Twin Primes Conjecture

Two prime numbers are called twins if one of them is equal to the other plus two units. Thus, the prime numbers 3 and 5 form a pair of twin primes. Other examples of twin prime pairs are 11 and 13 or 41 and 43.

The twin prime conjecture postulates the existence of infinitely many pairs of twin primes. Since it is a conjecture, it is still unproven.

The conjecture has been investigated by many number theorists. Most mathematicians believe that the conjecture is true, and they rely on numerical evidence and heuristic reasoning about the probability distribution of prime numbers.

In 1849, Alphonse de Polignac formulated a more general conjecture according to which, for every natural number k there are infinitely many pairs of primes whose difference is 2k. The case k=1 is the Twin Primes Conjecture.

Partial results

In 1940, Erdös showed that there exists a constant c < 1 and infinitely many primes p such that p - p < c ln(p), where p denotes the prime number that follows p. This result was successively improved: in 1986 Maier showed that a constant c < 0.25. Daniel Goldston, János Pintz and Cem Yildirim made a breakthrough in 2005 by proving that the result is valid for every constant c>0.

In 1973, Jing-run Chen published a proof that there are infinitely many primes p such that p+2 is a product of at most two factors cousins. To achieve this result he relied on the so-called riddle theory, and managed to treat the twin prime conjecture and Goldbach's conjecture in a similar way.

Hardy–Littlewood conjecture

There is also a generalization of the twin prime conjecture, known as the Hardy-Littlewood conjecture, about the distribution of twin primes, analogous to the prime number theorem. Let π₂(x) denote the number of primes p less than x such that p+2 is also prime. Define the constant of prime numbers C₂ as the following Euler product

C2= p≥ ≥ 3p(p− − 2)(p− − 1)2≈ ≈ 0,66016118158468695739278121100145...{displaystyle C_{2}=prod _{pgeq 3}{frac {p(p-2)}{(p-1)^{2}}}approx 0,6601181584686957398121100145...}

for primes greater than or equal to three. The conjecture says that:

in the same sense that the quotient of the two expressions tends to 1 as x tends to infinity; that is to say:

limx→ → +∞ ∞ π π 2(x)2C2∫ ∫ 2xdt(ln t)2=1{displaystyle lim _{xto +infty }{frac {pi _{2}(x)}{displaystyle 2C_{2}int _{2}{x}{dt over (ln t)^{2}}}}}{1}

This conjecture can be justified (but not proven) by assuming, informally speaking, that the event that n is not divisible by p and the event that n+2 is not divisible by p are statistically dependent only to the extent that the fact that n is not divisible by p makes p|n+2 an event between p-1 equally likely events, and not one event among p equally likely events. The numerical evidence behind the Hardy-Littlewood conjecture is certainly impressive.