350 KEPORTS ON THE STATE OF SCIENCE. — 1915, 



Prime Numbers. By G. H. Haedy, F.B.S. 



(Ordered by the General Committee to be printed in extenso.) 



The Theory of Numbers has always been regarded as one of the most 

 obviously useless branches of Pure Mathematics. The accusation is one 

 against which there is no valid defence ; and it is never more just than 

 when directed against the parts of the theory which are more particularly 

 concerned with primes. A science is said to be useful if its development 

 tends to accentuate the existing inequalities in the distribution of wealth, 

 or more directly promotes the destruction of human life. The theory 

 of prime numbers satisfies no such criteria. Those who pursue it will, 

 if they are wise, make no attempt to justify their interest in a subject 

 so trivial and so remote, and will console themselves with the thought 

 that the greatest mathematicians of all ages have found in it a 

 mysterious attraction impossible to resist. 



The foundations of the theory were laid by Euclid. Among Euclid's 

 theorems two in particular are of fundamental importance. The first 

 (Euc. vii. 24) is that if a and h are both prime to c, then ab is also prime 

 to c. This theorem is the basis of the whole theory of the factorisation 

 of numbers, systematised later by Euler and by Gauss, and in particular 

 of the theorem that every number can be expressed in one and only one way 

 as a product of primes. The second theorem (Euc. ix. 20) is that the 

 mimber of primes is infinite : to this theorem I shall return in a moment. 



In modern times the theory has developed in two different directions. 

 In the first place there is what may be called roughly the theory of in- 

 dividual or isolated primes, a theory which it is difficult to define precisely, 

 but of which a general idea may be formed by considering a few of its 

 characteristic problems. How can we determine whether a given number 

 is prime ? what conditions are necessary and what sufficient ? Can 

 we define forms which represent prime numbers only ? Are there 

 infinitely many pairs of primes which differ by 2 ? Is (as Goldbach 

 asserted) every even number the sum of two primes ? This theory 1 

 shall dismiss very briefly. We know a number of very beautiful theorems 

 of this character. I need only mention Wilson's theorem, Format's 

 theorem, and the extensions of the latter by Lucas. But on the whole 

 the record of research in this direction is a record of failure. The difii- 

 culties are too great for the methods of analysis at our command, and 

 the problems remain unsolved. 



Very different results are revealed when we turn to the second 

 principal branch of the modern theory, the theory of the average or 

 asymptotic distribution of primes. This theory (though one of its most 

 famous problems is still unsolved) is in some ways almost complete, and 

 certainly represents one of the most remarkable triumphs of modern 

 analysis. The theory centres round one theorem, the Primzahlsatz or 

 Prime Number Theorem ; and it is to the history of this theorem, which 

 may almost be said to embody the history of the whole subject, that I 

 shall devote the remainder of this lecture.* 



The problem may be stated crudely as follows : How many primes 



* A full account of the history of the theorem will be found in Landau'f? Iland- 

 buch der Lehre von der Verteilung der Primzahlen (Teubner, 1909). 



