Coprime integers

Innumber theory,twointegers $a$ and $b$ arecoprime,relatively primeormutually primeif the only positive integer that is adivisorof both of them is 1.^[1]Consequently, anyprime numberthat divides $a$ does not divide $b$ ,and vice versa. This is equivalent to theirgreatest common divisor(GCD) being 1.^[2]One says also $a$ is prime to $b$ or $a$ is coprime with $b$ .

The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of areduced fractionare coprime, by definition.

Notation and testing

When the integers $a$ and $b$ are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula $gcd(a, b) = 1$ or $(a, b) = 1$ .In their 1989 textbookConcrete Mathematics,Ronald Graham,Donald Knuth,andOren Patashnikproposed an alternative notation $a\perp b$ to indicate that $a$ and $b$ are relatively prime and that the term "prime" be used instead of coprime (as in $a$ isprimeto $b$ ).^[3]

A fast way to determine whether two numbers are coprime is given by theEuclidean algorithmand its faster variants such asbinary GCD algorithmorLehmer's GCD algorithm.

The number of integers coprime with a positive integer $n$ ,between 1 and $n$ ,is given byEuler's totient function,also known as Euler's phi function, $φ (n)$ .

Asetof integers can also be called coprime if its elements share no common positive factor except 1. A stronger condition on a set of integers is pairwise coprime, which means that $a$ and $b$ are coprime for every pair $(a, b)$ of different integers in the set. The set ${2, 3, 4}$ is coprime, but it is not pairwise coprime since 2 and 4 are not relatively prime.

Properties

The numbers 1 and −1 are the only integers coprime with every integer, and they are the only integers that are coprime with 0.

A number of conditions are equivalent to $a$ and $b$ being coprime:

Noprime numberdivides both $a$ and $b$ .
There exist integers $x, y$ such that $ax + by = 1$ (seeBézout's identity).
The integer $b$ has amultiplicative inversemodulo $a$ ,meaning that there exists an integer $y$ such that $by \equiv 1 (mod a)$ .In ring-theoretic language, $b$ is aunitin thering⁠ $\mathbb {Z} /a\mathbb {Z}$ ⁠ofintegers modulo $a$ .
Every pair ofcongruence relationsfor an unknown integer $x$ ,of the form $x \equiv k (mod a)$ and $x \equiv m (mod b)$ ,has a solution (Chinese remainder theorem); in fact the solutions are described by a single congruence relation modulo $ab$ .
Theleast common multipleof $a$ and $b$ is equal to their product $ab$ ,i.e. $lcm(a, b) = ab$ .^[4]

As a consequence of the third point, if $a$ and $b$ are coprime and $br \equiv bs (mod a)$ ,then $r \equiv s (mod a)$ .^[5]That is, we may "divide by $b$ "when working modulo $a$ .Furthermore, if $b 1, b 2$ are both coprime with $a$ ,then so is their product $b 1 b 2$ (i.e., modulo $a$ it is a product of invertible elements, and therefore invertible);^[6]this also follows from the first point byEuclid's lemma,which states that if a prime number $p$ divides a product $bc$ ,then $p$ divides at least one of the factors $b, c$ .

As a consequence of the first point, if $a$ and $b$ are coprime, then so are any powers $a k$ and $b m$ .

If $a$ and $b$ are coprime and $a$ divides the product $bc$ ,then $a$ divides $c$ .^[7]This can be viewed as a generalization of Euclid's lemma.

Figure 1. The numbers 4 and 9 are coprime. Therefore, the diagonal of a 4 × 9 lattice does not intersect any otherlattice points

The two integers $a$ and $b$ are coprime if and only if the point with coordinates $(a, b)$ in aCartesian coordinate systemwould be "visible" via an unobstructed line of sight from the origin $(0, 0)$ ,in the sense that there is no point with integer coordinates anywhere on the line segment between the origin and $(a, b)$ .(See figure 1.)

In a sense that can be made precise, theprobabilitythat two randomly chosen integers are coprime is $6/ π 2$ ,which is about 61% (see§ Probability of coprimality,below).

Twonatural numbers $a$ and $b$ are coprime if and only if the numbers $2 a - 1$ and $2 b - 1$ are coprime.^[8]As a generalization of this, following easily from theEuclidean algorithminbase $n > 1$ :

\gcd \left(n^{a}-1,n^{b}-1\right)=n^{\gcd(a,b)}-1.

