Innumber theory, aformula for primes is aformula generating theprime numbers, exactly and without exception. Formulas for calculating primes do exist; however, they are computationally very slow. A number of constraints are known, showing what such a "formula" can and cannot be.

A simple formula is

${\displaystyle f(n)=\lfloor \frac{n!mod(n+1)}{n}\rfloor (n-1)+2}$

for positiveinteger${\displaystyle n}$, where${\displaystyle \lfloor \rfloor }$ is thefloor function, which rounds down to the nearest integer.ByWilson's theorem,${\displaystyle n+1}$ is prime if and only if${\displaystyle n!\equiv n(\mathrm{mod}n+1)}$. Thus, when${\displaystyle n+1}$ is prime, the first factor in the product becomes one, and the formula produces the prime number${\displaystyle n+1}$. But when${\displaystyle n+1}$ is not prime, the first factor becomes zero and the formula produces the prime number 2.^[1] This formula is not an efficient way to generate prime numbers because evaluating${\displaystyle n!mod(n+1)}$ requires about${\displaystyle n-1}$ multiplications and reductions modulo${\displaystyle n+1}$.

In 1964, Willans gave the formula

${\displaystyle p_n=1+\sum _{i=1}^{2^n}\lfloor {\left(\frac{n}{\sum _{j=1}^i\lfloor {\left(\cos \frac{(j-1)!+1}{j}\pi \right)}^2\rfloor }\right)}^{1/n}\rfloor }$

for the${\displaystyle n}$th prime number${\displaystyle p_n}$.^[2] This formula reduces to^[3][4]

that is, it tautologically defines${\displaystyle p_n}$ as the smallest integer${\displaystyle m}$ for which theprime-counting function${\displaystyle \pi (m)}$ is at least${\displaystyle n}$. This formula is also not efficient. In addition to the appearance of${\displaystyle (j-1)!}$, it computes${\displaystyle p_n}$ by adding up${\displaystyle p_n}$ copies of${\displaystyle 1}$; for example,

${\displaystyle p_5=1+1+1+1+1+1+1+1+1+1+1+0+0+\cdots +0=11.}$

The articlesWhat is an Answer? byHerbert Wilf (1982)^[5] andFormulas for Primes byUnderwood Dudley (1983)^[6] 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${\displaystyle \mathrm{m}\mathrm{o}\mathrm{d}}$ as a function:^[7]

${\displaystyle p_n=\sum _{i=0}^{n^2}\left(1\dot{-}\left(\left(\sum _{j=0}^i(j\dot{-}1){!}^{2}modj\right)\dot{-}n\right)\right)}$.

Here,${\displaystyle \dot{-}}$ is themonus operator, and${\displaystyle xmod0}$ is defined to be${\displaystyle x}$.

Prunescu and Sauras-Altuzarra (2024)^[8] published the formula

,

where the arithmetic term for${\displaystyle n!}$ was given as

${\displaystyle n!=\lfloor \frac{{\left(2^{(n+1)(n+2)}\right)}^n}{\lfloor {\left(\frac{1+2^{2^{(n+1)(n+2)}}}{2^n}\right)}^{2^{(n+1)(n+2)}}\rfloor {mod2}^{2^{(n+1)(n+2)}}}\rfloor }$.

Notes

Prunescu & Shunia (2024)^[9] quoted that publication, but

• incorrectly stated that,
• misspelled (LaTeX error?) the arithmetic term for${\displaystyle n!}$

Because the set of primes is acomputably enumerable set, byMatiyasevich's theorem, it can be obtained from a system ofDiophantine equations.Jones et al. (1976) found an explicit set of 14 Diophantine equations in 26 variables, such that a given numberk + 2 is primeif and only if that system has a solution in nonnegative integers:^[10]

