MATHEMATICS 



unsolved. After what I have already said perhaps 

 none of my readers will raise any question as 

 to the ultimate utility, from any point of view 

 whatever, of such a general solution. 



In connection with problems of this nature, a 

 point of much interest, generally unappreciated by 

 the non-mathematician, emerges. Pure Mathe- 

 matics is often regarded in certain respects truly 

 as a kind of sublimated arithmetic. But this 

 view must not be carried too far, and I take this 

 occasion to point out a serious pitfall, which is not 

 sufficiently well known. I shall begin with a few 

 remarks upon prime numbers. A prime number 

 is defined as one which has no divisor smaller 

 than itself (except, of course, the number i, 

 which divides everything). The first few prime 

 numbers are 



i, 2, 3, 5,7, n, 13, 17, 19, 23, ... 



The study of prime numbers is one of the most 

 interesting, and yet difficult, branches of Pure 

 Mathematics. It illustrates what I have just said. 

 A formula has been known for some years, which 

 gives very closely the number of prime numbers 

 less than a given number. It has been tested by 

 arithmeticians, who by direct counting verified it 

 up to given numbers of many thousands, and it 

 was believed to continue indefinitely as a truthful 

 formula. Mr Littlewood has shewn recently that 



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