Twin Prime Conjecture
The twin prime conjecture asks whether there are infinitely many pairs of prime numbers that differ by exactly 2. The statement is short enough to explain in one sentence. Proving it has turned out to be one of the most persistent open problems in number theory.
This page gives the cleanest nontechnical overview of the conjecture itself: the exact statement, what infinity means in this context, why the problem is so hard, what nearby progress has been proved, and what still remains outside theorem territory.
The twin prime conjecture claims that there are infinitely many twin-prime pairs. A twin-prime pair is a pair of prime numbers with a gap of exactly 2, such as (11, 13), (17, 19), or (29, 31). The conjecture says that this exact gap-2 pattern never runs out permanently. It is not a claim about one long streak near the beginning of the number line. It is a claim about endless recurrence no matter how far you go.
Infinite does not mean that twin primes are common or evenly spaced. It means that no matter how far you go along the number line, there should still be more twin-prime pairs beyond that point. For example, if someone checked all twin primes below one million, the conjecture would still be asking whether more pairs exist above one million, above one billion, and beyond every other finite cutoff. The claim is about endless continuation, not regular frequency.
The statement only mentions prime pairs separated by 2, but a proof would need to control infinitely many local divisibility constraints at once. Numbers that look promising can be ruined by divisibility by 3, 5, 7, or larger primes, and those obstructions interact across the whole number line. That is why the conjecture is harder than proving that primes never end, and harder than proving that some small gap recurs infinitely often without naming the exact gap 2.
Modern results show that primes come within some bounded distance of each other infinitely often. Zhang's breakthrough, and the later Polymath and Maynard-Tao advances, prove that small prime gaps recur at arbitrarily large scales. This is major progress, but it is still weaker than proving that the exact gap of 2 occurs infinitely often. The difference between bounded gaps and twin primes is one of the most important distinctions on the site.
No theorem currently proves that infinitely many gap-2 prime pairs exist. This is the key misconception to avoid. Seeing many twin primes in a finite range, or even proving that some bounded gap recurs infinitely often, does not settle the exact twin-prime conjecture. The conjecture remains open because the final step from 'some fixed gap' to 'the specific gap 2' has not been proved.
The expectation comes from strong heuristics, statistical models, and enormous finite computation, not from a finished proof. Twin primes appear often enough in practice to make endless continuation plausible, and conjectural models such as Hardy-Littlewood predict that they should keep appearing, although more rarely, as numbers grow. On this site that expectation is always treated as heuristic or conjectural, not as established theorem-level fact.
The site cannot prove or disprove the conjecture, but it can help you see why the pattern is compelling and why finite evidence is not the same thing as a proof. The Lab makes gap-2 structure visible, Explorer lets you inspect concrete examples row by row, and Analysis summarizes how pairs, centers, and gaps behave in a chosen range. Theory and the Zhang page then connect those observations back to the larger mathematical story.
Use these links to keep reading or jump back into the live number views.
Use the shorter page when you want a clean yes-or-no explanation before reading the larger conjecture context.
The Zhang page explains the most famous modern theorem that moved the field forward without finishing the conjecture.
Use the prime-gaps page when you want the conjecture placed inside the larger language of prime spacing.
Use the Lab when you want the conjecture tied back to visible structure.
Explorer is the clearest place to move from the conjecture statement to concrete gap-2 pairs in a live range.