Chúng ta có thể sử dụng quan sát này để có được một chứng minh rằng có số nguyên tố vô cùng nhiều. Vì nếu có chỉ là số nguyên tố k 2, 3,. . . , Vn, sau đó chúng tôi có n! 1, kể từ khi mỗi nguyên tố có thể góp phần vào yếu tố của 2 "- 1. | FACTORIAL FACTORS 115 We can use this observation to get another proof that there are infinitely many primes. For if there were only the k primes 2 3 . . . pk then we d have n 2n k 2nk for all n 1 since each prime can contribute at most a factor of 2 - 1. But we can easily contradict the inequality n 2nk by choosing n large enough say n Then n 2nk 222kk TLn 2 contradicting the inequality n nn 2 that we derived in . There are infinitely many primes still. We can even beef up this argument to get a crude bound on the number of primes not exceeding n. Every such prime contributes a factor of less than 2n to n so as before n 2n7r n . If we replace n here by Stirling s approximation which is a lower bound and take logarithms we get n7i n nlg n e i lg 27m. hence rt n lg n e . This lower bound is quite weak compared with the actual value because logn is much smaller than n logn when n is large. But we didn t have to work very hard to get it and a bound is a bound. RELATIVE PRIMALITY Like perpendicular lines don t have a common direction perpendicular numbers don t have common factors. When gcd m n the integers m and n have no prime factors in common and we say that they re relatively prime. This concept is so important in practice we ought to have a special notation for it but alas number theorists haven t come up with a very good one yet. Therefore we cry Hear us 0 Mathematicians of the World Let US notwaitanylonger We can makemanyformulas clearer BY DEFINING A new notation now Let us AGREE TO write m _L n AND TO SAY m IS prime to n IF m and n ARE RELATIVELY PRIME. In other words let us declare that are integers and gcd m n 116 NUMBER THEORY A fraction m n is in lowest terms if and only if m i n. Since we reduce fractions to lowest terms by casting out the largest common factor of numerator and denominator we suspect that in general m gcd m n 1 n gcd m n and indeed this is true. It follows from a more general law gcd km kn kgcd m n proved in .
