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</ref> It is also known as '''Carmichael's λ function''', the '''reduced totient function''', and the '''least universal exponent function'''.
The order of the multiplicative group of integers modulo {{mvar | n}} is {{math | ''φ''(''n'')}}, where {{mvar | φ}} is [[Euler's totient function]]. Since the order of an element of a finite group divides the order of the group, {{math | ''λ''(''n'')}} divides {{math | ''φ''(''n'')}}. The following table compares the first 36 values of {{math | ''λ''(''n'')}} {{OEIS|id=A002322}} with [[Euler's totient function]]and {{mvarmath | ''φ''(''n'')}} (in '''bold''' if they are different; the {{mvar | n}}s such that they are different are listed in {{oeis|A033949}}).
{| class="wikitable" style="text-align: center;"
==Numerical examples==
# Carmichael's function at 5 is 4,* {{math | ''λn''(5) {{=}} 45}},. becauseThe forset anyof numbernumbers <math>0<a<5</math>less than and coprime to 5, i.e.is <{{math>a\in \| {1, 2, 3, 4\}~,</math>}}. thereHence isEuler's <math>a^mtotient \equivfunction 1has \,(\text{mod } 5)</math> with <math>m=4,</math> namely,value {{math | 1<sup>1⋅4</sup>''φ''(5) {{=}} 1<sup>4</sup> ≡ 1 (mod 5)}}, {{mathand |the 2<sup>4</sup>value {{=}}of 16Carmichael's ≡ 1 (mod 5)}}function, {{math | 3<sup>4</sup> {{=}} 81 ≡ 1 ''λ''(mod 5)}}, andmust {{mathbe |a 4<sup>2⋅2</sup>divisor {{=}}of 16<sup>2</sup> ≡ 1<sup>2</sup> (mod 5)}}4. The Anddivisor this1 {{mathdoes |not ''m'' {{=}} 4}} issatisfy the smallestdefinition exponentof withCarmichael's thisfunction property, becausesince <math>2a^2 =41 \not\equiv 1\pmod{5}</math> except for <math>a\,(equiv1\textpmod{mod } 5)}</math>. (andNeither does 2 since <math>2^2 \equiv 3^2 =\equiv 94 \not\equiv 1 \,(\textpmod{mod 5} 5)</math> as well.)<br />Moreover, Euler's [[totient function]] at 5 is 4,Hence {{math | ''φλ''(5) {{=}} 4}}. Indeed, because there are exactly <math>1^4 numbers less than and coprime to 5 (1,\equiv 2,^4\equiv 3, and^4\equiv 4). [[Euler's theorem]] assures that ^4\equiv1\pmod{{math | ''a''<sup>45}</supmath> ≡ 1 (mod 5)}} for all {{mvar | a}} coprime to 5, and 4 is the smallest such exponent. Both 2 and 3 are primitive {{mvar | λ}}-roots modulo 5 and also [[Primitive root modulo n|primitive roots]] modulo 5.
# Carmichael's function at 8 is 2,* {{math | ''λn''(8) {{=}} 28}},. becauseThe forset anyof numbernumbers {{mvarless |than a}}and coprime to 8, i.e.is <{{math>a\in \| {1, 3, 5, 7\}~,</math> it holds}}. thatHence {{math | ''aφ''<sup>2</sup> ≡ 1 (mod 8)}}. Namely, {{math | 1<sup>1⋅2</sup> {{=}} 1<sup>2</sup> ≡ 1 (mod 8)4}}, {{math | 3<sup>2</sup> {{=}} 9 ≡ 1 (mod 8)}},and {{math | 5<sup>2</sup> {{=}} 25 ≡ 1 ''λ''(mod 8)}} andmust {{mathbe |a 7<sup>2</sup>divisor {{=}}of 49 ≡ 1 (mod 8)}}4.<br />Euler'sIn [[totient function]] at 8 is 4,fact {{math | ''φλ''(8) {{=}} 42}}, becausesince there are exactly 4 numbers less than and coprime to 8 (<math>1,^2\equiv 3,^2\equiv 5, and^2\equiv 7). Moreover, [[Euler's theorem]] assures that ^2\equiv1\pmod{{math | ''a''<sup>48}</supmath> ≡ 1 (mod 8)}} for all {{mvar | a}} coprime to 8, but 4 is not the smallest such exponent. The primitive {{mvar | λ}}-roots modulo 8 are 3, 5, and 7. There are no primitive roots modulo 8.
== Recurrence for {{math | ''λ''(''n'')}} ==
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