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The power spectrum describes how the variance of a random field or stationary process is distributed over spatial wavenumber or temporal frequency.

For equal hoop and bead masses, with the hoop suspended from its circumference, the quadratic mass and stiffness matrices are proportional toThe dimensionless squared frequencies are the roots of $2c^2-5c+2$, namely $c=1/2$ and $c=2$.

A positive integer is a sum of two integer squares exactly when every prime factor congruent to $3$ modulo $4$ has even exponent in its prime factorization.

For positive integers $a,b$, the Euclidean algorithm repeatedly performs division with remainder: write $a=qb+r$ with $0\le r<b$, then replace $(a,b)$ by $(b,r)$ until the second entry is zero. The final nonzero entry is their greatest common divisor. A remainder preserves the set of common divisors: an integer divides $a$ and $b$ exactly when it divides $b$ and $a-qb$. The positive remainders strictly decrease, so the calculation terminates.

The complete calculation for $252$ and $198$ isHence $\gcd (252,198)=18$. The fraction $252/198$ reduces to $14/11$ by dividing numerator and denominator by that common factor. The same remainder calculation has a geometric interpretation: remove squares from a $252$-by-$198$ rectangle, then repeat on the remaining strip. Each division step records how many equal squares fit and the width of the new strip.

The procedure does more than test individual divisors. Each step replaces the whole common-divisor problem by a smaller equivalent one. Euclid's Book VII, Proposition 2 describes the corresponding repeated-subtraction construction; batching those subtractions into division with remainder gives the present formulation. ^[ref]

For coprime integers $a,n$ with $n>1$, the multiplicative order is the least positive integer $d$ such thatIt counts the distinct powers in the cyclic subgroup generated by the congruence class of $a$. Thus it is an order of a group element in the unit group $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$.^[ref]

For example, the modular exponentiation $2^k\mathrm{mod}7$, starting at $k=0$, produces $1,2,4,1,2,4,\dots$. The first return to one takes three steps, so ${\mathrm{ord}}_7(2)=3$. Starting at the congruence class three and repeatedly multiplying by two instead gives $3,6,5,3,\dots$: these are the other three units modulo seven, outside the cyclic subgroup generated by two.

Euler's theorem guarantees a return to one at exponent $\phi (n)$, where $\phi$ is Euler's totient function. Thus the multiplicative order exists for every unit.^[ref] In fact,By Lagrange's theorem, the size of the cyclic subgroup divides the size of the unit group, which is Euler's totient function. For two modulo seven, this is $3\mid 6$; the multiplicative order need not equal Euler's totient function. If $a\equiv 1(\mathrm{mod}n)$, its multiplicative order is one.

The coprime integers hypothesis is necessary: if $a^d=1+kn$, any common divisor of $a$ and $n$ divides one. For instance, powers of two modulo eight give $1,2,4,0,0,\dots$, with no positive-exponent return to one. This sequence eventually settles, but two has no multiplicative order modulo eight.

For an integer $n>1$, the Carmichael function $\lambda (n)$ is the exponent of a finite group for the unit group $U(n)=(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$: the least positive integer $m$ for whichHere the greatest common divisor condition selects the units. Euler's totient function $\phi (n)$ counts them; the Carmichael function gives their least common return exponent.^[ref]

Modulo eight, the units are $1,3,5,7$, and their squares are all congruent to one. Their multiplicative orders are $1,2,2,2$, so $\lambda (8)=2$ although $\phi (8)=4$. Modulo fifteen, the units $1,2,4,7,8,11,13,14$ have multiplicative orders $1,4,2,4,4,2,4,2$. For example, the powers of two return through $1,2,4,8,1$. Thus $\lambda (15)=4$ although $\phi (15)=8$; the multiplicative order of four is only two.

The three quantities are related byA common return exponent must be a multiple of every multiplicative order, hence of their least common multiple. Conversely, that least common multiple makes every unit return to one. Lagrange's theorem makes each multiplicative order divide the group order $\phi (n)$, proving the final divisibility. The Carmichael function applies to all units at once, while the multiplicative order concerns a chosen unit.

