Os primeiros 20 primos gêmeos

The first precise claim about primes two apart belongs to Alphonse de Polignac, who proposed in 1849 that every even number is the gap between infinitely many pairs of consecutive primes. The case of gap 2 is the twin prime conjecture. No case of Polignac's conjecture — not one — has ever been proved.

The first real theorem came from Norway. In 1915 Viggo Brun showed that the sum of the reciprocals of the twin primes converges, in sharp contrast to the sum over all primes, which Euler had shown diverges in 1737. Convergence means twin primes are sparse in a precise, quantifiable sense, and the machinery Brun built to prove it — the Brun sieve — became the foundation of modern sieve theory. The limiting value, now called Brun's constant, is about 1.9022; it is known only from computation, not in closed form.

Hardy and Littlewood pushed in the opposite direction in 1923, conjecturing that the number of twin pairs below x grows like 2C₂x/(ln x)², with C₂ ≈ 0.6602. The formula tracks the computed counts closely, and remains a conjecture.

Then came 2013. On 17 April, Yitang Zhang — then a lecturer at the University of New Hampshire, and largely unknown in the field — announced a proof that some even number below 70 million is the gap between infinitely many pairs of consecutive primes. It was the first finite bound of any kind, and it arrived from nowhere. Terence Tao organised a Polymath project to drive the bound down in public; James Maynard, then a postdoc, found an independent and simpler route at almost the same moment. Within a year of Zhang's announcement the bound stood at 246, where it still stands. Assuming the Elliott–Halberstam conjecture it would be 12, and assuming a generalised form of it, 6. Getting it to 2 would settle the twin prime conjecture.

Propriedades principais

• A twin prime pair is two primes differing by 2: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), …
• Apart from (3, 5), every pair has the form (6n−1, 6n+1), so the number between the twins is always a multiple of 6.
• 5 is the only prime belonging to two pairs, since it sits in both (3, 5) and (5, 7).
• The sum of any twin prime pair other than (3, 5) is divisible by 12.
• There are 8 pairs below 100, 35 below 1,000, 205 below 10,000, 1,224 below 100,000, 8,169 below one million and 58,980 below ten million.
• Brun proved in 1915 that the sum of the reciprocals of the twin primes converges, to roughly 1.9022 — so twin primes are much thinner on the ground than primes.
• Whether there are infinitely many twin pairs is unknown. The best result, from Zhang in 2013 and then Maynard, Tao and the Polymath collaboration, is that some gap no larger than 246 recurs infinitely often.
• The largest pair known was announced in September 2016 by PrimeGrid: 2996863034895 × 2^1290000 ± 1, with 388,342 digits each.