Some basics of Number Theory for RMO: part III: Fermat’s Little Theorem

Fermat’s Little Theorem:

The fact that there are only a finite number of essentially different numbers in arithmetic to a modulus m means that there are algebraic relations which are satisfied by every number in that arithmetic. There is nothing analogous to these relations in ordinary arithmetic.

Suppose we take any number x and consider its powers x, x^{2}, x^{3}, \ldots. Since there are only a finite number of possibilities of these to the modulus m, we must eventually come to one which we have met before, say x^{h} \equiv x^{k} {\pmod m}, where k <h. If x is relatively prime to m, the factor x^{k} can be cancelled, and it follows that x^{l} \equiv 1 {\pmod m}, where l \equiv {h-k}. Hence, every number x which is relatively prime to m satisfies some congruence of this form. The least exponent l for which x^{l} \equiv 1 {\pmod m} will be called the order of x to the modulus m. If x is 1, its order is obviously 1. To illustrate the definition, let us calculate the orders of a few numbers to the modulus 11. The powers of 2, taken to the modulus 11, are

2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, \ldots

Each one is twice the preceding one, with 11 or a multiple of 11 subtracted where necessary to make the result less than 11. The first power of 2 which is \equiv 1 is 2^{10}, and so the order of 2 \pmod {11} is 10. As another example, take the powers of 3:

3, 9, 5, 4, 1, 3, 9, \ldots

The first power of 3 which is equivalent to 1 is 3^{5}, so the order of 3 \pmod {11} is 5. It will be found that the order of 4 is again 5, and so also is that of 5.

It will be seen that the successive powers of x are periodic; when we have reached the first number l for which x^{l} \equiv 1, then x^{l+1} \equiv x and the previous cycle is repeated. It is plain that x^{n} \equiv 1 {\pmod m} if and only if n is a multiple of the order of x. In the last example, 3^{n} \equiv 1 {\pmod 11} if and only if n is a multiple of 5. This remains valid if n is 0 (since 3^{0} = 1), and it remains valid also for negative exponents, provided 3^{-n}, is interpreted as a fraction (mod 11) in the way explained earlier (an earlier blog article).

In fact, the negative powers of 3 (mod 11) are obtained by prolonging the series backwards, and the table of powers of 3 to the modulus 11 is:

\begin{array}{cccccccccccccc}    n & = & \ldots & -3 & -2 & -1 & 0 & 1 &2 & 3 & 4 & 5 & 6 & \ldots \\    3^{n} & \equiv & \ldots & 9 & 5 & 4 & 1 & 3 & 9 & 5 & 4 & 1 & 3 & \ldots \end{array}

Fermat discovered that if the modulus is a prime, say p, then every integer x not congruent to 0 satisfies

x^{p-1} \equiv 1 {\pmod p}….call this as equation A.

In view of what we have seen above, this is equivalent to saying that the order of any number is a divisor of p-1. The result A was mentioned by Fermat in a letter to Frenicle de Bessy of 18 October 1640, in which he also stated that he had a proof. But, as with most of Fermat’s discoveries, the proof was not published or preserved. The first known proof seems to have been given by Leibniz (1646-1716). He proved that x^{p} \equiv x {\pmod p}, which is equivalent to A, by writing x as a sum 1+ 1 + 1 + \ldots + 1 of x units (assuming x positive), and then expanding (1+1+ \ldots + 1)^{p} by the multinomial theorem. The terms 1^{p} + 1^{p} + \ldots + 1^{p} give x, and the coefficients of all the other terms are easily proved to be divisible by p.

Quite a different proof was given by Ivory in 1806. If x \not\equiv 0 {\pmod p}, the integers

x, 2x, 3x, \ldots, (p-1)x

are congruent in some order to the numbers

1, 2, 3, \ldots, p-1.

In fact, each of these sets constitutes a complete set of residues except that 0 (zero) has been omitted from each. Since the two sets are congruent, their products are congruent, and so

(x)(2x)(3x) \ldots ((p-1)x) \equiv (1)(2)(3)\ldots (p-1){(\pmod p)}

Cancelling the factors 2, 3, ….(p-1), as is permissible we obtain the above relation A.

One merit of this proof is that it can be extended so as to apply to the more general case when the modulus is no longer a prime.

The generalization of the result A to any modulus was first given by Euler in 1760. To formulate it, we must begin by considering how many numbers in the set 0, 1, 2, …, (m-1) are relatively prime to m. Denote this number by \phi(m). When m is a prime, all the numbers in the set except 0 (zero) are relatively prime to m, so that \phi(p) = p-1 for any prime p. Euler’s generalization of Fermat’s theorem is that for any modulus m,

x^{\phi(m)} = 1 {\pmod m}…relation B

provided only that x is relatively prime to m.

To prove this, it is only necessary to modify Ivory’s method by omitting from the numbers 0, 1, 2, \ldots, (m-1) not only the number 0, but all numbers which are not relatively prime to m. These remain \phi(m) numbers, say

a_{1}, a_{2}, \ldots, a_{\mu}, where \mu = \phi(m).

Then, the numbers

a_{1}x, a_{2}x, \ldots, a_{\mu}x

are congruent, in some order, to the previous numbers, and on multiplying and cancelling a_{1}, a_{2}, \ldots, a_{\mu} (as is permissible) we obtain x^{p} \equiv 1 {\pmod m}, which is relation B.

To illustrate this proof, take m=20. The numbers less than 20 and relatively prime to 20 are :

1, 3, 7, 9, 11, 13, 17, 19.

So that \phi(20) = 8. If we multiply these by any number x which is relatively prime to 20, the new numbers are congruent to the original numbers in some other order. For example, if x is 3, the new numbers are congruent respectively to

3, 9, 1, 7, 13, 19, 11, 17 {\pmod 20};

and the argument proves that 3^{8} \equiv 6561.

Reference:

  1. The Higher Arithmetic, H. Davenport, Eighth Edition.
  2. Elementary Number Theory, Burton, Sixth Edition.
  3. A Friendly Introduction to Number Theory, J. Silverman

Shared for those readers who enjoy expository articles.

Nalin Pithwa.

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