A formula for primes in mathematics is a mathematical formula generating the prime numbers, exactly and without exception. No easily-computable such formula is known. A great deal is known about what, more precisely, such a "formula" can and cannot be.

Prime formulas and polynomial functions

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It is known that no non-constant polynomial function P(n) exists that evaluates to a prime number for all integers n. The proof is simple: Suppose such a polynomial existed. Then P(1) would evaluate to a prime p, so $P(1)\; \backslash equiv\; 0\; \backslash pmod\; p$. But for any k, $P(1+kp)\; \backslash equiv\; 0\; \backslash pmod\; p$ also, so $P(1+kp)$ cannot also be prime (as it would be divisible by p) unless it were p itself, but the only way $P(1+kp)\; =\; P(1)$ for all k is if the polynomial function is constant.

Using more algebraic number theory, one can show an even stronger result: no non-constant polynomial function P(n) exists that evaluates to a prime number for almost all integers n.

The quadratic polynomial

P(n) = n² + n + 41

is prime for all non-negative integers less than 40. The primes for n = 0, 1, 2, 3... are 41, 43, 47, 53, 61, 71... 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. In fact if 41 divides n it divides P(n) too. The phenomenon is related to the Ulam spiral, which is also implicitly quadratic, and the class number.

There are polynomials in several variables whose positive values as the variables range over all positive integers are exactly the primes. The first such polynomial discovered was the following polynomial in 26 variables of degree 25:

$(k\; +\; 2)(1\; -\; (wz\; +\; h\; +\; j\; -\; q)^2$

- ((gk + 2g + k + 1)(h + j) + h - z)^2 - (2n + p + q + z - e)^2 - (16(k + 1)^3 (k + 2)(n + 1)^2 + 1 - f^2)^2 - (e^3(e + 2)(a + 1)^2 + 1 - o^2)^2 - ((a^2 - 1)y^2 + 1 - x^2)^2 - (16r^2y^4(a^2 - 1) + 1 - u^2)^2 - (((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2)^2 - (n + l + v - y)^2 - ((a^2 - 1)l^2 + 1 - m^2)^2 - (ai + k + 1 - l - i)^2 - (p + l(a - n - 1) + b(2an + 2a - n^2 - 2n - 2) - m)^2 - (q + y(a - p - 1) + s(2ap + 2a + p^2 - 2p - 2) - x)^2 - (z + pl(a - p) + t(2ap - p^2 - 1) - pm)^2)

It is known that there is such a polynomial in only 10 variables (but with a very high degree).^*

It is known, based on Dirichlet's theorem on arithmetic progressions, that linear polynomial functions $L(n)\; =\; an\; +\; b$ produce infinitely many primes as long as a and b are relatively prime (though no such function will assume prime values for all values of n). It is not known, however, whether there exists a polynomial of degree greater than or equal to 2 that assumes an infinite number of values that are prime.

Formula based on a system of Diophantine equations

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A set of Diophantine equations in 26 variables can be used to obtain primes: A given number k + 2 is prime iff. the following system of Diophantine equations has a solution in the natural numbers (Riesel, 1994):

$0\; =\; wz\; +\; h\; +\; j\; -\; q$

$0\; =\; (gk\; +\; 2g\; +\; k\; +\; 1)(h\; +\; j)\; +\; h\; -\; z$

$0\; =\; 16(k\; +\; 1)^3(k\; +\; 2)(n\; +\; 1)^2\; +\; 1\; -\; f^2$

$0\; =\; 2n\; +\; p\; +\; q\; +\; z\; -\; e$

$0\; =\; e^3(e\; +\; 2)(a\; +\; 1)^2\; +\; 1\; -\; o^2$

$0\; =\; (a^2\; -\; 1)y^2\; +\; 1\; -\; x^2$

$0\; =\; 16r^2y^4(a^2\; -\; 1)\; +\; 1\; -\; u^2$

$0\; =\; n\; +\; l\; +\; v\; -\; y$

$0\; =\; (a^2\; -\; 1)l^2\; +\; 1\; -\; m^2$

$0\; =\; ai\; +\; k\; +\; 1\; -\; l\; -\; i$

$0\; =\; ((a\; +\; u^2(u^2\; -\; a))^2\; -\; 1)(n\; +\; 4dy)^2\; +\; 1\; -\; (x\; +\; cu)^2$

$0\; =\; p\; +\; l(a\; -\; n\; -\; 1)\; +\; b(2an\; +\; 2a\; -\; n^2\; -\; 2n\; -\; 2)\; -\; m$

$0\; =\; q\; +\; y(a\; -\; p\; -\; 1)\; +\; s(2ap\; +\; 2p\; -\; p^2\; -\; 2p\; -\; 2)\; -\; x$

$0\; =\; z\; +\; pl(a\; -\; p)\; +\; t(2ap\; -\; p^2\; -\; 1)\; -\; pm.$

A general theorem of Matiyasevich says that the range of every recursively enumerable function may be given by the solutions to a set of Diophantine equations in 9 variables, but in general of very high degree. This implies the existence of a set of Diophantine equations in 9 variables that has the same property as the above one. J.P. Jones showed that there also exists such a set of equations of degree only 4, but in 58 variables. ^*.

