## Topological Vector Spaces by H.H. Schaefer

By H.H. Schaefer

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**Example text**

Hence every number x which is relatively prime to m satisfies some congruence of this form. The least exponent l for which x l ≡ 1 (mod 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, . . Each one is twice the preceding one, with 11 or a multiple of 11 subtracted where necessary to make the result less than 11.

3) 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 (3) was mentioned by Fermat in a letter to Fr´enicle 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 ≡ x (mod p), which is equivalent to (3), by writing x as a sum 1 + 1 + · · · + 1 of x units (assuming x positive), and then expanding (1 + 1 + · · · + 1) p by the multinomial theorem.

The conjecture seems to have been based on numerical evidence. 084 . . Numerical evidence of this kind may, of course, be quite misleading. But here the result suggested is true, in the sense that the ratio of π(X ) to X/ log X tends to the limit 1 as X tends to infinity. This is the famous Prime Number Theorem, first proved by Hadamard and de la Vall´ee Poussin independently in 1896, by the use of new and powerful analytical methods. It is impossible to give an account here of the many other results which have been proved concerning the distribution of the primes.