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, isfor positive integer, where 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 prime if and only if. Thus, when is prime, the first factor in the product becomes one, and the formula produces the prime number. But when is not prime, the first factor becomes zero and the formula produces the prime number 2. This formula is not an efficient way to generate prime numbers because evaluating requires about multiplications and reductions modulo.
In 1964, Willans gave the formula
for the th prime number. This formula reduces to
that is, it tautologically defines as the smallest integer for which the prime-counting function is at least. This formula is also not efficient. In addition to the appearance of, it computes by adding up copies of ; for example,
The articles What is an Answer? by Herbert Wilf and Formulas for Primes by Underwood Dudley have further discussion about the worthlessness of such formulas.
A shorter formula based on Wilson's theorem was given by J. P. Jones in 1975, using as a function:
Here, is the monus operator, defined as, and is defined to be.
Recurrence relations for primes
Gandhi's formula
In 1971, Gandhi proved thatwhere, is the Möbius function and runs through all dividers of, the primorial of.
This expression for given by Gandhi results from an application of the Sieve of Eratosthenes, which operates on the exponents of the powers of 1/2 in the sum.
This formula should be seen as a recurrence relation for the prime numbers, expressing in terms of.
Gandhi-Tréfeu's formula
In 2025, a new expression for, with only primes and without Möbius function, was published:
, where denotes the fractional part of.
This last expression of also relies on an application of the Sieve of Eratosthenes, but in a different way from that followed by Gandhi; its author establishes that the integers that escape the Sieve of Eratosthenes are of the form modulo , where , and deduces from this a generating expression for these integers.
Golomb's formula
Inspired by Gandhi's proof, Golomb proved the following recurrencewhere denotes the Riemann zeta function. It is based on the Euler product of zeta.Prime-representing constants
The notion of continued fraction can be used to define the constant from which we can recover the prime number sequence using the following recurrence relationship, and it follows that .An alternative construction was given by Fridman et al.. Given the constant , for, define the sequencewhere is the floor function. Then for,. The initial constant given in the article is precise enough for equation to generate the primes through 37, the twelfth prime.
The exact value of that generates all primes is given by the rapidly-converging series
where is the th prime, and is the primorial. The more digits of 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, ifthen
is a prime number for all positive integers. If the Riemann hypothesis is true, then the smallest such has a value of around 1.3063778838630806904686144926... and is known as Mills' constant. This value gives rise to the primes,,,.... Very little is known about the 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 in the formula. Tóth proved that there also exists a constant such that
is also prime-representing for.
In the case, the value of the constant begins with 1.24055470525201424067... The first few primes generated are:
Without assuming the Riemann hypothesis, Elsholtz developed several prime-representing functions similar to those of Mills. For example, if, then is prime for all positive integers. Similarly, if, then is prime for all positive integers.
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, ifthen
is prime for all. Wright gives the first seven decimal places of such a constant:. This value gives rise to the primes,, and. is even, and so is not prime. However, with,,, and are unchanged, while is a prime with 4932 digits. This sequence of primes cannot be extended beyond without knowing more digits of. 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 formwhere is the function rounding to the nearest integer. For example, with and, this gives 113, 367, 1607, 10177, 102217.... Using and with a certain number between 0 and one half, Plouffe found that he could generate a sequence of 50 probable primes. 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 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 would evaluate to a prime p, so. But for any integer k, also, so cannot also be prime unless it were p itself. But the only way for all k is if the polynomial function is constant.The same reasoning shows an even stronger result: no non-constant polynomial function P exists that evaluates to a prime number for almost all integers n.
Euler first noticed that the quadratic polynomial
is prime for the 40 integers n = 0, 1, 2,..., 39, with corresponding primes 41, 43, 47, 53, 61, 71,..., 1601. The differences between the terms are 2, 4, 6, 8, 10... For n = 40, it produces a square number, 1681, which is equal to 41 × 41, the smallest composite number for this formula for n ≥ 0. If 41 divides n, it divides P too. Furthermore, since P can be written as n + 41, if 41 divides n + 1 instead, it also divides P. 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. There are analogous polynomials for , corresponding to other Heegner numbers.
Given a positive integer S, there may be infinitely many c such that the expression n2 + n + c is always coprime to S. The integer c may be negative, in which case there is a delay before primes are produced.
Similarly, other polynomials produces finite sequences of prime numbers. In 2010, Dress and Landreau found the following polynomial representing a record-breaking 58 primes at consecutive values:More precisely, is prime for ranging from -42 to 15.
It is known, based on Dirichlet's theorem on arithmetic progressions, that linear polynomial functions produce infinitely many primes as long as a and b are relatively prime. Moreover, the Green–Tao theorem says that for any k there exists a pair of a and b, with the property that is prime for any n from 0 through k − 1. However, as of 2020 the best known result of such type is for k = 27:
is prime for all n 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 relationwhere gcd denotes the greatest common divisor of x and y. The sequence of differences an+1 − an 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 gcd 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, 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:The 14 equations 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. On the other hand, there also exists such a set of equations of degree only 4, but in 58 variables.