Applications

Mod operator

The mod operator, denoted by x mod y for absolute x and y, y ≠ 0, is authentic by the formula

x mod y is consistently amid 0 and y; i.e.

if y is positive,

and if y is negative,

If x is an accumulation and y is a absolute integer,

x mod y for a anchored y is a denticulate function.

editQuadratic reciprocity

Gauss's third affidavit of boxlike reciprocity, as adapted by Eisenstein, has two basal steps.1920

Let p and q be audible absolute odd prime numbers, and let

First, Gauss's antecedent is acclimated to appearance that the Legendre symbols are accustomed by

and

The additional footfall is to use a geometric altercation to appearance that

Combining these formulas gives boxlike advantage in the form

There are formulas that use attic to accurate the boxlike appearance of baby numbers mod odd primes p:21

editRounding

The accustomed rounding of the absolute amount x to the abutting accumulation can be bidding as The accustomed rounding of the abrogating amount x to the abutting accumulation can be bidding as

editTruncation

The truncation of a nonnegative amount is accustomed by The truncation of a nonpositive amount is accustomed by .

The truncation of any absolute amount can be accustomed by: , area sgn(x) is the assurance function.

editNumber of digits

The amount of digits in abject b of a absolute accumulation k is

with the appropriate ancillary of the blueprint aswell captivation accurate for .

editFactors of factorials

Let n be a absolute accumulation and p a absolute prime number. The backer of the accomplished ability of p that divides n! is accustomed by the formula22

Note that this is a bound sum, back the floors are aught if pk > n.

editBeatty sequence

The Beatty arrangement shows how every absolute aberrant amount gives acceleration to a allotment of the accustomed numbers into two sequences via the attic function.23

editEuler's connected (γ)

There are formulas for Euler's connected γ = 0.57721 56649 ... that absorb the attic and ceiling, e.g.24

and

editRiemann action (ζ)

The apportioned allotment action aswell shows up in basic representations of the Riemann zeta function. It is aboveboard to prove (using affiliation by parts)25 that if φ(x) is any action with a connected acquired in the bankrupt breach a, b,

Letting φ(n) = n−s for absolute allotment of s greater than 1 and absolution a and b be integers, and absolution b access beyond gives

This blueprint is accurate for all s with absolute allotment greater than −1, (except s = 1, area there is a pole) and accumulated with the Fourier amplification for {x} can be acclimated to extend the zeta action to the absolute circuitous even and to prove its anatomic equation.26

For s = σ + i t in the analytical band (i.e. 0 < σ < 1),

In 1947 van der Pol acclimated this representation to assemble an alternation computer for award roots of the zeta function.27

editFormulas for prime numbers

n is a prime if and alone if28

Let r > 1 be an integer, pn be the nth prime, and define

Then29

There is a amount θ = 1.3064... (Mills' constant) with the acreage that

are all prime.30

There is aswell a amount ω = 1.9287800... with the acreage that

are all prime.30

π(x) is the amount of primes beneath than or according to x. It is a aboveboard answer from Wilson's assumption that31

Also, if n ≥ 2,32

None of the formulas in this area is of any applied use.

editSolved problem

Ramanujan submitted this botheration to the Journal of the Indian Mathematical Society.33

If n is a absolute integer, prove that

(i)

(ii)

(iii)

editUnsolved problem

The abstraction of Waring's botheration has led to an baffling problem:

Are there any absolute integers k, k ≥ 6, such that34

Mahler35 has accepted there can alone be a bound amount of such k; none are known.