NATURE 



[January 5, 1922 



known skill of the author, who has added to our 

 libraries a most useful and interesting work. 

 Both he and the Wireless Press, which has pro- 

 duced the book, may be cordially congratulated 

 on the result of their labours. 



Fermat's Last Theorem. 



Three Lectures on Fermat's Last Theorem. 

 By L, J. Mordell. Pp. vii + 31. (Cambridge: 

 At the University Press, 192 1.) 45. net. 



THE " last theorem of Fermat " states that if 

 X, y, z, p denote positive integers, the equa- 

 tion :>cP + yP = zP is impossible if p exceeds 2 : thus 

 no cube can be the sum of two cubes, and so on. 

 If the theorem is true when p is 4, or an odd prime, 

 it is true for all other integral values of p. For 

 three centuries this theorem has baffled the efforts 

 of all who have attacked it, although it has 

 attracted the attention of all first-rate arith- 

 meticians, and a great number of amateurs. For 

 ^ = 3. 4> 5) 7 comparatively simple proofs have 

 been discovered; but so far none of these has 

 led to a complete generalisation. 



The first great advance in the theory was made 

 by Kummer, in connection with his researches on 

 cyclotomic integers. He showed that if the 

 theorem is false for any particular odd prime p, 

 then p must not be a factor of the numerator of 

 any one of the first Up-3) numbers of Bernoulli. 

 This very recondite test rules out all values of 

 p below 100 except 37, 59, 67. By additional 

 criteria Kummer was able to prove the theorem 

 for these exceptional primes, and hence for all 

 values of p from 3 to 100 inclusive. 



Not many years ago (1907) a prize of 100,000 

 marks was set aside for the first who succeeded 

 in giving a complete proof or disproof of the 

 theorem. Quite recently, new criteria, indepen- 

 dent of Kummer's, have been discovered, which 

 have to be satisfied by odd primes p for which 

 the theorem is false, and the simplest of these is 

 the condition 2^-^=1 (mod. p^), discovered by 

 Wilferich in 1909. Other tests of a more or less 

 similar kind have been accumulated, and the net 

 result is that any value of p for which the theorem 

 is false must exceed 7000. Gauss's tables of 

 quadratic forms warn us not to draw any con- 

 clusions from this result; in fact if N is any 

 assigned integer, however large, a proof that the 

 theorem is true unless ^>N gives us no infor- 

 mation about the truth or falsity of the theorem 

 in general. 



Mr. Mordell's lectures give a clear and interest- 

 ing account of the history and present state of 

 this subject. Lecture I. gives a statement of the 

 NO. 2723, VOL. 109] 



theorem, and a summary of the work done by 

 Kummer's predecessors; Lecture H". is on 

 Kummer's researches, and more recent investiga- 

 tions of similar type; and Lecture HL gives an 

 account of various results obtained by Libri, 

 Sophie Germain, and others. Full references are 

 given to the original papers, so that a reader 

 within reach of a good reference library can make 

 himself acquainted with details of all that has been 

 done hitherto. 



A perplexing circumstance, often alluded to, is 

 the fact that, in a private note, Fermat distinctly 

 asserted that he had proved the theorem. Now 

 Fermat was never convicted of a false assertion, 

 and only once of a wrong conjecture ; on the other 

 hand it is extremely improbable that Fermat's 

 proof, if he had one, was in any way analogous to 

 the work of Kummer and his successors. It is 

 not, perhaps, unreasonable to hope that a proof 

 may be found, some day, derived from Diophan- 

 tine analysis proper, combined with a process of 

 induction, and possibly with some application of 

 analytical geometry, or theory of equations, or 

 both. A really gifted youth, approaching the 

 problem without knowledge of modern analysis, 

 might throw a quite new and unexpected light 

 upon it. 



Mr. Mordell's pamphlet ought to do much to 

 stimulate our rising mathematicians, and we hope 

 that it will have a large circulation. 



G. B. M. 



Chemistry of Coke-oven and By product 

 Works. 



Coke-oven and By-product Works Che7nistry. 

 By T. Biddulph-Smith. Pp. x+ 180 + 7 plates. 

 (London: Charles Griffin and Co., Ltd., 1921.) 



215. 



THE author states in the preface that his 

 object in compiling this book is to furnish 

 a concise manual covering, so far as space will 

 allow, the general work required for the chemical 

 control of coke-oven and by-product works. As 

 regards the variety of subjects treated, he has 

 doubtless achieved his object, but it is to be re- 

 gretted that the apparent exigencies of space 

 have caused the manual to become so concise in 

 certain sections as to detract appreciably from 

 the value of the work as a whole. 



The most valuable section of the manual is that 

 relating to the coal-tar naphthas. There is no 

 doubt that the author has taken considerable 

 pains to collect together the work of some of our 

 best analytical chemists on methods of evaluating 

 the constituents of coal-tar naphthas — work which 



