Formula for primes

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.

Prime formulas and polynomial functions

There is no general equation for finding prime numbers. 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 $P(1+kp)$ 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.

Euler first noticed (in 1772) that 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; this polynomial is related to the Heegner number $163=4\cdot 41-1$, and there are analogous polynomials for $p=2,3,5,11,{\mbox{ and }}17$, 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 = 25:

6171054912832631 + 81737658082080n is prime for all n from 0 to 24 (Andersen 2008).

It is not known whether there exists a univariate polynomial of degree at least 2 that assumes an infinite number of values that are prime.

Formula based on a system of Diophantine equations

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 = $16r^{2}y^{4}(a^{2}-1)+1-u^{2}$ = 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:

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

ie:

$(k+2)(1-$

$[wz+h+j-q]^{2}-$

$[(gk+2g+k+1)(h+j)+h-z]^{2}-$

$[16(k+1)^{3}(k+2)(n+1)^{2}+1-f^{2}]^{2}-$

$[2n+p+q+z-e]^{2}-$

$[e^{3}(e+2)(a+1)^{2}+1-o^{2}]^{2}-$

$[(a^{2}-1)y^{2}+1-x^{2}]^{2}-$

$[16r^{2}y^{4}(a^{2}-1)+1-u^{2}]^{2}-$

$[n+l+v-y]^{2}-$

$[(a^{2}-1)l^{2}+1-m^{2}]^{2}-$

$[ai+k+1-l-i]^{2}-$

$[((a+u^{2}(u^{2}-a))^{2}-1)(n+4dy)^{2}+1-(x+cu)^{2}]^{2}-$

$[p+l(a-n-1)+b(2an+2a-n^{2}-2n-2)-m]^{2}-$

$[q+y(a-p-1)+s(2ap+2a-p^{2}-2p-2)-x]^{2}-$

$[z+pl(a-p)+t(2ap-p^{2}-1)-pm]^{2})$

$>0$

is a polynomial inequality in 26 variables, and the set of prime numbers is identical to the set of positive values taken on by this polynomial inequality 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).

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 that take only prime numbers as values for all positive integers n.

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. 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.

Formulas using the floor function (or mod operator) 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.

Following tables has shown some proper formulas and their result.

There is an Exception in some formulas in the following tables which for number 3 and 4 they don't function as it is expected.

Formula	n is Prime	n is Composite
${(n-1)!+1} \over {n}$	Positive integer	Improper fraction
${(n-1)!} \over {n}$	Improper fraction	Positive integer
$((n-1)!+1)\,\operatorname {mod} (n)$	$0$	$1$
$(n-1)!\,\operatorname {mod} (n)$	$n-1$	$0$
$2(n-1)!\,\operatorname {mod} (n)$	$n-2$	$0$
$\left\lfloor {{(n-1)!+1} \over {n}}\right\rfloor$	${(n-1)!+1} \over {n}$	${(n-1)!} \over {n}$
$\left\lfloor {{(n-1)!} \over {n}}\right\rfloor$	${{(n-1)!+1} \over {n}}-1$	${(n-1)!} \over {n}$
$\left\lfloor {{2(n-1)!} \over {n}}\right\rfloor$	${{2(n-1)!+2} \over {n}}-1$	${2(n-1)!} \over {n}$

Next table has substituted n by n+1.

Formula	n+1 is Prime	n+1 is Composite
${n!+1} \over {n+1}$	Positive integer	Improper fraction
${n!} \over {n+1}$	Improper fraction	Positive integer
$(n!+1)\,\operatorname {mod} (n+1)$	$0$	$1$
$n!\,\operatorname {mod} (n+1)$	$n$	$0$
$2(n!)\,\operatorname {mod} (n+1)$	$n-1$	$0$
$\left\lfloor {{n!+1} \over {n+1}}\right\rfloor$	${n!+1} \over {n+1}$	${n!} \over {n+1}$
$\left\lfloor {{n!} \over {n+1}}\right\rfloor$	${{n!+1} \over {n+1}}-1$	${n!} \over {n+1}$
$\left\lfloor {{2(n!)} \over {n+1}}\right\rfloor$	${{2(n!)+2} \over {n+1}}-1$	${2(n!)} \over {n+1}$

By using first formula of the above table , if

${\begin{aligned}r\colon \mathbb {N} &\to \mathbb {R} ^{+}\\n&\mapsto {(n-1)!+1 \over n}\end{aligned}}$

that is:

$r(n)={(n-1)!+1 \over n}$

then , for non-negative integers n ; r is positive integer when n is prime ; and r is improper fraction when n is composite:

In order to produce a prime test function , the following function called IsInteger can be defined:

${\begin{aligned}IsInteger\colon \mathbb {R} ^{+}&\to \{0,1\}\\x&\mapsto {\begin{cases}1,&{\mbox{if }}x\in \mathbb {Z} \\0,&{\mbox{if }}x\notin \mathbb {Z} \end{cases}}\end{aligned}}$

The following functions return 0 when x is not integer and return 1 when x is integer.

$IsInteger(x)=\left(\left\lfloor {x}\right\rfloor +\left\lfloor {-x}\right\rfloor +1\right)$

$IsInteger(n)=\left(\left\lfloor {{\left\lfloor {x}\right\rfloor } \over {x}}\right\rfloor \right)$

$IsInteger(n)=\left\lfloor \cos ^{2}\left({x}\pi \right)\right\rfloor$ : formula by C. P. Willans (Bowyer n.d.)

