Disproving the Twin Prime Conjecture would be disproving the Prime Number Theorem

(Assuming the truth of these four posts: 1,2,3,4.)

While you cannot necessarily prove directly the evidence of infinite twin primes; you can easily disprove the idea of a largest such set on the basis that a single set (5,7) or (11,13) exists at all given the infinitude of numbers, and specifically prime numbers of the form 6k+/-1. This indirectly proves infinite twin primes by contradiction.

Conjecture: As Hotchkiss, I assert that any attempt to disprove the twin prime conjecture reduces to an attempt to disprove the infinitude of prime numbers.

Proof:

Assumption: Suppose the Twin Prime Conjecture is false. This implies that there exists a largest twin prime pair (p, q) such that there are no twin prime pairs beyond this pair.

Consequence for the Infinitude of Primes:

Euclid’s Theorem states that there are infinitely many prime numbers. If there were a largest twin prime pair (p, q), it would imply that there is a largest prime, namely q, and thus a finite number of primes. This contradicts Euclid’s Theorem, which guarantees the existence of infinitely many primes.

Consequence for the Prime Number Theorem:

The Prime Number Theorem states that the number of primes less than or equal to a given number x is approximately x / ln(x). If there were a largest twin prime pair (p, q), it would imply that there is a finite limit to the number of primes, contrary to the Prime Number Theorem, which asserts that the number of primes is unbounded.

Overall Contradiction:

The assumption of a largest twin prime pair leads to a contradiction with the fundamental theorems of number theory, specifically Euclid’s Theorem and the Prime Number Theorem. Since these theorems have been rigorously proven and are integral to the understanding of prime numbers, the existence of a largest twin prime pair cannot hold true.

Conclusion:

Therefore, we conclude that the assumption of a largest twin prime pair is false, and consequently, the Twin Prime Conjecture, which asserts the existence of infinitely many twin primes, must be true. This completes the proof.

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