A Double Factorial based novel proof for the infinitude of primes from a Euclid Family Proofs

This paper presents a novel variation of Euclid's classic proof of the infinitude of prime numbers, utilizing the even double factorial construction. Assuming a finite set of all primes p₁, p₂, ..., pₖ, let P be the product of all these primes and define the even double factorial P!! as the product of all even numbers from 2 up to P (that is, 2 × 4 × 6 × ... × P). The paper shows that every prime pᵢ divides P!!, and consequently, the number N = P!! + 1 is not divisible by any of the assumed primes. Hence, any prime divisor of N must be a new prime not in the original list—leading to a contradiction if the list was assumed complete. This construction offers a structurally distinct alternative to classical Euclid-type proofs (such as those using P, pₖ!, or primorials), with the divisibility mechanism arising from the sequence of even numbers 2, 4, 6, ..., P. The paper also compares the growth rates of double factorials and ordinary factorials using Stirling's approximation.