Formula for primes on arvau

Wikipedia / Formula for primes

In number theory, a formula for primes is a formula that outputs prime numbers. Such formulas for calculating primes do exist; however, they are computationally very slow, compared to a simple algorithm for prime-finding. A number of constraints are known, showing what such a "formula" can and cannot be.

Formulas based on Wilson's theorem

A simple formula that produces all primes, albeit mostly interspersed by the prime number 2, is

for positive integer is the floor function, which rounds down to the nearest integer. The first few values of the function are 2, 2, 3, 2, 5, 2, 7, 2, 2, 2, 11...

The formula works because by Wilson's theorem is not prime, the first factor becomes zero and the formula produces the prime number 2..

In 1964, Willans gave the formula

. This formula reduces to

for which the prime-counting function; for example,

The articles What is an Answer? by Herbert Wilf (1982) and Formulas for Primes by Underwood Dudley (1983) have further discussion about the worthlessness of such formulas.

as a function:

is the monus operator.

Recurrence relations for primes

Gandhi's formula

is the Möbius function, the primorial. This formula should be seen as a recurrence relation.

given by Gandhi results from an application of a modified Sieve of Eratosthenes. There are analog recurrences where the process is done in a base.

was published:

denotes the fractional part.

It is based on a more clever use of the Sieve of Eratosthenes through the Chinese remainder theorem.

Golomb's formula

Inspired by Gandhi's proof, Golomb proved the following recurrence denotes the Riemann zeta function. It is based on the Euler product.

Prime-representing constants

The notion of continued fraction .

An alternative construction was given by Fridman et al. is the floor function given in the article is precise enough for equation () to generate the primes through 37, the twelfth prime.

The exact that generates all primes is given by the rapidly-converging series

that we know, the more primes equation () will generate. For example, we can use 25 terms in the series, using the 25 primes less than 100, to calculate the following more precise approximation:

This has enough digits for equation () to yield again the 25 primes less than 100.

Mills' formula

The first such formula known was established by , who proved that there exists a real number A such that, if

then

. If the Riemann hypothesis has a value of around 1.3063778838630806904686144926... and is known as Mills' constant., ... . This formula has no practical value, because there is no known way of calculating the constant without finding primes in the first place.

There is nothing special about the floor function such that

begins with 1.24055470525201424067... The first few primes generated are:

Without assuming the Riemann hypothesis, Elsholtz developed several prime-representing functions.

Wright's formula

A tetrationally growing prime-generating formula similar to Mills' comes from a theorem of E. M. Wright. He proved that there exists a real number α such that, if

then

. is even is a prime with 4932 digits. This sequence. Like Mills' formula, and for the same reasons, Wright's formula cannot be used to find primes.

Plouffe's formulas

In 2018 Simon Plouffe conjectured a set of formulas for primes. Similarly to the formula of Mills, they are of the form

, this gives 113, 367, 1607, 10177, 102217... a certain number between 0 and one half, Plouffe found that he could generate a sequence of 50 probable primes (with high probability of being prime). Presumably there exists an ε such that this formula will give an infinite sequence of actual prime numbers. The number of digits starts at 501 and increases by about 1% each time.

Prime formulas and polynomial functions

It is known that no non-constant polynomial function P(n) with integer coefficients exists that evaluates to a prime number for all integers n. The proof is as follows: suppose that such a polynomial existed. Then P(1) would evaluate to a prime p. But for any integer k cannot also be prime (as it would be divisible by p) unless it were p for all k is if the polynomial function is constant. The same reasoning shows an even stronger result: no non-constant polynomial function P(n) exists that evaluates to a prime number for almost all integers n.

Euler first noticed (in 1772) that the quadratic polynomial

, it produces a square number, the smallest composite number for this formula for divides , it divides too. Furthermore, since can be written as divides instead, it also divides . The phenomenon is related to the Ulam spiral, which is also implicitly quadratic, and the class number; this polynomial is related to the Heegner number (the lucky numbers of Euler), corresponding to other Heegner numbers.

Given a positive integer , there may be infinitely many such that the expression is always coprime to . The integer may be negative, in which case there is a delay before primes are produced.

Similarly, other polynomials (of higher degree) produces finite sequences of prime numbers. In 2010, Dress and Landreau found the following polynomial representing a record-breaking 58 primes at consecutive values: ranging from -42 to 15.

It is known, based on Dirichlet's theorem on arithmetic progressions produce infinitely many primes as long as and are relatively prime (though no such function will assume prime values for all values of ). Moreover, the Green–Tao theorem says that for any there exists a pair of a and b is prime for any . However, :

is prime for all from 0 through 26. It is not even known whether there exists a univariate polynomial of degree at least 2, that assumes an infinite number of values that are prime; see Bunyakovsky conjecture.

Rowland's prime-generating sequence

Another prime generator is defined by the recurrence relation

where denotes the greatest common divisor function. The sequence of differences starts with 1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 3, 1, 5, 3, ... . proved that this sequence contains only ones and prime numbers. However, it does not contain all the prime numbers, since the terms are always odd and so never equal to 2. The same paper conjectures that the sequence contains all odd primes: in fact, 587 is the smallest odd prime not appearing in the first 10,000 outcomes different from 1.

This recurrence is rather inefficient. In perspective, it is trivial to write an algorithm to generate all prime numbers (from the definition), and many more efficient algorithms are known. Thus, such recurrence relations are more a matter of curiosity than of practical use.

Prime-describing system of Diophantine equations

Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a system of Diophantine equations. found an explicit set of 14 Diophantine equations in 26 variables , such that a given number is prime if and only if that system has a solution in nonnegative integers:

can be used to produce a prime-generating polynomial inequality in 26 variables:

is a polynomial inequality in 26 variables, and the set of prime numbers is identical to the set of positive values taken on by the left-hand side as the variables range over the nonnegative integers.

A general theorem of Matiyasevich says that if a set is defined by a system of Diophantine equations, it can also be defined by a system of Diophantine equations in only 9 variables. Hence, there is a prime-generating polynomial inequality as above with only 10 variables. However, its degree is large (in the order of 1045). On the other hand, there also exists such a set of equations of degree only 4, but in 58 variables.