For example the following functions return 0 when n is composite and return 1 when n is prime.

${\begin{aligned}IsPrime\colon \mathbb {N} &\to \{0,1\}\\n&\mapsto {\begin{cases}1,&{\mbox{if }}n\in \mathbb {P} \\0,&{\mbox{if }}n\notin \mathbb {P} \end{cases}}\end{aligned}}$

And Examples are:

$IsPrime(n)=IsInteger(r(n))={\begin{cases}1,&{\mbox{if }}n\in \mathbb {P} \\0,&{\mbox{if }}n\notin \mathbb {P} \end{cases}}$

$IsPrime(n)={\frac {\sin ^{2}({\pi \over n}(n-1)!)}{\sin ^{2}({\pi \over n})}}$

Some formulas use above IsPrime(n) function to generate all prime numbers but not in an ascending order and some other, use them to build a formula for π(m) or p_n ; π(m) is the number of primes less than or equal to m.

For Example , the following functions yield all the primes, and only primes, for non-negative integers n:

$f(n)=2\left({n \over 2}\right)^{\left(\left\lfloor {{\left\lfloor {r(n)}\right\rfloor } \over {r(n)}}\right\rfloor \right)}=2\left({n \over 2}\right)^{IsPrime(n)}$

$f(n)=2+(n-2)\left(\left\lfloor {r(n)}\right\rfloor +\left\lfloor {-r(n)}\right\rfloor +1\right)=2+(n-2)IsInteger(r(n))=2+(n-2)IsPrime(n).$

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

A prime p is generated for n = p and 2 otherwise; that is, 2 is generated when n is composite.

In order to produce new prime numbers more often,we can substitute n by $2x+1$ or $2x-1$ which only produces odd numbers.

Also the following substitution will omit even numbers ,the number 5 and it's multiples.

$n:=10\left\lfloor {x \over {10}}\right\rfloor +5-3(-1)^{\left(\left\lfloor {x \over {2}}\right\rfloor \right)}-(-1)^{x}$

In general, we may define

$\pi (m)=\sum _{j=2}^{m}IsPrime(j)$

Examples are:

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

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.

a is n-th prime number if and only if these two conditions are satisfied:

1) a is prime meaning that IsPrime(a)=1

2) The number of primes less that or equal to a is n that is π(a)=n

The following function returns 1 when a is n-th prime else returns 0:

$IsnthPrime(a,n)=IsPrime(a)*notZero(\pi (a)-n)={\begin{cases}1,&{\mbox{if }}a=P(n)\\0,&{\mbox{if }}a\neq P(n)\end{cases}}$

Where

$notZero(x)={\begin{cases}1,&{\mbox{if }}x=0\\0,&{\mbox{if }}x\neq 0\end{cases}}$

The n-th prime number p_n can then be written as

$p(n)=\sum _{m=1}^{2^{n}}{m}*{IsnthPrime(m,n)}$

Examples are:

$p_{n}=1+\sum _{k=1}^{2^{n}}\left\lfloor \left\lfloor {n \over 1+\pi (k)}\right\rfloor ^{1 \over n}\right\rfloor .$

$p_{n}=1+\sum _{k=1}^{2(\lfloor n\ln(n)\rfloor +1)}\left(1-\left\lfloor {\pi (k) \over n}\right\rfloor \right).$ by Sebastián Martín-Ruiz.

Note the following equality:

$\left\lfloor \left\lfloor {n \over 1+\pi (k)}\right\rfloor ^{1 \over n}\right\rfloor ={\begin{cases}1,&{\mbox{if }}\pi (k)<=n-1\\0,&{\mbox{if }}\pi (k)>n-1\end{cases}}$

Another approach using the floor function

Another approach, by Sebastián Martín-Ruiz, does not use factorials and Wilson's theorem, but also heavily employs the floor function (Rivera n.d.):

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

Recurrence relation

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 a_{n + 1} − a_n starts with 1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1 (sequence A132199 in the OEIS). Rowlands (2008) proved that this sequence contains only ones and prime numbers.

See also

• FRACTRAN

References

• Andersen, Jens Kruse (2008), Primes in Arithmetic Progression Records, retrieved 2009-01-27.
• Bowyer, Adrian (n.d.), Formulae for Primes, retrieved 2008-07-09{{citation}}: CS1 maint: year (link).
• Jones, James P.; Sato, Daihachiro; Wada, Hideo; Wiens, Douglas (1976), "Diophantine representation of the set of prime numbers", American Mathematical Monthly, 83: 449–464, doi:10.2307/2318339.
• Jones, James P. (1982), "Universal diophantine equation", Journal of Symbolic Logic, 47: 549–571.
• Rivera, Carlos (n.d.), Problem 38. Sebastián Martín Ruiz- Prime formulas, retrieved 2008-07-09{{citation}}: CS1 maint: year (link).
• Rowland, Eric S. (2008), "A Natural Prime-Generating Recurrence", Journal of Integer Sequences, 11: 08.2.8, ISSN 1530-7638.

External links

• Weisstein, Eric W. "Prime Formulas". MathWorld.
• Weisstein, Eric W. "Prime-Generating Polynomial". MathWorld.
• Weisstein, Eric W. "Mill's Constant". MathWorld.
• A Venugopalan. Formula for primes, twinprimes, number of primes and number of twinprimes. Proceedings of the Indian Academy of Sciences – Mathematical Sciences, Vol. 92, No 1, September 1983, pp. 49-52. Page 49, 50, 51, 52, errata.