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Just a quick question. Do you think it makes more sense to introduce Euler's Theorem and then prove Fermat's Little Theorem as a corollary or prove Fermat's Little Theorem and generalise to Euler's Theorem?

Tsing Shi Tao
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  • I guess it would depend on what I was proving it for. – XRBtoTheMOON Nov 02 '17 at 22:47
  • In group theory you also have Lagrange ! – Maman Nov 02 '17 at 22:48
  • I would say Fermat little theorem is special because we can prove it by induction $a^p \equiv a \bmod p \implies (a+1)^p \equiv a+1 \bmod p$ using the binomial formula, and it is certainly the best way to introduce the rings $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$ rigorously. Only after, show Euler and Lagrange using that $(\mathbb{Z}/n\mathbb{Z})^\times$ is a group with $\varphi(n)$ elements – reuns Nov 03 '17 at 02:24
  • Thanks for your answers. It seems like it mostly depends on context and it's probably worth proving each one on its own to show how FLT is important but Euler's Theorem is the more useful general case. – Tsing Shi Tao Nov 05 '17 at 15:34

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Just an opinion.

The books I read have given Fermat theorem as a corollary of Euler theorem. It also makes sense as you can easily obtain a proof of Fermat theorem by Euler theorem. The concept of introducing Fermat theorem first is used by some lecturers in their lecture notes to indicate the importance of the theorem. But still I would prefer Euler theorem first.

BTW I use Niven Zukerman Montgomery

Vidyanshu Mishra
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I would prove Euler's theorem (first) and then Fermat's little theorem. Note the standard proof of Euler's theorem is (if n is fixed) and $T=\{a_1, a_2, \cdots, a_{\phi(n)}\}$ are the numbers less than (or equal to) and co-prime and $b$ is also co-prime to $n$, then consider $T'=\{ba_1, ba_2, \cdots, ba_{\phi(n)}\}$. It should be easy to show $T'$ and $T$ are in fact the same set, just written differently. Multiplying out all the elements of $T$, and all the elements of $T'$ and equating them, we see $b^{\phi(n)}=1$ $\text(mod) n$. When $n$ is prime, then (?)

Surajit131
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