SCIENCE PROGRESS 



RECENT ADVANCES IN SCIENCE 



FUBE MATHEMATICS. By F. Puryer White, M.A., St. John's 

 College, Cambridge. 



Theory of Numbers. — One of the most remarkable features of 

 pure mathematics during the past ten years has been the 

 development of the analytic theory of numbers by Hardy 

 and Littlewood in England, and by Landau and others in 

 Germany. The problems dealt with may be treated under 

 three heads. 



(i) The additive theory of numbers or " Partitio 

 numerorum." 



The proposition that every integer can be expressed as the 

 sum of at most m ^th powers of integers, where m is inde- 

 pendent of the ii.teger (but depends on k), is known as Waring's 

 Problem, having been stated, apparently as an empirical result, 

 by Waring in 1782. It had only been verified for a few special 

 values of k until 1909, when Hilbert obtained a general proof. 

 In 1919 Hardy and Littlewood {Q.J., 48, 1919, 272-93) published 

 an account of a new analytic method, using Dirichlet series, 

 which led to another solution and which gives information as 

 to the value of g{k), the least value of m that will do ; details 

 of the proof are given in a later paper {Gottingen Nach., 1920, 

 33~'54)- Minor improvements in the proof have since been 

 made by Landau (ibid., 1921, 88-92) and Weyl {ibid., 1921, 

 181-92). But the method also leads to another number 

 G(^), which is even more fundamental than g{k) ; it is the 

 least number m such that from a certain point onwards every 

 number can be expressed as the sum of m ^th powers. Thus 

 every large number can be expressed as the sum of at most 

 21 biquadrates (Hardy and Littlewood, Math. Zs., 9, 1921, 

 14-27), whereas for smaller numbers it is possible that 37 may 

 be required. In the Gottingen paper cited above, Hardy and 

 Littlewood showed that G{k)^Kk + i , where K=2*"\ but 

 more recently {Math. Zs., 12, 1922, 161-88) they have improved 

 this result to G{k)^{k - 2)K + 5. 



A general account of the methods employed has recently 



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