-prime-divisors proof.
Paillier-Blum Modulus: An efficient proof that shows a number is the product of two primes congruent with 3 mod 4.