650 SCIENCE PROGRESS 



The approximate formulae discovered are so accurate that for m = 200, 

 the coefficient — a number of thirteen figures — is found with an error of 0-004. 

 The second is Waring's Problem, where the error introduced by taking the 

 approximation suggested is of smaller order for large values of m than the 

 approximation, thus proving the theorem for large values of m, and also 

 finding a rough estimate of the number of nth powers required. 



It is because of the fact that new functions and new questions are so easily 

 suggested in the Theory of Numbers that it is so intimately associated with 

 many branches of pure mathematics. But, while it is easy enough to suggest 

 a problem, the solution is usually a matter of great difficulty ; the arithmetical 

 and analytical difficulties attending each new type of question being such as 

 to require a new chapter in mathematics. A simple illustration will make 

 this clear. 



Consider first the equation 



x^ -\- y* = z^ 



of which rational solutions were known to the Egyptians nearly 2,000 years B.C., 

 and of which rational solutions involving a parameter were known to Pytha- 

 goras and Plato. The general solution of this equation in co-prime integers 

 can be written as 



X = p^ - g^. y =^ 2pq, z = p^ + q\ 



and was practically known by Diophantus, and expHcitly by the Indian 

 mathematicians about 600 a.d. 

 Take next the equation 



x^ -\- y^ = z^, 



(or, for that matter, say 



x^ + ly^ = z^ 



where an obvious solution is x = 1, y = i, z = 2), which it seems likely 

 must have been discussed by mathematicians before the time of the Arab 

 Alkhodjandi, who gave a defective proof of its impossibihty at some date 

 before a.d. 972. For a long time, many persons must have wondered if such 

 equations could be solved in a similar manner by means of parameters p, q. 

 Their efforts, of course, would have been in vain, as the introduction of 

 elliptic functions, certainly one of the intellectual achievements of the 

 nineteenth century, makes almost obvious. Their difiiculties may illustrate 

 one still facing mathematicians after several hundred years. 

 Consider the equation 



y^ — ax^ -\- bx -\- c, 



which includes as a special case the famous equation miscalled the Pell 

 Equation, which has led Kronecker to surprising relations between the number 

 of classes of binary quadratics and elliptic functions. There is no difficulty 

 in finding not only all the integer solutions, but also tests for deciding if the 

 equation is possible. Take now the equation 



ya = 4X^ — g^x - g^, 



and practically the same argument applies to the equation 



yi = ax* + bx^ -\- cx"^ -\- dx -\- e 



or to (^ {x, y, z) =0 



if is a ternary cubic in x, y, z. If we know one set of rational values for 

 X, y, to which we shall refer as the rational point P {x, y) of the curve 



it is clear that in general we can find another rational point Q from the other 



