We prove the infinitude of prime numbers by the principle of contradiction, that is different from Euclid's proof in a way that it uses an explicit property of prime numbers. A sieve method that applies the inclusion-exclusion principle is used to give the property of the prime numbers in terms of the prime counting function.
"Sieving for the Primes to Prove Their Infinitude." Missouri J. Math. Sci. 29 (2) 176 - 183, November 2017. https://doi.org/10.35834/mjms/1513306829