There exist a variety of formulas for either producing the 
th prime as a function of 
or taking on only prime values. However, all such formulas
require either extremely accurate knowledge of some unknown constant, or effectively
require knowledge of the primes ahead of time in order to use the formula (Dudley
1969; Ribenboim 1996, p. 186). There also exist simple prime-generating
polynomials that generate only primes for the first (possibly large) number of
integer values.
There are also many beautiful formulas involving prime sums and prime products that can be done in closed form.
Considering examples of formulas that produce only prime numbers (although not necessarily the complete set of prime numbers 
), there exists a constant 
(OEIS A051021)
known as Mills' constant such that
 |
(1)
|
where 
is the floor function, is prime for all 
(Ribenboim 1996, p. 186). The first few values
of 
are 2, 11, 1361, 2521008887, ...
(OEIS A051254). It is not known if 
is irrational. There
also exists a constant 
(OEIS A086238)
such that
 |
(2)
|
(Wright 1951; Ribenboim 1996, p. 186) is prime for every 
. The first few values of 
are 3, 13, 16381, ... (OEIS A016104).
In the case of both 
and 
,
the numbers at 
grow so rapidly that an extremely precise value of 
or 
is needed in order to obtain the correct value, and values
for 
are effectively incomputable.
Explicit formulas exist for the 
th prime both as a function of 
and in terms of the primes 2, ..., 
(Hardy and Wright 1979, pp. 5-6, 344-345, and 414;
Guy 1994, pp. 36-41), a number of which are given below. However, it should
again be emphasized that these formulas are extremely inefficient, and in many (if
not all) cases, simply performing an efficient sieving would yield the primes much
more quickly and efficiently.
A prime-generating formula sometimes known as Willans' formula can be constructed as follows. Let
 |  | ![|_cos^2[pi((j-1)!+1)/j]_|](/images/equations/PrimeFormulas/Inline23.svg) |
(3)
|
 |  |  |
(4)
|
for 
an integer, where 
is again the floor function. This formula is a
consequence of Wilson's theorem and conceals the
prime numbers 
as those for which 
,
i.e., the values of 
are 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, ... (OEIS A080339).
Then
 |
(5)
|
and
 |  |  |
(6)
|
 |  |  |
(7)
|
where 
is the prime counting function (Willans
1964; Havil 2003, pp. 168-169).
Gandhi gave the formula in which 
is the unique integer such that
 |
(8)
|
where 
is the primorial function (Gandhi 1971, Eynden 1972,
Golomb 1974) and 
is the Möbius function. It is also true
that
 |
(9)
|
(Ribenboim 1996, pp. 180-182). Note that the number of terms in the summation to obtain the 
th
prime is 
,
so these formulas turn out not to be practical in the study of primes.
Hardy and Wright (1979, p. 414) give the formula
 |
(10)
|
for 
,
where
![f(x,y)={0 for x=y; 1/2[1+(x-y)/(|x-y|)] for x!=y](/images/equations/PrimeFormulas/NumberedEquation7.svg) |
(11)
|
and an "elementary" formula for the prime counting function is given by
![pi(n)=-1+sum_(j=3)^n[(j-2)!-j|_((j-2)!)/j_|]](/images/equations/PrimeFormulas/NumberedEquation8.svg) |
(12)
|
(correcting a sign error), where 
is the floor function.
A double sum for the 
th
prime 
is
![p_n=1+sum_(k=1)^(2(|_nlnn_|+1))[1-|_(sum_(j=2)^(k)1+|_s(j)_|)/n_|],](/images/equations/PrimeFormulas/NumberedEquation9.svg) |
(13)
|
where
 |
(14)
|
(Ruiz 2000).
An asymptotic formula for 
is given by
 |
(15)
|
(Cipolla 1902). This asymptotic expansion is the inverse of the logarithmic integral 
obtained by series reversion, where the inversion of 
gives 
because the prime number
theorem says that 
,
where 
is the prime counting function and the
inverse of 
is 
in the sense that 
.
However, the formula oscillates a great deal as illustrated above, where 
is the difference between the actual 
th prime and that given by the Cipolla formula. Interestingly,
truncating at 
gives the refined form of Rosser's theorem, which
is a strict inequality on 
. Salvy (1994) deals with more general cases.
B. M. Bredihin proved that
 |
(16)
|
takes prime values for infinitely many integral pairs 
(Honsberger 1976, p. 30). For example, 
, 
, 
, and so on. The primes of this form are 3, 11, 19,
41, 53, 59, 73, 83, 101, 107, 131, 137, 149, 163, ... (OEIS A079544;
Mitrinović and Sándor 1995, p. 11). The values of 
and 
for which 
is prime are plotted above, showing some interesting
patterns.
It is in general not difficult to artificially construct formulas that always generate prime numbers. For example, consider the formula
![f(x,y)=1/2(y-1)[|B^2(x,y)-1|-(B^2(x,y)-1)]+2,](/images/equations/PrimeFormulas/NumberedEquation13.svg) |
(17)
|
where
 |
(18)
|
is the factorial,
and 
and 
are positive integers (Honsberger 1976, p. 33).
This will always have 
and hence yield the value 2 unless ![x=[(p-1)!+1]/p](/images/equations/PrimeFormulas/Inline72.svg)
and 
, in which case it simplifies to
 |
(19)
|
The formula therefore generates odd primes exactly once each (Honsberger 1976, p. 33) at the values for which the Wilson
quotient ![x=[(p-1)!+1]/p](/images/equations/PrimeFormulas/Inline74.svg)
is an integer, i.e., 1, 1, 5, 103, 329891, 36846277, 1230752346353, ... (OEIS A007619).
The FRACTRAN game (Guy 1983, Conway and Guy 1996, p. 147) provides an unexpected means of generating the prime numbers based on 14 fractions, but it is actually just a concealed version of a sieve.
numero primo." Napoli
Rend. 3, 132-166, 1902.Conway, J. H. and Guy, R. K.
Prime is Greater than 
for 
." Math. Comput. 68, 411-415, 1999.Eynden,
C. V. "A Proof of Gandhi's Formula for the 
th Prime." Amer. Math. Monthly 79, 625,
1972.Finch, S. R. "Hardy-Littlewood Constants." §2.1
in 
th Prime." Proc. Washington State University Conferences
on Number Theory. pp. 96-107, 1971.Gardner, M. "Patterns
and Primes." Ch. 9 in 
th Prime Number." Math. Gaz. 48,
413-415, 1964.Wright, E. M. "A Prime-Representing Function."
Amer. Math. Monthly 58, 616-618, 1951.