What is most motivating way to introduce Fermat's Little Theorem

Question

What is the best way to introduce Fermat’s Little Theorem (F$l$T) to students? What can I use as an opening paragraph which will motivate and have an impact on why students should learn this theorem and what are the applications of F$l$T? Are there any good resources on this topic?

Answer 1

I wrote an article for NCTM's grades 8-14 journal, The Mathematics Teacher, which is about building towards ${\rm F}\ell{\rm T}$ by thinking through ways in which the following statement can be generalized:

If you subtract a natural number from its square, then the result is even.

I wrote more about this in some earlier StackExchange posts, e.g., MESE 907. Since then, the article has appeared; its citation and link are:

Dickman, B. (2017). Enriching divisibility: multiple proofs and generalizations. The Mathematics Teacher, 110(6), pp. 416-423. Link (without pay-wall).

The write-up is less concerned with specific applications, and has as its goal, instead, to indicate how the process of generalizing a routine number theoretic proposition can lead to theorems like ${\rm F}\ell{\rm T}$.

Answer 2

Look at patterns in decimal expansions: what is the period of the repeating decimal of $1/n$? From numerical data, the period is at most $n-1$, and you only get equality when $n = p$ is prime (but not conversely: $1/11$ has decimal period $2$, not $10$). If you look at the period of the decimal of $1/p$ for primes $p$ besides $2$ and $5$, you find that although the period is not always $p-1$ it does always appear to divide $p-1$.

That divisibility (sharper than an inequality) is basically Fermat's little theorem. The period of the decimal for $1/n$ is the order of $10 \bmod n$ (for $n$ relatively prime to $10$), so the fact that the period of the decimal for $1/p$ when $p$ is a prime (besides $2$ or $5$) is a factor of $p-1$ is equivalent to saying $10^{p-1} \equiv 1 \bmod p$. This is possibly how Gauss was led to Fermat's little theorem.

Answer 3

Beauty. I really enjoy the combinatorial necklace proof, which alone could motivate it.

Another way to think of it is a representation which I first learned of in Wagon and Bressoud, where you look at a table of all powers of the integers (or units) modulo $p$ and color-code entries by least nonnegative residue. Then the last column of the table is all one color, just like the first column; the first column is $a^0\equiv 1$, but the last is $a^{p-1}\equiv 1$; so we have a visualization of FlT (among many other results).

Try it out! (Quick Sage code below, you may wish to relabel the rows or columns.)

@interact
def power_table_plot(p=(7,prime_range(50))):
    P=matrix_plot(matrix(p-1,[mod(a,p)^b for a in range(1,p) for b in srange(p)]),cmap='jet')
    show(P,figsize=6)

Answer 4

Generally We enter the subject(FLT) with a problem like $a^m \equiv ? \pmod{n}$.

After the training like these:

$3^{100}\equiv ? \pmod{5}$, $2^{80}\equiv ? \pmod{7}$, ... etc

Students are important to 'We need to smallest positive $m$ that $3^{m}\equiv 1 \pmod{5}$ or $2^{m}\equiv 1 \pmod{7}$, ... etc. (in fact $m$ is order of $3$ in $\mod 5$.) But sometimes we are in difficulty for finding the order. Especially for bigger $p$ prime numbers in $ \mod{p} $. If we ask to students $2^{100}\equiv ? \pmod{101}$, they will spend a lot of time. Thus, we can give FLT to students. If students do not use FLT, they will conclude that it will be very difficult.

Furthermore, if $\gcd (a,p)=1 $ then $a^{p-1}\equiv 1 \pmod{p}$ but $p-1$ coudn't be order of $a$ in $\mod p$. So, we can say $\text{ord}(a) \mid {p-1}$.

Last of all; because of The World need to the order(!), FLT is important for us:)