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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 Carmichael number is a composite number $N>1$ for whichEvery unit modulo $N$ satisfies the same modular congruence supplied by Fermat's little theorem for a prime number. The name refers to Robert Daniel Carmichael. The coprime qualification is essential: a nonunit can expose a prime factor through a greatest common divisor calculation. A Carmichael number is therefore a Fermat pseudoprime to every coprime base, rather than to every nonzero residue class.^[ref]

For example,The factors are distinct prime numbers. Their predecessor numbers $2,10,16$ all divide $560$, so Korselt criterion applies. More explicitly, for any integer $a$ coprime to $561$, Fermat's little theorem givesThe Chinese remainder theorem combines these into $a^{560}\equiv 1(\mathrm{mod}561)$. This proves all coprime bases pass from three short divisibility checks, rather than from a long table of sampled bases.

Equivalently, a composite $N$ is a Carmichael number exactly when $a^N\equiv a(\mathrm{mod}N)$ for every integer $a$, including nonunits. The equivalence follows from Korselt criterion: modulo each prime factor, either $a$ is zero or its nonzero power returns to one, and squarefreeness combines those congruences. This does not say a nonunit satisfies $a^{N-1}\equiv 1$. For instance $a=3$ gives $3^{560}\equiv 375(\mathrm{mod}561)$, while $3^{561}\equiv 3(\mathrm{mod}561)$; the greatest common divisor already supplies the factor three.^[ref]

The Carmichael function supplies another compact description: a composite $N$ is Carmichael exactly when $\lambda (N)\mid N-1$. For $561$, $\lambda (561)=\mathrm{lcm}(2,10,16)=80$ and $560=7\cdot 80$. In contrast, Euler's totient function gives $\phi (561)=320$, which does not divide $560$. The least common return exponent, rather than the number of units, controls this test.

The obstruction is useful to know when interpreting a Fermat primality test: even checking every coprime base cannot certify primality. The passing-base fraction for the Fermat test makes this failure quantitative. Stronger primality tests require additional structure, such as the squaring conditions in a Miller-Rabin primality test.

The historical record predates the name. Václav Šimerka's 1885 paper lists $561,1105,1729,2465,2821,6601,8911$ on printed page 224, together with factorizations. This is an early published list of examples, independently of their later name.^[ref]

For a unit modulo n $a$ and a nonnegative integer $k$, division with remainder writes $k=q\phi (n)+r$ with $0\le r<\phi (n)$. Euler's theorem then givesThis replaces a potentially enormous exponent by a small remainder. The exponent itself is reduced modulo Euler's totient function, while the powers are reduced modulo $n$: these are different integer moduli.

For a worked modulo-fifteen calculation, seven is coprime to fifteen and $\phi (15)=8$. ThereforeUse exponentiation by squaring on the remaining exponent $5=(101{)}_2$, reducing after each multiplication:Thus $7^{2029}\mathrm{mod}15=7$. Modular exponentiation supplies the computation; Euler's theorem supplies the optional shortening of its exponent. Exponentiation by squaring also works for nonunit bases, without using this shortening.

The guaranteed exponent $\phi (n)$ is a common return exponent and an upper bound on the shortest period; it need not be that period. The multiplicative order $d={\mathrm{ord}}_n(a)$ is the least positive return exponent for the chosen unit. The Carmichael function $\lambda (n)$ is the least positive return exponent that works for every unit. Their precise relation isEither smaller return exponent can replace $\phi (n)$ in exponent reduction for units. Modulo fifteen, ${\mathrm{ord}}_{15}(7)=\lambda (15)=4<8=\phi (15)$, so $2029=507\cdot 4+1$ gives $7^{2029}\equiv 7$ immediately. For base four the multiplicative order is only two, smaller even than $\lambda (15)$. A return at exponent eight alone does not prove that the multiplicative order is eight.^[ref]

If the reduced exponent is zero, the rule gives $a^k\equiv a^0=1$, and still requires a unit. Applying it to the nonunit three would falsely predict $3^8\equiv 3^0=1(\mathrm{mod}15)$; the actual residue is six. For negative exponents, first interpret the power using a modular multiplicative inverse, as in modular exponentiation.

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]

For the Fermat primality test, a passing base satisfies $a^{n-1}\equiv 1(\mathrm{mod}n)$. Passing proves only this modular congruence, not that $n$ is a prime number. An odd composite number that passes a coprime base is a Fermat pseudoprime to that base. A Carmichael number passes every coprime base, so even testing all the units would not certify that $n$ is a prime number. ^[ref]

This small table uses only the bases $2,3,5,11$, each reduced modulo $n$ first. A numeric entry is $a^{n-1}\mathrm{mod}n$: one passes, while any other numeric entry gives a Fermat witness. A "factor" entry records a proper greatest common divisor and stops before modular exponentiation. The prime/composite classification is independently established by trial division or the displayed prime factorization, not by the passing entries. These four bases are illustrative, with no asserted universal coverage or error probability.

For $341=11\cdot 31$, $\gcd (2,341)=1$ and $2^{10}=1024=3\cdot 341+1$, so $2^{340}=(2^{10}{)}^{34}\equiv 1(\mathrm{mod}341)$. Yet $\gcd (3,341)=1$ and $3^{340}\equiv 56(\mathrm{mod}341)$, so three is a Fermat witness. One coprime base passes and another fails: being a Fermat pseudoprime depends on the base. Choosing eleven instead reveals the proper factor through $\gcd (11,341)=11$.

For $561=3\cdot 11\cdot 17$, every coprime base passes: each of $2,10,16$ divides $560$, so Fermat's little theorem gives $a^{560}\equiv 1$ modulo each prime factor; the Chinese remainder theorem combines the three modular congruences. Equivalently, the Carmichael function gives $\lambda (561)=80\mid 560$. Bases two and five have greatest common divisor one and pass, but $\gcd (3,561)=3$, $\gcd (11,561)=11$ and $\gcd (17,561)=17$ expose factors. The qualification "coprime" is therefore crucial even for a Carmichael number.

The Fermat primality test is useful because a Fermat witness can prove compositeness using repeated squaring without finding a factor. Its failure to certify a passing candidate motivates stronger primality testing methods such as the Miller-Rabin primality test.

Euler's theorem states that $a^{\varphi (m)}\equiv 1(\mathrm{mod}m)$ whenever $\gcd (a,m)=1$, where $\varphi (m)$ counts the invertible residue classes. For each prime-power divisor, the same theorem gives $a^{\varphi (p_i^{e_i})}\equiv 1(\mathrm{mod}{p}_i^{e_i})$. Since the exponent defined in the source is a multiple of every one of these exponents, raising that congruence to the corresponding integer power givesThe prime powers are pairwise coprime, so their product divides $a^{\lambda (m)}-1$, proving the congruence modulo $m$.

For the displayed composite integer,Thus every integer coprime to 561 satisfies $\boxed{a^{560}\equiv 1(\mathrm{mod}561)}$. This is a Carmichael number. The source's prime-power least common multiple need not be the minimal universal exponent for all moduli; for powers of two its definition can exceed the usual Carmichael function. No minimality claim is needed here.

A mathematician associated with Carmichael numbers and the Carmichael function.^[ref]