CHAPTER XVIII. 
THEORY OF PRIME NUMBERS. 
Existence of an Infinitude of Primes. 
Euclid^ noted that, if p were the greatest prime, and M = 2-S-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 contra- 
diction. 
L. Euler^ deduced the theorem from the [invahd] equation 
s -=n(i--) , 
„=in V p/ 
the left member being infinite and the right finite if there be only a finite 
number of primes. Euler^ concluded from the same equation that "the 
number of primes exceeds the number of squares." 
Euler^ modified Euclid's^ argument slightly. The number of integers 
<M and prime to M is <t>(M) =2-4. . . (p — 1), so that they include integers 
which are either primes >p or have prime factors >p. 
The theorem follows from Tchebychef's^^^ proof of Bertrand's postulate. 
L. Kronecker^ noted that we may rectify Euler's^ proof by using 
2 -^=nfl-i) ' (s>l). 
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. Jaensch® repeated Euler's- argument, also ignoring convergency. 
E. Kummer'^ gave essentially Euler's'* argument. 
J. Perott^ noted that, if pi,. . ., p„ are the primes ^N, there are 2" 
integers ^N which are not divisible by a square, and 
2">iV- 
"-mx'-^i^'^'-iyi- 
Hence there exist infinitely many primes. 
L. Gegenbauer^" proved the theorem by means of 2"ii*n~*. 
lElementa, IX, 20; Opera (ed., Heiberg), 2, 1884, 388-91. 
^Introductio in analysin infinitorum, 1, Ch. 15, Lausanne, 1748, p. 235; French transl. by 
J. B. Labey, 1, 218. 
'Comm. Acad. Petrop., 9, 1737, 172-4. 
^Posthumous paper. Coram. Arith. Coll., 2, 518, Nos. 134-6; Opera Postuma, I, 1862, 18. 
'Vorlesungen uber Zahlentheorie, I, 1901, 269-273, Lectures of 1875-6. 
*Die Schwierigeren Probl. Zahlentheorie, Progr. Rastenburg, 1876, 2. 
'Monatsber. Ak. Wiss. Berlin fiir 1878, 1879, 777-8. 
sBull. sc. math, et astr., (2), 5, 1881, I, 183-4. 
8«Sitzungsber. Ak. Wiss. Wien (Math.), 95, II, 1887, 94-6; 97, Ila, 1888, 374-7. 
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