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Will Orrick (talk | contribs) "Extension for powers of two" is a misleading section heading and out of place, as this section proves a core part of Theorem 1, and is not an extension of anything. I've renamed and moved this, and provided the rest of the proof of Theorem 1. |
→Carmichael's theorems: Remove a 't ' typo Tags: Mobile edit Mobile web edit |
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If {{mvar | g}} is one of the primitive {{mvar | λ}}-roots guaranteed by the theorem, then <math>g^m\equiv1\pmod{n}</math> has no positive integer solutions {{mvar | m}} less than {{math | ''λ''(''n'')}}, showing that there is no positive {{math | ''m'' < ''λ''(''n'')}} such that <math>a^m\equiv 1\pmod{n}</math> for all {{mvar | a}} relatively prime to {{mvar | n}}.
The second statement of Theorem 2 does not imply that all primitive {{mvar | λ}}-roots modulo {{mvar | n}} are congruent to powers of a single root {{mvar | g}}.<ref>Carmichael (1914) p.56</ref> For example, if {{math | ''n'' {{=}} 15}}, then {{math | ''λ''(''n'') {{=}} 4}} while <math>\varphi(n)=8</math> and <math>\varphi(\lambda(n))=2</math>. There are four primitive {{mvar | λ}}-roots modulo 15, namely 2, 7, 8, and 13 as <math>1\equiv2^4\equiv8^4\equiv7^4\equiv13^4</math>. The roots 2 and 8 are congruent to powers of each other and the roots 7 and 13 are congruent
For a contrasting example, if {{math | ''n'' {{=}} 9}}, then <math>\lambda(n)=\varphi(n)=6</math> and <math>\varphi(\lambda(n))=2</math>. There are two primitive {{mvar | λ}}-roots modulo 9, namely 2 and 5, each of which is congruent to the fifth power of the other. They are also both primitive <math>\varphi</math>-roots modulo 9.
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