CarmichaelLambda

The Carmichael function $\lambda(n)$: the exponent of the multiplicative group mod $n$ — the least $m$ with $a^m\equiv1\pmod n$ for every $a$ coprime to $n$.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageCarmichaelLambda✓↗

CarmichaelLambda(n)the exponent of $(\mathbb{Z}/n)^\times$

SageMathcarmichael_lambda(n)SymPyreduced_totient(n)

Typed signatures

Type	Package	Applies
(integer) -> integer	compute-engine	always
(integer) -> integer	number-theoryreplaces compute-engine	always

Domain: Number theory

Details

• compute-engine has this natively; the gap closed here is Wolfram's $\lambda(-n)=\lambda(n)$ and threading over a list, which the native handler left unevaluated.
• $\lambda(n) \mid \varphi(n)$ always, and the two agree exactly when $(\mathbb{Z}/n)^\times$ is cyclic. See Totient.

Plot

Examples

Basic #

The exponent of the group of units mod 10 is 4

live ↗

In$\mathrm{CarmichaelLambda}(10)$Out$4$

Scope #

Listable

live ↗

In$\mathrm{CarmichaelLambda}(\bigl\lbrack8, 15\bigr\rbrack)$Out$\bigl\lbrack2, 4\bigr\rbrack$

Negative n: $\lambda(-n)=\lambda(n)$

live ↗

In$\mathrm{CarmichaelLambda}(-100)$Out$20$

Listable over a Range

live ↗

In$\mathrm{CarmichaelLambda}(1..10)$Out$\bigl\lbrack1, 1, 2, 2, 4, 2, 6, 2, 6, 4\bigr\rbrack$

The sign of the argument is ignored

live ↗

In$\mathrm{CarmichaelLambda}(-10)$Out$4$

Threads over a list

live ↗

In$\mathrm{CarmichaelLambda}(\bigl\lbrack2, 4, 7\bigr\rbrack)$Out$\bigl\lbrack1, 2, 6\bigr\rbrack$

Applications #

561 is a Carmichael number: $\lambda(561)$ leaves remainder 1

live ↗

In$561\bmod\mathrm{CarmichaelLambda}(561)$Out$1$

Smaller than $\varphi(12)=4$: units mod 12 have exponent 2

live ↗

In$\mathrm{CarmichaelLambda}(12)$Out$2$

Properties #

If $m\mid n$ then $\lambda(m)\mid\lambda(n)$

live ↗

In$\bigl\lbrack8\mid24, \mathrm{CarmichaelLambda}(8)\mid\mathrm{CarmichaelLambda}(24)\bigr\rbrack$Out$\bigl\lbrack\mathrm{True}, \mathrm{True}\bigr\rbrack$

$a^{1+\lambda(n)}\equiv a \pmod n$ for squarefree $n$

live ↗

In$3^{1+\mathrm{CarmichaelLambda}(17)}\bmod17$Out$3$

Every element's multiplicative order divides $\lambda(n)$

live ↗

In$\mathrm{MultiplicativeOrder}(4, 9)\mid\mathrm{CarmichaelLambda}(9)$Out$\mathrm{True}$

The trivial case, $\lambda(1)=1$

live ↗

In$\mathrm{CarmichaelLambda}(1)$Out$1$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

The only unit mod 2 is 1, so $\lambda(2)=1$

live ↗

In$\mathrm{CarmichaelLambda}(2)$Out$1$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

For an odd prime $p$, $\lambda(p) = p-1$

live ↗

In$\mathrm{CarmichaelLambda}(3)$Out$2$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

Units mod 4 form a cyclic group of order 2

live ↗

In$\mathrm{CarmichaelLambda}(4)$Out$2$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

$\lambda(5) = 4$: the units mod 5 form a cyclic group

live ↗

In$\mathrm{CarmichaelLambda}(5)$Out$4$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

Multiplicative over coprime parts: $\mathrm{lcm}(1,2)=2$

live ↗

In$\mathrm{CarmichaelLambda}(6)$Out$2$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

$\lambda(n)$ always divides $\varphi(n)$

live ↗

In$\mathrm{CarmichaelLambda}(8)\mid\mathrm{Totient}(8)$Out$\mathrm{True}$

Mod 7 there is a primitive root, so $\lambda = \varphi = 6$

live ↗

In$\mathrm{CarmichaelLambda}(7)$Out$6$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

Powers of 2 above 4 fall short of $\varphi$: $\lambda(8)=2$, not 4

live ↗

In$\mathrm{CarmichaelLambda}(8)$Out$2$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

Odd prime powers are cyclic, so $\lambda(9)=\varphi(9)=6$

live ↗

In$\mathrm{CarmichaelLambda}(9)$Out$6$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

$\lambda(11) = 10$, the full order of the unit group

live ↗

In$\mathrm{CarmichaelLambda}(11)$Out$10$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

$\lambda(13) = 12$: 2 is a primitive root mod 13

live ↗

In$\mathrm{CarmichaelLambda}(13)$Out$12$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

Equals $\lambda(7)=6$ since 2 contributes nothing

live ↗

In$\mathrm{CarmichaelLambda}(14)$Out$6$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

$\mathrm{lcm}(\lambda(3),\lambda(5))=\mathrm{lcm}(2,4)=4$

live ↗

In$\mathrm{CarmichaelLambda}(15)$Out$4$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

Power of 2 beyond 4: $2^{k-2}$, here 4

live ↗

In$\mathrm{CarmichaelLambda}(16)$Out$4$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

$\lambda(17) = 16$, a power of 2 for a Fermat prime

live ↗

In$\mathrm{CarmichaelLambda}(17)$Out$16$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

Edge values: negatives mirror positives, and 0 gives 0

live ↗

In$\mathrm{CarmichaelLambda}(\bigl\lbrack-2, -1, 0, 1, 2\bigr\rbrack)$Out$\bigl\lbrack1, 1, 0, 1, 1\bigr\rbrack$

$\lambda(\mathrm{lcm}(m,n))$ is the lcm of $\lambda(m)$ and $\lambda(n)$

live ↗

In$\bigl\lbrack\operatorname{lcm}(\mathrm{CarmichaelLambda}(\bigl\lbrack3, 8\bigr\rbrack)), \mathrm{CarmichaelLambda}(\operatorname{lcm}(3, 8))\bigr\rbrack$Out$\bigl\lbrack\bigl\lbrack2, 2\bigr\rbrack, 2\bigr\rbrack$

Zero is a degenerate input and gives 0

live ↗

In$\mathrm{CarmichaelLambda}(0)$Out$0$

Known value · Wolfram MathematicalFunctionData, CarmichaelLambda

About

Examples	30
Checked against	Wolfram Language 30 agreeSageMath 21 agree5 errorSymPy 21 agree4 errormathlib4 22 agree6 error
Crosswalk	Wolfram LanguageSageMathSymPy

Implementation

mappedsympypackages/oracle/src/mappings.ts

reduced_totient is SymPy's Carmichael function.

mappedsagepackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

ArithmeticFunction.carmichael, the exponent of the multiplicative group mod n; natural n only.

See also: Totient, MultiplicativeOrder