Formula for primes

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In mathematics, a formula for primes is a 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.

1 Prime formulas and polynomial functions
2 Formula based on a system of Diophantine equations
3 Formulas using the floor function
4 Other formulas
5 References
6 External links

[edit] Prime formulas and polynomial functions

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 $P(1) \equiv 0 \pmod p$ . But for any k, $P(1+kp) \equiv 0 \pmod p$ also, so $P (1 + k p)$ cannot also be prime (as it would be divisible by p) unless it were p itself, but the only way $P (1 + k p) = P (1)$ 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.

It is known, based on Dirichlet's theorem on arithmetic progressions, that linear polynomial functions $L (n) = a n + b$ 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 univariate polynomial of degree greater at least 2 that assumes an infinite number of values that are prime.

[edit] Formula based on a system of Diophantine equations

A set of Diophantine equations in 26 variables can be used to obtain primes. Jones et al. (1976) proved that a given number k + 2 is prime if and only if the following system of 14 Diophantine equations has a solution in the natural numbers:

0 = w z + h + j - q

0 = (g k + 2 g + k + 1)(h + j) + h - z

0 = 16(k + 1) 3 (k + 2)(n + 1) 2 + 1 - f 2

0 = 2 n + 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 = 16 r 2 y 4 (a 2 - 1) + 1 - u 2

0 = n + l + v - y

0 = (a 2 - 1) l 2 + 1 - m 2

0 = a i + k + 1 - l - i

0 = ((a + u 2 (u 2 - a)) 2 - 1)(n + 4 d y) 2 + 1 - (x + c u) 2

0 = p + l (a - n - 1) + b (2 a n + 2 a - n 2 - 2 n - 2) - m

0 = q + y (a - p - 1) + s (2 a p + 2 p - p 2 - 2 p - 2) - x

0 = z + p l (a - p) + t (2 a p - p 2 - 1) - p m .

This can be used to produce a prime-generating polynomial. Denote the right-hand side of the above equations by α₁, …, α₁₄. Then

$(k-2)(1-\alpha_1^2+\alpha_2^2+\cdots+\alpha_{14}^2)$

is a polynomial in 26 variables, and the set of prime numbers is identical to the set of positive values taken on by this polynomial as the variables a, b, …, z 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 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 (Jones 1982).

[edit] Formulas using the floor function

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

[edit] 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

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

is a prime number for all positive integers n. The smallest such A has a value of around 1.3063... 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.

[edit] 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

$\pi(m) = \sum_{j=2}^m \frac { \sin^2 ( {\pi \over j} (j-1)!^2 ) } { \sin^2( {\pi \over j} ) }$

or, alternatively,

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

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

$p_n = 1 + \sum_{m=1}^{2^n} \left\lfloor \left\lfloor { n \over 1 + \pi(m) } \right\rfloor^{1 \over n} \right\rfloor.$

[edit] 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

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

and then

$p_n = 1 + \sum_{k=1}^{2(\lfloor n \ln(n)\rfloor+1)} \left(1 - \left\lfloor{\pi(k) \over n} \right\rfloor\right).$

[edit] Other formulas

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

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

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.)

[edit] References

Jones, James P.; Daihachiro Sato & Hideo Wada et al. (1976), "Diophantine representation of the set of prime numbers", American Mathematical Monthly 83: 449–464.
Jones, James P. (1982), "Universal diophantine equation", Journal of Symbolic Logic 47: 549–571.