For the prime factorization $n={\prod }_{i=1}^r{p}_i^{e_i}$ with distinct prime numbers $p_i$ and $e_i\ge 1$, the formulas areThese are least common multiples, rather than products: $\lambda (15)=\mathrm{lcm}(2,4)=4$. The convention $\lambda (1)=1$ handles the trivial case separately.^[ref]

The Chinese remainder theorem for unit groups explains the least common multiple: a power returns to one in every prime power component exactly when it is a multiple of each component's exponent of a finite group. For an odd prime power, the unit group is a cyclic group of group order $p^{e-1}(p-1)$. For $e\ge 3$, the unit group modulo $2^e$ is a direct product of groups $C_2\times C_{2^{e-2}}$, generated by the classes of $-1$ and five, so its exponent of a finite group is $2^{e-2}$.^[ref]

This gives a uniform reduction for modular exponentiation: for a unit $a$ and a nonnegative integer $k$, reduce $k$ modulo $\lambda (n)$. For example, $2^{1001}\equiv 2(\mathrm{mod}15)$ because $1001\equiv 1(\mathrm{mod}4)$. The unit hypothesis matters: $3^2\equiv 1(\mathrm{mod}8)$, but $2^2\not\equiv 1(\mathrm{mod}8)$. Computing the formula assumes the prime factorization is available. The same common-exponent viewpoint explains Carmichael numbers: a composite number $n$ passes the Fermat primality test for every unit base exactly when $\lambda (n)\mid n-1$. For example, $\lambda (561)=\mathrm{lcm}(2,10,16)=80\mid 560$.

A rational number is a quotient of two integers with nonzero denominator.

The identity $r^2-s^2=(r-s)(r+s)$ converts a square difference into a factorization.

An integer has a square factor when some prime square divides it.

A squarefree integer is not divisible by the square of any prime.

For a prime number $p$ and an integer $a$ not divisible by $p$, Fermat's little theorem states the modular congruenceEquivalently, for every integer $a$, including $a<0$ and multiples of $p$,The first formulation follows from the second by multiplying by a modular multiplicative inverse of $a$. Conversely, multiply the first by $a$ when $p\nmid a$; when $p\mid a$, both sides of the second are zero modulo $p$. The divisibility condition in the first formulation is essential.^[ref]

For $p=7$, $3^6=729=7\cdot 104+1$, so $3^6\equiv 1(\mathrm{mod}7)$ and $3^7\equiv 3(\mathrm{mod}7)$. Also $(-2{)}^6=64\equiv 1(\mathrm{mod}7)$, while $7^6\equiv 0(\mathrm{mod}7)$: a zero residue class cannot satisfy the first formulation. It does satisfy $7^7\equiv 7\equiv 0(\mathrm{mod}7)$. A failure at a zero residue class therefore says nothing against a candidate's being a prime number.

The prime number $p=2$ is included. Its only nonzero residue class is one: every odd integer $a$ satisfies $a^{2-1}=a\equiv 1(\mathrm{mod}2)$. Every integer also satisfies $a^2\equiv a(\mathrm{mod}2)$, because $a(a-1)$ is even. Testing the first formulation with an even integer would violate its hypothesis.

The useful surprise is that every nonzero residue class returns to one at the same exponent $p-1$, although its shortest return can be earlier. This gives exponent reduction for units: for a nonnegative integer $k$, $a^k\equiv a^{k\mathrm{mod}(p-1)}(\mathrm{mod}p)$ when $p\nmid a$. It also computes a modular multiplicative inverse as $a^{p-2}$ modulo $p$, including $p=2$. Repeated squaring makes these modular exponentiations practical.

In group theory, the nonzero residue classes form the multiplicative group of integers modulo n with $n=p$, of group order $p-1$. Lagrange's theorem says that the multiplicative order of $a$ divides $p-1$, giving the guaranteed return. Euler's theorem extends this to a general integer modulus $n$, using the units and Euler's totient function $\phi (n)$; the Carmichael function gives the least common return exponent for all units. A nonzero residue class modulo a composite number need not be a unit.

