Twin Prime Conjecture (American Way)

This paper proposes a claimed proof of the Twin Prime Conjecture, which states that there exist infinitely many pairs of prime numbers that differ by 2. Beginning with the known initial set of twin primes, the argument constructs an inductive framework in which any given twin prime pair ( 𝑝 1 , 𝑞 1 ) can be extended to a larger pair ( 𝑝 2 , 𝑞 2 ) with 𝑝 2 > 𝑝 1 . The method relies on the author’s “Taha’s Detector Prime Number” (TDPN), a criterion based on factorial expressions intended to distinguish primes from composite numbers. By assuming that for every integer greater than a given prime there exists a test determining primality, the paper concludes that additional twin prime pairs must exist. Through this iterative process, the set of twin primes is asserted to be infinite. The work aims to contribute to ongoing discussions surrounding one of number theory’s most enduring open problems.