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Will Orrick (talk | contribs) The revised lead defines lambda using an expression involving lambda, which may also cause confusion. Minimization takes place over a set, which is how the previous wording was meant to be read. Added slight clarification of this point. |
→Carmichael's theorems: +anchor for old links (e.g. at RSA (cryptosystem) |
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== Carmichael's theorems ==
{{anchor|Carmichael's theorem}}
Carmichael proved two theorems that, together, establish that if {{math | ''λ''(''n'')}} is considered as defined by the recurrence of the previous section, then it satisfies the property stated in the introduction, namely that it is the smallest positive integer {{mvar | m}} such that <math>a^m\equiv 1\pmod{n}</math> for all {{mvar | a}} relatively prime to {{mvar | n}}.
{{Math theorem |name=Theorem 1|math_statement=If {{mvar | a}} is relatively prime to {{mvar | n}} then <math>a^{\lambda(n)}\equiv 1\pmod{n}</math>.<ref>Carmichaael (1914) p.40</ref>}}
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