Coprimality in sets

Asetof integers $S=\{a_{1},a_{2},\dots,a_{n}\}$ can also be calledcoprimeorsetwise coprimeif thegreatest common divisorof all the elements of the set is 1. For example, the integers 6, 10, 15 are coprime because 1 is the only positive integer that divides all of them.

If every pair in a set of integers is coprime, then the set is said to bepairwise coprime(orpairwise relatively prime,mutually coprimeormutually relatively prime). Pairwise coprimality is a stronger condition than setwise coprimality; every pairwise coprime finite set is also setwise coprime, but the reverse is not true. For example, the integers 4, 5, 6 are (setwise) coprime (because the only positive integer dividingallof them is 1), but they are notpairwisecoprime (because $gcd(4, 6) = 2$ ).

The concept of pairwise coprimality is important as a hypothesis in many results in number theory, such as theChinese remainder theorem.

It is possible for aninfinite setof integers to be pairwise coprime. Notable examples include the set of all prime numbers, the set of elements inSylvester's sequence,and the set of allFermat numbers.

Coprimality in ring ideals

Twoideals $A$ and $B$ in acommutative ring $R$ are called coprime (orcomaximal) if $A+B=R.$ This generalizesBézout's identity:with this definition, twoprincipal ideals( $a$ ) and ( $b$ ) in the ring of integers⁠ $\mathbb {Z}$ ⁠are coprime if and only if $a$ and $b$ are coprime. If the ideals $A$ and $B$ of $R$ are coprime, then $AB=A\cap B;$ furthermore, if $C$ is a third ideal such that $A$ contains $BC$ ,then $A$ contains $C$ .TheChinese remainder theoremcan be generalized to any commutative ring, using coprime ideals.

Probability of coprimality

Given two randomly chosen integers $a$ and $b$ ,it is reasonable to ask how likely it is that $a$ and $b$ are coprime. In this determination, it is convenient to use the characterization that $a$ and $b$ are coprime if and only if no prime number divides both of them (seeFundamental theorem of arithmetic).

Informally, the probability that any number is divisible by a prime (or in fact any integer) $p$ is⁠ ${\tfrac {1}{p}};$ ⁠for example, every 7th integer is divisible by 7. Hence the probability that two numbers are both divisible by $p$ is⁠ ${\tfrac {1}{p^{2}}},$ ⁠and the probability that at least one of them is not is⁠ $1-{\tfrac {1}{p^{2}}}.$ ⁠Any finite collection of divisibility events associated to distinct primes is mutually independent. For example, in the case of two events, a number is divisible by primes $p$ and $q$ if and only if it is divisible by $pq$ ;the latter event has probability⁠ ${\tfrac {1}{pq}}.$ ⁠If one makes the heuristic assumption that such reasoning can be extended to infinitely many divisibility events, one is led to guess that the probability that two numbers are coprime is given by a product over all primes,

\prod _{{\text{prime }}p}\left(1-{\frac {1}{p^{2}}}\right)=\left(\prod _{{\text{prime }}p}{\frac {1}{1-p^{-2}}}\right)^{-1}={\frac {1}{\zeta (2)}}={\frac {6}{\pi ^{2}}}\approx 0.607927102\approx 61\%.

Here $ζ$ refers to theRiemann zeta function,the identity relating the product over primes to $ζ (2)$ is an example of anEuler product,and the evaluation of $ζ (2)$ as $π 2 /6$ is theBasel problem,solved byLeonhard Eulerin 1735.

There is no way to choose a positive integer at random so that each positive integer occurs with equal probability, but statements about "randomly chosen integers" such as the ones above can be formalized by using the notion ofnatural density.For each positive integer $N$ ,let $P N$ be the probability that two randomly chosen numbers in $\{1,2,\ldots,N\}$ are coprime. Although $P N$ will never equal $6/ π 2$ exactly, with work^[9]one can show that in the limit as $N\to \infty,$ the probability $P N$ approaches $6/ π 2$ .