${\displaystyle {\alpha }_0=wz+h+j-q=0}$

${\displaystyle {\alpha }_1=(gk+2g+k+1)(h+j)+h-z=0}$

${\displaystyle {\alpha }_2=16(k+1{)}^3(k+2)(n+1{)}^2+1-f^2=0}$

${\displaystyle {\alpha }_3=2n+p+q+z-e=0}$

${\displaystyle {\alpha }_4=e^3(e+2)(a+1{)}^2+1-o^2=0}$

${\displaystyle {\alpha }_5=(a^2-1)y^2+1-x^2=0}$

${\displaystyle {\alpha }_6=16r^2{y}^4(a^2-1)+1-u^2=0}$

${\displaystyle {\alpha }_7=n+\ell +v-y=0}$

${\displaystyle {\alpha }_8=(a^2-1){\ell }^2+1-m^2=0}$

${\displaystyle {\alpha }_9=ai+k+1-\ell -i=0}$

${\displaystyle {\alpha }_{10}=((a+u^2(u^2-a){)}^2-1)(n+4dy{)}^2+1-(x+cu{)}^2=0}$

${\displaystyle {\alpha }_{11}=p+\ell (a-n-1)+b(2an+2a-n^2-2n-2)-m=0}$

${\displaystyle {\alpha }_{12}=q+y(a-p-1)+s(2ap+2a-p^2-2p-2)-x=0}$

${\displaystyle {\alpha }_{13}=z+p\ell (a-p)+t(2ap-p^2-1)-pm=0}$

The 14 equations${\displaystyle {\alpha }_0,\dots ,{\alpha }_{13}}$ can be used to produce a prime-generating polynomial inequality in 26 variables:

${\displaystyle (k+2)(1-{\alpha }_0^2-{\alpha }_1^2-\cdots -{\alpha }_{13}^2)>0.}$

That is,

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 variablesa,b, ...,z range over the nonnegative integers.

A general theorem ofMatiyasevich 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.^[11] 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.^[12]

The first such formula known was established by W. H. Mills (1947), who proved that there exists areal numberA such that, if

${\displaystyle d_n=A^{3^n}}$

then

${\displaystyle \lfloor d_n\rfloor =\lfloor A^{3^n}\rfloor }$

is a prime number for all positive integers${\displaystyle n}$.^[13] If theRiemann hypothesis is true, then the smallest such${\displaystyle A}$ has a value of around 1.3063778838630806904686144926... (sequenceA051021 in theOEIS) and is known asMills' constant.^[14] This value gives rise to the primes${\displaystyle \lfloor d_1\rfloor =2}$,${\displaystyle \lfloor d_2\rfloor =11}$,${\displaystyle \lfloor d_3\rfloor =1361}$, ... (sequenceA051254 in theOEIS). Very little is known about the constant${\displaystyle A}$ (not even whether it isrational). 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 thefloor function in the formula. Tóth proved that there also exists a constant${\displaystyle B}$ such that

${\displaystyle \lceil B^{r^n}\rceil }$

is also prime-representing for${\displaystyle r>2.106\dots }$.^[15]

In the case${\displaystyle r=3}$, the value of the constant${\displaystyle B}$ begins with 1.24055470525201424067... The first few primes generated are:

${\displaystyle 2,7,337,38272739,56062005704198360319209,}$

${\displaystyle 176199995814327287356671209104585864397055039072110696028654438846269,\dots }$

Without assuming the Riemann hypothesis, Elsholtz developed several prime-representingfunctions similar to those of Mills. For example, if${\displaystyle A=1.00536773279814724017\dots }$, then${\displaystyle \lfloor A^{{10}^{10n}}\rfloor }$ is prime for all positive integers${\displaystyle n}$. Similarly, if${\displaystyle A=3.8249998073439146171615551375\dots }$, then${\displaystyle \lfloor A^{3^{13n}}\rfloor }$ is prime for all positive integers${\displaystyle n}$.^[16]

Atetrationally growing prime-generating formula similar to Mills' comes from a theorem ofE. M. Wright. He proved that there exists a real numberα such that, if

