2
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if $n=p_1^{a_1}p_2^{a_2}p_3^{a_3}\dots p_r^{a_r}$

and

$\lambda(n) = lcm[(p_1-1)(p_1^{a_1-1}),(p_2-1)(p_2^{a_2-1}),(p_3-1)(p_3^{a_3-1}),\dots,(p_r-1)(p_r^{a_r-1})]$

then

$k^{\lambda{n}} \equiv 1(\mod{n}) $

for all $k$ such that $gcd(k,n) = 1 $

where $p_1,p_2,p_3,\dots,p_n$ are prime numbers and $\lambda(n)$ is Carmichael function

please provide a proof for the above Carmichael theorem

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3
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By Euler's Theorem, $k^{\varphi(p_i^{a_i})}\equiv 1\pmod{p_i^{a_i}}$. Note that $\varphi(p_i^{a_i})$ divides $\lambda(n)$. It follows that $k^{\lambda(n)}\equiv 1\pmod{p_i^{a_i}}$ for all $i\le r$, and therefore $k^{\lambda(n)}\equiv 1\pmod{n}$.

Remark: The definition given in the OP is not quite the usual definition of the Carmichael function, and is not equivalent to it. (Everything is fine except for the prime $2$.)

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