Editing Floor and ceiling functions (section)

===Formulas for prime numbers===
The floor function appears in several formulas characterizing prime numbers.  For example, since
<math display=block>\left\lfloor\frac{n}{m} \right\rfloor -\left\lfloor\frac{n-1}{m}\right\rfloor = \begin{cases}
1 &\text{if } m \text{ divides } n \\
0 &\text{otherwise},
\end{cases}</math>
it follows that a positive integer ''n'' is a prime [[if and only if]]<ref>Crandall & Pomerance, Ex. 1.3, p. 46. The infinite upper limit of the sum can be replaced with ''n''. An equivalent condition is ''n''&nbsp;&gt;&nbsp;1 is prime if and only if
<math display=block>\sum_{m=1}^{\lfloor \sqrt n \rfloor} \left(\left\lfloor\frac{n}{m}\right\rfloor-\left\lfloor\frac{n-1}{m}\right\rfloor\right) = 1.</math></ref>

:<math>\sum_{m=1}^\infty \left(\left\lfloor\frac{n}{m}\right\rfloor-\left\lfloor\frac{n-1}{m}\right\rfloor\right) = 2.</math>

One may also give formulas for producing the prime numbers.  For example, let ''p''<sub>''n''</sub> be the ''n''-th prime, and for any integer ''r''&nbsp;>&nbsp;1, define the real number ''α'' by the sum

:<math>\alpha = \sum_{m=1}^\infty p_m r^{-m^2}.</math>

Then<ref>Hardy & Wright, § 22.3</ref>

:<math>p_n = \left\lfloor r^{n^2}\alpha \right\rfloor - r^{2n-1}\left\lfloor r^{(n-1)^2}\alpha\right\rfloor.</math>

A similar result is that there is a number ''θ'' = 1.3064... ([[Mills' constant]]) with the property that

:<math>\left\lfloor \theta^3 \right\rfloor, \left\lfloor \theta^9 \right\rfloor, \left\lfloor \theta^{27} \right\rfloor, \dots</math>

are all prime.<ref name="Ribenboim, p. 186">Ribenboim, p. 186</ref>

There is also a number ''ω'' = 1.9287800... with the property that

:<math>\left\lfloor 2^\omega\right\rfloor, \left\lfloor 2^{2^\omega} \right\rfloor, \left\lfloor 2^{2^{2^\omega}} \right\rfloor, \dots</math>

are all prime.<ref name="Ribenboim, p. 186"/>

Let {{pi}}(''x'') be [[Prime-counting function|the number of primes less than or equal to ''x'']]. It is a straightforward deduction from [[Wilson's theorem]] that<ref>Ribenboim, p. 181</ref>

:<math>\pi(n) = \sum_{j=2}^n\Biggl\lfloor\frac{(j-1)!+1}{j} - \left\lfloor\frac{(j-1)!}{j}\right\rfloor\Biggr\rfloor.</math>

Also, if ''n'' ≥ 2,<ref>Crandall & Pomerance, Ex. 1.4, p. 46</ref>

:<math>\pi(n) = \sum_{j=2}^n \left\lfloor \frac{1} {\displaystyle\sum_{k=2}^j\left\lfloor\left\lfloor\frac{j}{k}\right\rfloor\frac{k}{j} \right\rfloor} \right\rfloor.</math>

None of the formulas in this section are of any practical use.<ref>Ribenboim, p. 180 says that "Despite the nil practical value of the formulas ... [they] may have some relevance to logicians who wish to understand clearly how various parts of arithmetic may be deduced from different axiomatzations ... "</ref><ref>Hardy & Wright, pp. 344—345 "Any one of these formulas (or any similar one) would attain a different status if the exact value of the number α ... could be expressed independently of the primes. There seems no likelihood of this, but it cannot be ruled out as entirely impossible."</ref>