${\displaystyle g_0=\alpha }$ and

${\displaystyle g_{n+1}=2^{g_n}}$ for${\displaystyle n\ge 0}$,

then

${\displaystyle \lfloor g_n\rfloor =\lfloor 2^{\dots {}^{2^{2^{\alpha }}}}\rfloor }$

is prime for all${\displaystyle n\ge 1}$.^[17] Wright gives the first seven decimal places of such a constant:${\displaystyle \alpha =1.9287800}$. This value gives rise to the primes${\displaystyle \lfloor g_1\rfloor =\lfloor 2^{\alpha }\rfloor =3}$,${\displaystyle \lfloor g_2\rfloor =13}$, and${\displaystyle \lfloor g_3\rfloor =16381}$.${\displaystyle \lfloor g_4\rfloor }$ iseven, and so is not prime. However, with${\displaystyle \alpha =1.9287800+8.2843\cdot {10}^{-4933}}$,${\displaystyle \lfloor g_1\rfloor }$,${\displaystyle \lfloor g_2\rfloor }$, and${\displaystyle \lfloor g_3\rfloor }$ are unchanged, while${\displaystyle \lfloor g_4\rfloor }$ is a prime with 4932 digits.^[18] Thissequence of primes cannot be extended beyond${\displaystyle \lfloor g_4\rfloor }$ without knowing more digits of${\displaystyle \alpha }$. Like Mills' formula, and for the same reasons, Wright's formula cannot be used to find primes.

Given the constant${\displaystyle f_1=2.920050977316\dots }$ (sequenceA249270 in theOEIS), for${\displaystyle n\ge 2}$, define the sequence

${\displaystyle f_n=\lfloor f_{n-1}\rfloor (f_{n-1}-\lfloor f_{n-1}\rfloor +1)}$

1

where${\displaystyle \lfloor \rfloor }$ is thefloor function.Then for${\displaystyle n\ge 1}$,${\displaystyle \lfloor f_n\rfloor }$ equals the${\displaystyle n}$th prime:${\displaystyle \lfloor f_1\rfloor =2}$,${\displaystyle \lfloor f_2\rfloor =3}$,${\displaystyle \lfloor f_3\rfloor =5}$, etc.^[19] The initial constant${\displaystyle f_1=2.920050977316}$ given in the article is precise enough for equation (1) to generate the primes through 37, the${\displaystyle 12}$th prime.

Theexact value of${\displaystyle f_1}$ that generatesall primes is given by the rapidly-convergingseries

${\displaystyle f_1=\sum _{n=1}^{\mathrm{\infty }}\frac{p_n-1}{P_n}=\frac{2-1}{1}+\frac{3-1}{2}+\frac{5-1}{2\cdot 3}+\frac{7-1}{2\cdot 3\cdot 5}+\cdots ,}$

where${\displaystyle p_n}$ is the${\displaystyle n}$th prime, and${\displaystyle P_n}$ is the product of all primes less than${\displaystyle p_n}$. The more digits of${\displaystyle f_1}$ that we know, the more primes equation (1) 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:

${\displaystyle f_1\simeq \mathrm{2.920050977316134712092562917112019.}}$

This has enough digits for equation (1) to yield again the 25 primes less than 100.

As with Mills' formula and Wright's formula above, in order to generate a longer list of primes, we need to start by knowing more digits of the initial constant,${\displaystyle f_1}$, which in this case requires a longer list of primes in its calculation.

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

${\displaystyle \left\{a_0^{r^n}\right\}}$

where${\displaystyle \{\}}$ is the function rounding to the nearest integer. For example, with${\displaystyle a_0\approx 43.80468771580293481}$ and${\displaystyle r=5/4}$, this gives 113, 367, 1607, 10177, 102217... (sequenceA323176 in theOEIS). Using${\displaystyle a_0={10}^{500}+961+\epsilon }$ and${\displaystyle r=1.01}$ with${\displaystyle \epsilon }$ a certain number between 0 and one half, Plouffe found that he could generate a sequence of 50probable 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.^[20][21]

