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## Christening Fermat’s Little Theorem

I am writing an article on Fermat's work in number theory and feel uncomfortable everytime I have to write "Fermat's Little Theorem": it's clumsy and belittles the fundamental character of Fermat's result. "Fermat's Theorem" is too ambiguous, and I don't really like acronyms such as Flt or Flit. Has anyone ever seen a better name for this result (or a new suggestion)?

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Are you thinking of something like "Fermat's primality test"? – Gjergji Zaimi Jul 27 2010 at 13:58
"Fermat's First Theorem"? Roughly as historically accurate as "Fermat's Last Theorem"... – Cam McLeman Jul 27 2010 at 18:56
Franz, "Fermat's Little Theorem" in no way belittles the result but "Fermat's little Theorem" (which I've seen many times) does! – Wadim Zudilin Jul 27 2010 at 22:03
You have the ability to rename without harm. For example: ' "insert congruence here " (9) is also known as Fermat's Little Theorem; for this article, we shall refer to it as one of "Congruence 9", "K-9", or "Fred" . ' Something like that should work. Gerhard "Results Renamed While You Wait" Paseman, 2010.07.27 – Gerhard Paseman Jul 27 2010 at 23:41
Victor, that's a good idea for Franz: "Petit théorème de Fermat"! Franz, you can call it "Fermat's Petit Theorem" (!) with a nice abbreviation FPT reminding on Gjergji's "Fermat's primality test". Pourquoi pas?! – Wadim Zudilin Jul 28 2010 at 7:57

I think you shouldn't change the name. It's universally known as Fermat's Little Theorem, and especially if you're writing a survey or historical article, you're not in a place to try to revolutionize established mathematical nomenclature. There are many instances of unfortunate terminology in mathematics, but in my opinion, once they are in general use, they become part of the lore and the culture. I would make exceptions only in a few cases, such as:

a) it's on the level of adjectives such as "good" and "admissible", b) it's crediting the wrong person (Cayley numbers, Burnside's lemma), or c) it's very recent, with the inventor implicitly begging to attach his name to it

And if your life work is going to become known as "Lemmermeyer's dirty trick", well, take it with humor.

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Stick to the standard name, however much it bothers you right now. (Is there any major mathematical language in which it is not called his little theorem?). I do have a different suggestion too. The first time you introduce the congruence, please describe it using correct quantifiers: $a^{p-1} = 1 \bmod p$ for all $a \not= 0 \bmod p$ (or $a^p = a \bmod p$ for all $a \bmod p$). That is the essential point. Last fall I asked my class what Fermat's little theorem is (we had discussed it in the previous class). One said "if $p$ is prime then for some $a$, $a^{p-1} = 1 \bmod p$." Ouch. – KConrad Jul 28 2010 at 0:09
I also have another suggestion: show that Fermat's little theorem is used in RSA. Nearly all books I have seen that discuss RSA use Euler's theorem, which compels the authors to say RSA requires that the message to be encoded is rel. prime to the modulus. This is wrong: RSA works for all numbers. That is, if p and q are different primes and d and e are positive integers such that de = 1 mod(p−1)(q−1) then $x^{de} = x \bmod pq$ for all integers $x$. To prove this, work mod $p$ and mod $q$ separately and use Fermat. Even the original RSA paper uses Fermat! – KConrad Jul 28 2010 at 0:20

The MathWorld entry for Fermat's Little Theorem claims "Fermat's simple theorem" has been used as an alternate name. Also the entry for Fermat's Congruence consists of a link to the Fermat's Little Theorem page.

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Compared to Fermat's two squares theorem, or Fermat's four squares theorem, Fermat's Little theorem is indeed Little.

Not to mention the hard-to-prove Fermat Last Theorem, which goes under FLT; so that acronym, or a contraction Flt isn't suitable as it will cause confusion.

Therefore one might as well stick with "Fermat's Little Theorem" itself. I have given above a reasoning that it is comparatively little.

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I disagree. The other theorems of Fermat you have mentioned are perhaps more beautiful (debatable), but Fermat's "Little" Theorem is by far more fundamental and more important than these theorems. – danseetea Jul 27 2010 at 14:54
I think Anweshi means little in the sense that it is easy to prove. Although the warning of Grothendieck comes to mind... – Andrea Ferretti Jul 27 2010 at 15:16
@Andrea Ferretti: I'm curious -- what warning of Grothendieck? – Daniel Litt Jul 27 2010 at 15:51
Daniel, I think Andrea might be referring to some of the discussion at the following question, especially Coudy's answer: mathoverflow.net/questions/28788/… – Dan Ramras Jul 27 2010 at 17:18
Exactly that one. Although it appears that the quote on the snobism of young people is due to Whitehead. – Andrea Ferretti Jul 27 2010 at 17:25

Der kleine Fermatsche Satz? This is how it started.

Fermat's congruence. This would be in line with Kummer's congruence, or Clausen-von-Staudt congruence etc.

The success of this name is foreseeable.

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I agree with Gjergji Zaimi's comment above: Both "Fermat test" and "Fermat primality test" are short and descriptive.

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But "the Fermat primality test" seems to refer to something a bit different from the usual statement of Fermat's Little Theorem: en.wikipedia.org/wiki/Fermat_primality_test. – Dan Ramras Jul 27 2010 at 17:13
It seems the theorem is the necessary condition behind the test. Perhaps "Fermat pseudoprime criterion"? – S. Carnahan Jul 27 2010 at 22:56
Dan is right. Don't label the congruence that is called Fermat's Little Theorem by the standard name for one of its applications. – KConrad Jul 28 2010 at 0:03