PROBLEMS OF MODERN SCIENCE 



years old, to the effect that no whole numbers can 

 be found to satisfy 



Euler stated that though he could not prove this, 

 there was no reason to doubt its truth. Nobody 

 has proved it yet, and still there is no reason to 

 doubt it, and much less reason even than in the 

 time of Euler. 



All such things are really centred round what 

 is called Waring's problem, enunciated in 1770, 

 which is essentially a problem of partitions. A 

 number N may be portioned off into a set of 

 whole numbers which add up to N the ordinary 

 partition to which I have referred above or it 

 may be partitioned into squares, cubes, fourth 

 powers, and so on, of whole numbers. We may 

 seek to write, for instance, 



N = xS+ x 2 n + . . . + x r n 



the sum of the nth powers of r integers. What 

 is the minimum number of such integers necessary 

 for any specified value of #, and any value of N ? 

 This is essentially Waring's problem, and 1909 was 

 a red-letter year in its history, for in that year 

 much was accomplished by Landau, Hilbert, and 

 others towards its solution. Professor Hardy, of 

 Oxford, and Mr Littlewood, of Cambridge, have 

 done much since, but the general problem remains 



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