It is known that no non-constant polynomial functionP(n) with integer coefficients exists that evaluates to a prime number for all integersn. The proof is as follows: suppose that such a polynomial existed. ThenP(1) would evaluate to a primep, so${\displaystyle P(1)\equiv 0(\mathrm{mod}p)}$. But for any integerk,${\displaystyle P(1+kp)\equiv 0(\mathrm{mod}p)}$ also, so${\displaystyle P(1+kp)}$ cannot also be prime (as it would be divisible byp) unless it werep itself. But the only way${\displaystyle P(1+kp)=P(1)=p}$ for allk is if the polynomial function is constant.The same reasoning shows an even stronger result: no non-constant polynomial functionP(n) exists that evaluates to a prime number foralmost all integersn.

Euler first noticed (in 1772) that thequadratic polynomial

${\displaystyle P(n)=n^2+n+41}$

is prime for the 40 integersn = 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... Forn = 40, it produces asquare number, 1681, which is equal to 41 × 41, the smallestcomposite number for this formula forn ≥ 0. If 41 dividesn, it dividesP(n) too. Furthermore, sinceP(n) can be written asn(n + 1) + 41, if 41 dividesn + 1 instead, it also dividesP(n). The phenomenon is related to theUlam spiral, which is also implicitly quadratic, and theclass number; this polynomial is related to theHeegner number${\displaystyle 163=4\cdot 41-1}$. There are analogous polynomials for${\displaystyle p=2,3,5,11\text{ and }17}$ (thelucky numbers of Euler), corresponding to other Heegner numbers.

Given a positive integerS, there may be infinitely manyc such that the expressionn² +n +c is always coprime toS. The integerc may be negative, in which case there is a delay before primes are produced.

It is known, based onDirichlet's theorem on arithmetic progressions, that linear polynomial functions${\displaystyle L(n)=an+b}$ produce infinitely many primes as long asa andb arerelatively prime (though no such function will assume prime values for all values ofn). Moreover, theGreen–Tao theorem says that for anyk there exists a pair ofa andb, with the property that${\displaystyle L(n)=an+b}$ is prime for anyn from 0 throughk − 1. However, as of 2020,^[update] the best known result of such type is fork = 27:

${\displaystyle 224584605939537911+18135696597948930n}$

is prime for alln from 0 through 26.^[22] It is not even known whether there exists aunivariate polynomial of degree at least 2, that assumes an infinite number of values that are prime; seeBunyakovsky conjecture.

Another prime generator is defined by therecurrence relation

${\displaystyle a_n=a_{n-1}+gcd(n,a_{n-1}),a_1=7,}$

where gcd(x,y) denotes thegreatest common divisor ofx andy. The sequence of differencesa_n+1 −a_n 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, ... (sequenceA132199 in theOEIS).Rowland (2008) proved that this sequence contains only ones and prime numbers. However, it does not contain all the prime numbers, since the terms gcd(n + 1,a_n) are alwaysodd 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.^[23]

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

Formula fromPaulo Ribenboim:${\displaystyle f(n)=\sum _{k=2}^{2^n}k\cdot \frac{1}{\left|k-2\sum _{i=1}^{k-1}\left[-\left\{\frac{k}{i}\right\}\right]\right|}\cdot \frac{1}{\left|n-\sum _{j=2}^k\frac{1}{\left|j-2\sum _{i=1}^{j-1}\left[-\left\{\frac{j}{i}\right\}\right]\right|}\right|}}$^[24][25]

• Prime number theorem

• Regimbal, Stephen (1975), "An explicit Formula for the k-th prime number",Mathematics Magazine,48 (4), Mathematical Association of America:230–232,doi:10.2307/2690354,JSTOR 2690354

• Venugopalan, A (September 1983), "Formula for primes, twinprimes, number of primes and number of twinprimes",Proceedings of the Indian Academy of Sciences—Mathematical Sciences,92 (1):49–52,doi:10.1007/BF02866907 (errata)

• Weisstein, Eric W., "Prime Formulas" ("Prime-Generating Polynomial") atMathWorld.

• Prime numbers

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