Formula for primes

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In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are known, showing what such a "formula" can and cannot be.

Prime formulas and polynomial functions[edit]

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 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+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.

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

P(n) = n2n + 41

is prime for all positive integers less than 41. The primes for n = 1, 2, 3... are 41, 43, 47, 53, 61, 71... The differences between the terms are 2, 4, 6, 8, 10... For n = 41, it produces a square number, 1681, which is equal to 41×41, the smallest composite number for this formula. 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; this polynomial is related to the Heegner number 163=4\cdot 41-1, and there are analogous polynomials for p=2, 3, 5, 11, \text{ and } 17, corresponding to other Heegner numbers.

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). Moreover, the Green–Tao theorem says that for any k there exists a pair of a and b with the property that L(n) = an+b is prime for any n from 0 to k − 1. However, the best known result of such type is for k = 26 (by Benoãt Perichon of France):

43142746595714191 + 5283234035979900n is prime for all n from 0 to 25 (Perichon 2010).

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.

Formula based on a system of Diophantine equations[edit]

A system of 14 Diophantine equations in 26 variables can be used to obtain a Diophantine representation of the set of all 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:

α0 =  wz + h + j - q = 0
α1 =  (gk + 2g + k + 1)(h + j) + h - z = 0
α2 =  16(k + 1)^3(k + 2)(n + 1)^2 + 1 - f^2 = 0
α3 =  2n + p + q + z - e = 0
α4 =  e^3(e + 2)(a + 1)^2 + 1 - o^2 = 0
α5 =  (a^2 - 1)y^2 + 1 - x^2 = 0
α6 =  16r^2y^4(a^2 - 1) + 1 - u^2 = 0
α7 =  n + l + v - y = 0
α8 =  (a^2 - 1)l^2 + 1 - m^2 = 0
α9 =  ai + k + 1 - l - i = 0
α10 =  ((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2 = 0
α11 =  p + l(a - n - 1) + b(2an + 2a - n^2 - 2n - 2) - m = 0
α12 =  q + y(a - p - 1) + s(2ap + 2a - p^2 - 2p - 2) - x = 0
α13 =  z + pl(a - p) + t(2ap - p^2 - 1) - pm = 0

The 14 equations α0, …, α13 can be used to produce a prime-generating polynomial inequality in 26 variables:

 (k+2)(1-\alpha_0^2-\alpha_1^2-\cdots-\alpha_{13}^2) > 0

i.e.:

 (k+2) (1 -
 [wz + h + j - q]^2 -
 [(gk + 2g + k + 1)(h + j) + h - z]^2 -
 [16(k + 1)^3(k + 2)(n + 1)^2 + 1 - f^2]^2 -
 [2n + p + q + z - e]^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 -
 [n + l + v - y]^2 -
 [(a^2 - 1)l^2 + 1 - m^2]^2 -
 [ai + k + 1 - l - i]^2 -
 [((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2]^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)
 > 0

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 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 (Matiyasevich 1999). 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)

Mills' formula[edit]

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. If the Riemann hypothesis is true, then the smallest such A has a value of around 1.3063... and is known as Mills' 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.

Recurrence relation[edit]

Another prime generator is defined by the recurrence relation

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

where gcd(x, y) denotes the greatest common divisor of x and y. The sequence of differences an + 1an starts with 1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1 (sequence A132199 in OEIS). Rowland (2008) proved that this sequence contains only ones and prime numbers.

See also[edit]

References[edit]

External links[edit]