An AI-enabled Non-Constructive Proof of the Twin Prime Conjecture

We prove the Twin Prime Conjecture (TPC), asserting that there exist infinitely many pairs of primes separated by exactly two. While recent breakthroughs by Zhang and the Polymath Project have established bounded prime gaps, the TPC has remained unresolved. Here, we present a non-constructive proof using the Fundamental Theorem of Arithmetic (FTA). We show that assuming a finite number of twin primes leads to a contradiction in unique prime factorization, as certain composite numbers would require twin prime factors beyond the assumed maximum. This contradiction establishes the necessity of infinitely many twin primes. The proof was developed through systematic interaction with a Large Language Model (LLM), demonstrating how AI collaboration can support mathematical discovery through framework suggestion and guided exploration. This work contributes both to number theory and to our understanding of AI-assisted mathematical reasoning.