CHAPTER XVIIL THEORY OF PRIME NUMBERS. EXISTENCE OF AN INFINITUDE OF PRIMES. Euclid1 noted that, if p were the greatest prime, and M= 2-3-5. . .p is the product of all the primes ^p, then M+l is not divisible by one of those primes and hence has a prime factor >p, thus involving a contradiction. L. Euler2 deduced the theorem from the [invalid] equation n,i n \ p/ the left member being infinite and the right finite if there be only a finite number of primes. Euler3 concluded from the same equation that "the number of primes exceeds the number of squares/' Euler4 modified Euclid's1 argument slightly. The number of integers < M and prime to M is $(M) =24. . . (p—1), so that they include integers which are either primes >p or have prime factors >p. The theorem follows from TchebychefV61 proof of Bertrand's postulate. L. Kronecker6 noted that we may rectify Euler's2 proof by using where p ranges over all primes >1. If there were only a finite number of p's, the product would remain finite when s approaches unity, while the sum increases indefinitely. He also gave the proof a form leading to an interval from m to n within which there exists a new prime however great m is taken. R. Jaensch6 repeated Euler's2 argument, also ignoring convergency. E. Kummer7 gave essentially Euler 's4 argument. J. Perott8 noted that, if PI,. . ., pn are the primes ^AT, there are 2n integers ^N which are not divisible by a square, and Hence there exist infinitely many primes. L. Gegenbauer8a proved the theorem by means of SJI*n~"a. 'Elementa, IX, 20; Opera (ed., Heiberg), 2, 1884, 388-91. 2Introductio in analysin infinitorum, 1, Ch. 15, Lausanne, 1748, p. 235; French transl. by J. B. Labey, 1, 218. 8Comm. Acad. Petrop., 9, 1737, 172-4. 4Posthumous paper, Comm. Arith. Coll., 2, 518, Nos. 134-6; Opera Postuma, I, 1862, 18. 6Vorlesungen iiber Zahlentheorie, I, 1901, 269-273, Lectures of 1875-6. •Die Schwierigeren Probl. Zahlentheorie, Progr. Rastenburg, 1876, 2. 7Monatsber. Ak. Wiss. Berlin ftir 1878, 1879, 777-8. "Bull. BC. math, et astr:, (2), 5, 1881, 1, 183-4. »«Sitzungsber. Ak. Wiss. Wien (Math.), 95, II, 1887, 94-6; 97, Ha, 1888, 374-7.