Leonhard Euler's paper presented in 1732 and published in 1738 states a historical formulation: Theorem 1 says that powers with exponent $n-1$ leave zero or one upon division by a prime number $n$. The preceding discussion states the difference-of-powers version and explicitly says that he does not have its proof.^[ref]

Euler's totient function is the arithmetic functionIt counts the congruence classes modulo $n$ represented by integers coprime to $n$. In particular, $\phi (1)=1$, since $\gcd (1,1)=1$. For $12$, the coprime integers among $1,\dots ,12$ are $1,5,7,11$, so $\phi (12)=4$; they form a reduced residue system. ^[ref]

Counting these congruence classes is useful because they are exactly the classes with modular multiplicative inverses. Their number supplies the exponent in Euler theorem. Möbius inversion gives the product formulawhere $p$ runs over distinct prime factors of $n$. Thus $\phi (p^a)=p^{a-1}(p-1)$ for prime powers with $a\ge 1$, and $\phi (12)=12(1-1/2)(1-1/3)=4$. The product for $n=1$ is an empty product, so it gives the same convention $\phi (1)=1$.

For a prime number $p$ and a positive integer $n$, the prime valuation is the exponent of $p$ in the prime factorization of $n$, equivalentlyThus $v_2(360)=3$ because $8$ divides $360$ but $16$ does not; $v_3(360)=2$, $v_5(360)=1$ and $v_7(360)=0$. In particular $v_p(1)=0$. Multiplication adds prime valuations, $v_p(ab)=v_p(a)+v_p(b)$, and the fundamental theorem of arithmetic makes comparison of prime valuations an exact test for divisibility between positive integers. This definition extends to a nonzero integer by taking its absolute value; for zero one commonly sets $v_p(0)=+\infty$.

A binary quadratic form is $a{x}^2+bxy+c{y}^2$; integral changes of variables organize forms into equivalence classes.

Diophantine approximation studies how closely numbers can be approximated by rational or algebraic numbers subject to bounds on their arithmetic complexity.

Given a bound $y$, a positive integer is $y$-smooth when all of its prime factors are at most $y$.

An integer is a perfect power when it has the form $x^k$ for integers $x$ and $k>1$.

A composite integer $N$ is Carmichael exactly when it is square-free and $p-1$ divides $N-1$ for every prime $p$ dividing $N$.

A composite $N$ is Carmichael when $a^{N-1}\equiv 1(\mathrm{mod}N)$ for every $a$ coprime to $N$.

Quadratic reciprocity compares two questions: is the prime number $p$ a quadratic residue modulo $q$, and is $q$ a quadratic residue modulo $p$? Let $p,q$ be distinct odd primes. In terms of the Legendre symbol,The two Legendre symbols have opposite signs exactly when both primes are $3$ modulo $4$; otherwise they are equal. Their values are nonzero because the primes are distinct. Equal signs can mean two quadratic residues or two quadratic nonresidues: reciprocity compares the answers rather than declaring either answer positive.^[ref]

For a direct computation, the nonzero quadratic residues modulo $3,5,7$ are respectively $\{1\}$, $\{1,4\}$, and $\{1,2,4\}$. Squaring just $1,\dots ,(p-1)/2$ suffices because $x$ and $p-x$ have the same square modulo an odd prime $p$.The pair $3,7$ has a sign flip, whereas $5,7$ has two negative signs. These small quadratic residue tables make the distinction tangible.

The first supplementary law for quadratic reciprocity handles numerator $-1$, and the second supplementary law for quadratic reciprocity handles numerator $2$:For an odd prime $p$, $-1$ is a quadratic residue exactly when $p\equiv 1(\mathrm{mod}4)$, while $2$ is a quadratic residue exactly when $p\equiv 1,7(\mathrm{mod}8)$. The remaining odd congruence classes modulo $8$, namely $3,5$, make $2$ a quadratic nonresidue. These laws fill the cases excluded from swapping distinct odd primes.^[ref]