126 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

For example, 2115 1

CONGRUENCE OF POWERS

301. Fermat’s Theorem.
If m is a positiv prime number and a —>> m,

then
a™! = 1 (mod m) ().
By the hypothesis allm>1 § 164.
Hence
a 1
2a 2
3ar = 3 (mod m) § 298.
(m — 1)21 w1

Whatever the order of the numbers on the right, the product
of the numbers on the left is congruent to the product of the
numbers on the right. § 215.

Hence [1:2:3++-(m—1)Jam»1=1:2-3:++ (m — 1) (modm)

But, since m is a prime number, each of the numbers 1, 2, 3,

-+, m — 1 is prime to m. § 165.
Hence their product is prime to m (?).

Therfor a™ ! =1 (mod m) § 281.

(1) See Tannery, l. c., p. 468.
(2) Theory of Integers, § 632.