606 HISTORY OP THE THEORY OF NUMBERS. [CHAP, xxi 



Three integers whose sum is a cube and sum of any two less the third a 

 cube (2, 1895, 86-7). Three positive integers the product of the first by 

 the sum of the other two a square and sum of their cubes a square 395 (p. 196). 

 Four positive integers each less double the cube of their sum a cube (7, 

 1900, 49-50). Three positive integers whose sum, sum of squares, and sum 

 of cubes, are all squares (9, 1902, 145-6), or all cubes (24, 1917, 240). 



R. F. Davis and others 396 made X 3 +7 2 +Z 2 , F 3 +X 2 +Z 2 , Z 3 +X 2 +Y 2 

 all squares. Take X = 2(l-y}, Y = 2(l+y), Z = 2(l-y 2 ). Then the first 

 two equal the squares of 2(2=F?/+?/ 2 ). The third is a square if y = 4/3. 



A. Martin 397 solved A 2 + 2 +C 2 = D, A 3 + 3 +C 3 = Z) 3 . As the solutions 

 of the latter he employed the products of a by the values given by Young. 56 

 Take n 2 +2 = (n r) 2 . Then SA 2 = D becomes a quartic for q whose solu- 

 tion, found as usual, is a very long expression for q. Take r = 3, whence 

 n = 7/6. Then g = a//3, where a = 81420385, = 11290752. Take a = 6/3 2 . 

 Then A, B, C are integers each of 17 digits: 



A = 11868013975030087, 5 = 16269106368215226, = 88837226814909894. 



M. Rignaux 398 noted a solution of the last problem involving parameters 

 m, n, g such that w 2 +2n 2 =D; in his numerical example, A and C are 

 negative, while A, B, C contain only 6 or 7 digits. 



P. Tannery and H. Brocard 399 noted that 3, 4, 5 yield by multiplication 



54+72+90 = 6 3 , 54 3 +72 3 +90 3 = 108 3 . 

 E. B. Escott 400 gave numbers without common factor: 



3+4-6 = l 3 , 3 3 +4 3 -6 3 = -5 3 ; 



36+37-46 = 3 3 , 36 3 +37 3 -46 3 = -3 3 . 



H. Brocard 401 gave 9+15-16 = 2 3 , 9 3 +15 3 -16 3 = 2 3 and 24+2-18 = 2 3 , 



24 3 +2 3 -18 3 = 20 3 . 



A. Ge*rardin 402 noted that, if x+y+z, 2x 2 and 2z 3 are all cubes, x, y, z 

 are in neither geometrical nor arithmetical progression. He and others 403 

 noted special sets of integral solutions of x+y+z = c 2 , x 2 +y 2 +z 2 = b 3 ; also 

 values making s 3 xy, s 3 yz, s 3 xz all squares or all cubes, where 

 s = x+y+z (ibid., 23, 1916, 5-6); s 3 x, s* y, s*z all squares (pp. 157-9); 

 xyz+x 2 , xyz+y 2 , xyz+z 2 all squares (24, 1917, 37-8); s 2 x-y, s 2 yz, 

 s 2 -x-z all cubes (22, 1915, 220). 



395 Also, Math. Quest. Educ. Times, 24, 1913, 63-4. 

 398 Math. Quest. Educ. Times, 64, 1896, 26. 



397 Math. Magazine, 2, 1898, 254-5. 



398 L'interme'diaire des math., 24, 1917, 79-80. He corrected a misprint in a citation of 



Martin's solution, correctly quoted in 7, 1900, 162. 

 899 L'interme'diaire des math., 6, 1899, 190. 



400 Ibid., 7, 1900, 141. 



401 Ibid., 10, 1903, 14. 



402 Sphinx-Oedipe, 9, 1914, 38-9. 



403 L'interme'diaire des math., 22, 1915, 172; 23, 1916, 93. 



