CHAP, xxiii] SINGLE EQUATIONS OF DEGREE w>4. 695 



G. Cordone 159 investigated polynomials U, V in x which satisfy 



identically in x, where the Pi(x) are polynomials in x. 



E. Maillet 160 considered recurring series u , HI, of rational terms with 

 the generating equation /(z) =x q +a,ix q ~ 1 -i ----- ha 3 = and law of recurrence 



(2) M B+fl +aitt B+fl _H ----- \-a q u n = 0, 



where a\, , a q are rational. An algebraic equation with rational coeffi- 

 cients is irreducible if and only if all the recurring series of rational terms 

 having the equation as their generating equation admit the corresponding 

 law of recurrence as an irreducible law. To apply this to diophantine 

 equations, let 



2 ' U n 



i ' U n +q-\ 



become F(u n , u n+ i, , u n+q -\) when w n+2g _ 2 , , u n+q are expressed in 

 terms of w n + g -i, -, u n by means of (2). It is known that the law (2) 

 is reducible if and only if A g (0) = 0. Hence F(u Q , -, u q -i) = has rational 

 solutions if and only if f(x] = is reducible. If u , , u q ~i give a rational 

 solution, the same argument shows that u n , , u n + q -i give a rational 

 solution for n arbitrary. We get all the rational solutions by taking in 

 turn all the maximum divisors x(x) = x t -\- - - -\-c t , with rational coefficients, 

 of /Or), i. e., a divisor not dividing any other divisor of /(re), and forming all 

 the recurring series of rational terms having x(z) =0 as generating equation 

 and any rational numbers as the first t terms u , Ui, , u t -i. Among the 

 recurring series which together give all the rational solutions of F=G, 

 those which give only a finite number of solutions are the ones whose 

 generating functions are divisors 0(z), with rational coefficients, of f(x), 

 such that 6(x) = has as its roots only distinct roots of unity. For example, 

 let q = 3 and f(x) =x*y. Then 



F(u , HI, u 2 )=v z u%+yui+u 3 2 3vu Q UiU2. 



Let 7 be the cube of a rational number 5, so that / is reducible. The 

 maximum divisors are x 8 and x z -\-8x-\-d 2 . To the first correspond the 

 solutions u , du , 8 2 u , where u is any rational number. To the second 

 correspond U Q) u\, ^(ui+duo), where u and u\ are any rational numbers. 

 If 7 is not the cube of a rational number, there is no rational solution of 

 ^=0. Let (2) be an irreducible law for u , u\, and let a q = 1. Then 

 A(0) =0 + 0, F(u n , -, Mn+g-i) = 0, so that we have rational solutions of 

 the latter. There are similar results for integral solutions when the a's are 

 integral. 



D. Hilbert 161 treated the diophantine equation D = l, where 



Giornale di Mat., 33, 1895, 106, 218. 



160 Assoc. franQ. av. ec., 24, II, 1895, 233-42. 



161 Gottingen Nachrichten (Math.), 1897, 48-52. Cf. Eisenstein 266 of Ch. XXII for n=3. 



