CHAP, xxiii] SYSTEMS OF EQUATIONS OP DEGREE n>4. 701 



integral solutions of f(x\, , a? M ) =0, , F(XI, , x ft }=0, and we set 

 Xi = didi in all possible ways and write 77 = +! or --1 according as di- -d^ 

 is a product of an even or odd number of primes (equal or distinct), then 

 2 77 is the number of sets of solutions of the given equations in which each 

 Xi is a square. 



H. Delorme 198 noted that the system x* m = ay- n +l, x 2p+l = by* q+1 +c is 

 insolvable if o+l and c are divisible by 3, while 6 is not [since impossible 

 modulo 3]. 



A. B. Evans 199 found four integers (ax 5 , , efo 5 ) whose sum is a sixth 

 power and the sum of any three a fifth power. Take a+b+c+d = x. 

 Then the conditions are x a = p b , , x d = s 5 . Thus x = % (p 5 + q 5 + r 5 + s 5 ) 

 is an integer if p = 3m, g=3m+l, r = 3ra+2, s = 3m+3, and then a, b, c, d 

 are also integers. 



A. Desboves 200 called a a congruent number of order m if the system 



has integral solutions. For m = 2, the quotient of the expression found 

 for a by 16 is xy(x*-y 2 )(x 4 -QxY+y 4 )/2. Taking x = 2, y = l, the latter 

 becomes 21. The least congruent number of order 2 is 21. A. Ge'rardin 201 

 remarked that it seems more logical to call a a congruent number of order m 

 if x m ay m = D hold simultaneously. Cf. papers 210, 222, and Ch. XVI. 



L. Gegenbauer 202 considered a set of positive integral solutions Xi, -,#" 

 of the system of equations fi(xi, , # M ) =0, , f r (xi, , &) = 0, and 

 any divisor 5 of xl, and called the product 5? 5 a divisor-product 

 belonging to the set x\, , z. Let x(#) be a function for which 



= x(x) x(y) for all values x, y satisfying a definite condition. Let 

 X(ri) = Sx(d), where d ranges over all divisors of n. Then 



where on the left the summation extends over those sets of solutions x\, 

 , z which satisfy the condition mentioned, while on the right the summa- 

 tion extends over all the divisor-products belonging to these sets of solutions. 

 If we take x(#) = +1 or ~~1> according as # is a product of an even or odd 

 number of primes (equal or distinct) and note that Sx(cO = +l or 0, 

 according as n is a square or not, we obtain the theorem stated by Liouville. 197 

 Other special cases are obtained by taking xO*0 to be the number fa(x) of 

 sets of k integers <x and prime to x, or p(x) of Vol. I, Ch. 19, and noting 

 that 2<f> k (d) =n k , S/i(d) =0 if n>l. 



Several writers 203 found two integers whose sum, difference and difference 

 of squares are all twelfth powers (square, cube and biquadrate). Else- 

 where 204 was added the condition that the product of the nine roots of these 

 powers shall be a square, cube and biquadrate. 



. Ann. Math., (2), 1, 1862, 455-7. 

 199 Math. Quest. Educ. Times, 25, 1876, 76. 

 200 Nouv. Ann. Math., (2), 18, 1879, 490. 



201 L'intermediaire des math., 22, 1915, 101. 



202 Sitzungsber. Akad. Wiss. Wien (Math.), 95, II, 1887, 606-9. 

 208 Amer. Math. Monthly, 2, 1895, 128-9. 



204 Math. Quest. Educ. Times, 60, 1894, 37-38. 