More generally, the probability of $k$ randomly chosen integers being setwise coprime is⁠ ${\tfrac {1}{\zeta (k)}}.$ ⁠

Generating all coprime pairs

Thetree rootedat (2, 1). The root (2, 1) is marked red, its three children are shown in orange, third generation is yellow, and so on in the rainbow order.

All pairs of positive coprime numbers $(m, n)$ (with $m > n$ ) can be arranged in two disjoint completeternary trees,one tree starting from $(2, 1)$ (for even–odd and odd–even pairs),^[10]and the other tree starting from $(3, 1)$ (for odd–odd pairs).^[11]The children of each vertex $(m, n)$ are generated as follows:

Branch 1: $(2m-n,m)$
Branch 2: $(2m+n,m)$
Branch 3: $(m+2n,n)$

This scheme is exhaustive and non-redundant with no invalid members. This can be proved by remarking that, if $(a,b)$ is a coprime pair with $a>b,$ then

if $a>3b,$ then $(a,b)$ is a child of $(m,n)=(a-2b,b)$ along branch 3;
if $2b<a<3b,$ then $(a,b)$ is a child of $(m,n)=(b,a-2b)$ along branch 2;
if $b<a<2b,$ then $(a,b)$ is a child of $(m,n)=(b,2b-a)$ along branch 1.

In all cases $(m,n)$ is a "smaller" coprime pair with $m>n.$ This process of "computing the father" can stop only if either $a=2b$ or $a=3b.$ In these cases, coprimality, implies that the pair is either $(2,1)$ or $(3,1).$

Applications

In machine design, an even, uniformgearwear is achieved by choosing the tooth counts of the two gears meshing together to be relatively prime. When a 1:1gear ratiois desired, a gear relatively prime to the two equal-size gears may be inserted between them.

In pre-computercryptography,someVernam ciphermachines combined several loops of key tape of different lengths. Manyrotor machinescombine rotors of different numbers of teeth. Such combinations work best when the entire set of lengths are pairwise coprime.^[12]^[13]^[14]^[15]

Generalizations

This concept can be extended to other algebraic structures than⁠ $\mathbb {Z};$ ⁠for example,polynomialswhosegreatest common divisoris 1 are calledcoprime polynomials.

Notes

^Eaton, James S. (1872).A Treatise on Arithmetic.Boston: Thompson, Bigelow & Brown. p. 49.Retrieved10 January2022.Two numbers aremutuallyprime when no whole number butonewill divide each of them
^Hardy & Wright 2008,p. 6
^Graham, R. L.; Knuth, D. E.; Patashnik, O. (1989),Concrete Mathematics/ A Foundation for Computer Science,Addison-Wesley, p. 115,ISBN 0-201-14236-8
^Ore 1988,p. 47
^Niven & Zuckerman 1966,p. 22, Theorem 2.3(b)
^Niven & Zuckerman 1966,p. 6, Theorem 1.8
^Niven & Zuckerman 1966,p.7, Theorem 1.10
^Rosen 1992,p. 140
^This theorem was proved byErnesto Cesàroin 1881. For a proof, seeHardy & Wright 2008,Theorem 332
^Saunders, Robert & Randall, Trevor (July 1994), "The family tree of the Pythagorean triplets revisited",Mathematical Gazette,78:190–193,doi:10.2307/3618576.
^Mitchell, Douglas W. (July 2001), "An alternative characterisation of all primitive Pythagorean triples",Mathematical Gazette,85:273–275,doi:10.2307/3622017.
^ Klaus Pommerening. "Cryptology: Key Generators with Long Periods".
^ David Mowry. "German Cipher Machines of World War II". 2014. p. 16; p. 22.
^ Dirk Rijmenants. "Origins of One-time pad".
^ Gustavus J. Simmons. "Vernam-Vigenère cipher".

References

Hardy, G.H.;Wright, E.M.(2008),An Introduction to the Theory of Numbers(6th ed.),Oxford University Press,ISBN 978-0-19-921986-5
Niven, Ivan; Zuckerman, Herbert S. (1966),An Introduction to the Theory of Numbers(2nd ed.), John Wiley & Sons
Ore, Oystein (1988) [1948],Number Theory and Its History,Dover,ISBN 978-0-486-65620-5
Rosen, Kenneth H. (1992),Elementary Number Theory and its Applications(3rd ed.), Addison-Wesley,ISBN 978-0-201-57889-8