Is $n^{800}= 1 \mod{800}$ given that $n$ is relatively prime to 800?

Question

I was playing around with some numbers and it seems that for a lot of numbers that are relatively prime to 800 $$n^{800} = 1 \mod{800}$$ is this true in general, or is it just coincidental? Is it true that $$n^k=1 \mod{k}$$ where $n,k$ are relatively prime?

See Euler's Theorem and Carmichael's Lambda function – Bill Dubuque Jan 31 '19 at 23:54 — Bill Dubuque, Jan 31 '19 at 23:54
See also OEIS sequence A124240 – Robert Israel Feb 01 '19 at 00:02 — Robert Israel, Feb 01 '19 at 00:02
Thanks Robert, that is very interesting! – Peter Foreman Feb 01 '19 at 00:24 — Peter Foreman, Feb 01 '19 at 00:24

lhf · Accepted Answer · 2019-02-01T00:20:42.293

2

Nice observation! It is true for $800$ but not in general.

The explanation is given by Carmichael's theorem: $a^{\lambda(m)} \equiv 1 \bmod m$ for all $a$ coprime with $m$.

Since $\lambda(800)=40$, we have $a^{800} = (a^{40})^{20} \equiv 1^{20} = 1 \bmod 800$ for all $a$ coprime with $m$.

edited Feb 01 '19 at 00:20

answered Feb 01 '19 at 00:13

lhf

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Is $n^{800}= 1 \mod{800}$ given that $n$ is relatively prime to 800?

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