mm 
795] numbers. 597 
one and only one of these prime residues. For instance, the prime residues may be 
the numbers less than IV and prime to it. In particular, if IV is a prime number p, 
then the residues may be the p— 1 numbers, 1, 2, 3, — 1. 
In all that follows, the letter p, in the absence of any statement to the contrary, 
will be used to denote an odd prime other than unity. A theorem for p may hold 
good for the even prime 2, but it is in general easy to see whether this is so or not. 
11. Fermat’s theorem, a^ -1 —1 = 0 (mod.y>). The generalized theorem is —1 = 0 
(mod. IV). The proof of the generalized theorem is as easy as that of the original 
theorem. Consider the series of the </>(IV) numbers a, b, c,..., each less than IV and 
prime to it; let x be any number prime to IV, then each of the numbers xa, xb, x c,..., 
is prime to IV, and no two of them are congruent to the modulus IV, that is, we 
cannot have x (a — b) = 0 (mod. IV); in fact, x is prime to A r , and the difference a — b 
of two positive numbers each less than A r will be less than IV. Hence the numbers 
xa, xb, xc,..., are in a different order congruent to the numbers a, b, c, ... ; and 
multiplying together the numbers of each set we have a&W abc ... = abc ... (mod. IV), or, 
since a, b, c, ..., are each prime to IV, and therefore also the product abc... is prime 
to IV, we have xfrW = 1, or say x^W — 1 = 0 (mod. IV). 
In particular, if A be a prime number =p, then </> (IV) is =p — l, and the 
theorem is xP~ Y — 1 = 0 (mod. p), x being now any number not divisible by p. 
12. The general congruence f(x) = 0 (mod. p). f(x) is written to denote a rational 
and integral function with integer coefficients which may without loss of generality 
be taken to be each of them less than p ; it is assumed that the coefficient A of the 
highest power of x is not = 0. If there is for x an integer value a such that 
f(a) = 0 (mod. p, throughout), then a is said to be a root of the congruence f(x)= 0; 
we may, it is clear, for a substitute any value whatever a' = a + kp, or say any value 
a which is = a, but such value a' is considered not as a different root but as the same 
root of the congruence. We have thus /(a) = 0; and therefore f{x)=f{x)—f(a), 
= (x — a) f x (x), where _/i (x) is a function of like form with f(x), that is, with integer 
coefficients, but of the next inferior order n— 1. Suppose there is another root b of the 
congruence, that is, an integer value b such that f(b) = 0 ; we have then (6 — a) (b) = 0, 
and b — a is not = 0 (for then b would be the same root as a). Hence f x (b) = 0, and 
f(x) = {x — a){f 1 {x)~f l (b)}, = {x — a)(x — b)f 2 (x), where f 2 (x) is an integral function 
such as f(x), but of the order n— 2; and so on, that is, if there exist n different 
(non-congruent) roots of the congruence f(x) = 0, then f(x) = A (x — a) (x — b) ... (x - k), 
and the congruence may be written A(x — a)(x—b) ... (x — k) = 0. And this cannot be 
satisfied by any other value l; for if so we should have A(l —a) (l— b) ...(l—k) = 0, 
that is, some one of the congruences (I — a) = 0, &c., would have to be satisfied, and 
l would be the same as one of the roots a, b, c, ..., k. That is, a congruence of the 
order n cannot have more than n roots, and if it have precisely n roots a, b, c, ..., k, 
then the form is / (x) = A (x — a) (x — b)... (x — k), = 0. 
Observe that a congruence may have equal roots, viz. if the form be 
f(x) = A (x — a) a (x — by..., =0,