Formulas using the floor function

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Using the floor function $\backslash lfloor\; x\backslash rfloor$ (defined to be the largest integer less than or equal to the real number x), one can construct several formulas for the nth prime.

Mills's formula

The first such formula known was established in 1947 by W. H. Mills, who proved that there exists a real number A such that

$\backslash lfloor\; A^\{3^\{n\}\}\backslash ;\backslash rfloor$

is a prime number for all positive integers n. The smallest such A has a value of around 1.3094... and is known as Mills's constant. This formula has no practical value, because very little is known about the constant (not even whether it is rational), and there is no known way of calculating the constant without finding primes in the first place.

Floor function formulas based on Wilson's theorem

By using Wilson's theorem, we may generate several other formulas, given below. These formulas also have little practical value: most primality tests are far more efficient.

In general, we may define

$\backslash pi(m)\; =\; \backslash sum\_\{j=2\}^m\; \backslash frac$

{ \sin^2 ( {\pi \over j} (j-1)!^2 ) } { \sin^2( {\pi \over j} ) } or, alternatively,

$\backslash pi(m)\; =\; \backslash sum\_\{j=2\}^m\; \backslash left\backslash lfloor\; \{(j-1)!\; +\; 1\; \backslash over\; j\}\; -\; \backslash left\backslash lfloor\{(j-1)!\; \backslash over\; j\}\backslash right\backslash rfloor\; \backslash right\backslash rfloor.$

These definitions are equivalent; π(m) is the number of primes less than or equal to m. The n-th prime number p_n can then be written as

$p\_n\; =\; 1\; +\; \backslash sum\_\{m=1\}^\{2^n\}\; \backslash left\backslash lfloor\; \backslash left\backslash lfloor\; \{\; n\; \backslash over\; 1\; +\; \backslash pi(m)\; \}\; \backslash right\backslash rfloor^\{1\; \backslash over\; n\}\; \backslash right\backslash rfloor.$

Another approach using the floor function

Another approach does not use factorials and Wilson's theorem, but also heavily employs the floor function (S. M. Ruiz 2000): first define

$\backslash pi(k)\; =\; k\; -\; 1\; +\; \backslash sum\_\{j=2\}^k\; \backslash left\backslash lfloor\; \{2\; \backslash over\; j\}\; \backslash left(1\; +\; \backslash sum\_\{s=1\}^\{\backslash left\backslash lfloor\backslash sqrt\{j\}\backslash right\backslash rfloor\}\; \backslash left(\backslash left\backslash lfloor\{\; j-1\; \backslash over\; s\}\backslash right\backslash rfloor\; -\; \backslash left\backslash lfloor\{j\; \backslash over\; s\}\backslash right\backslash rfloor\backslash right)\; \backslash right)\backslash right\backslash rfloor$

and then

$p\_n\; =\; 1\; +\; \backslash sum\_\{k=1\}^\{2(\backslash lfloor\; n\; \backslash ln(n)\backslash rfloor+1)\}\; \backslash left(1\; -\; \backslash left\backslash lfloor\{\backslash pi(k)\; \backslash over\; n\}\; \backslash right\backslash rfloor\backslash right).$

Other formulas

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The following function yields all the primes, and only primes, for non-negative integers n:

$f(n)\; =\; 2\; +\; (2n!\; \backslash ,\backslash operatorname\{mod\}\; (n+1)).$

This formula is based on Wilson's theorem; the number two is generated many times and all other primes are generated exactly once by this function. (In fact a prime p is generated for n = (p − 1) and 2 otherwise; that is, 2 is generated when n + 1 is composite.)

Formula for twin primes

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In the Proceedings of the Indian Academy of Sciences(Math. Sci.), Vol.92, No.1,September 1983 pp 49-52, an explicit formula for the (n+1)th Prime which is the same as nth Prime as n could be substituted for n+1 for a given integer. In other words, if one wants to find the 6th Prime Number, then the integer n=5. The paper also gives a formula for the (n+1)th Twin prime which is the same as the nth Twin Prime. Further, the paper gives formulae for number of Primes and Twin Primes less than any given Prime. The proof is written in a cumbersome format to avoid plagiarism and therefore it is difficult to follow. The following are the pages of the paper. http://www.ias.ac.in/jarch/mathsci/92/00000050.pdf http://www.ias.ac.in/jarch/mathsci/92/00000051.pdf http://www.ias.ac.in/jarch/mathsci/92/00000052.pdf http://www.ias.ac.in/jarch/mathsci/92/00000053.pdf http://www.ias.ac.in/jarch/mathsci/93/00000068.pdf

Even though the papers gives a formula for the number of Twin Primes less then a given Prime Pn, the Twin Pirme conjecture remains unsolved. The author Venu Atiyolil stopped research in Number Theory since the publication of the above paper and changed the field of interest. A hint, as to how to prove or disprove the twin Prime conjecture is mentioned in the paper.

References

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J.P. Jones, Hideo Wada, Daihachiro Sato and Douglas Wiens, Diophantine representation of the set of prime numbers, Amer. Math. Monthly 83 (1976), 449-464.

J.P. Jones, Universal diophantine equation, Journal of Symbolic Logic 47 (1982), 549-571.

External links

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• Sebastian Martin-Ruiz Prime Formulas http://www.primepuzzles.net/problems/prob_038.htm

Prime numbers