602 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxi 



A. S. Werebrusow 370 found an infinitude of solutions of the last question. 

 Teilhet 371 gave a more general treatment of the problem. 



A. Gerardin 372 treated the system x 3 +hy 3 = a*+hb*, x+hy = a+hb, and 

 found many solutions, such as 

 x, a = (9m 2 -l)a 2:: Fl8ma/3-3/3 2 ; y, b = (9m 2 -l)a 2 6a/3-f-3/3 2 ; fc = 3m. 



In 1'intermediaire des mathematicians are discussed the problems: 



= z 3 , P(x+y)+Qy = w 3 , 22,1915,145-6. 



P, (y- x -)2- y = w P t p>3, 196; 23, 1916, 68-9; 24, 1917, 85-6. 



?, (z+y) 3 +2/ = 6 3 , 23,1916,141-2. 



x 3 -hy 3 =D, x 3 +hy 3 =D, 22, 1915, 53, 232; 24, 1917, 39. 

 x 3 +y 3 =a 3 -b 3 , x 3 -7/ = c 3 +d 3 , 26, 1919, 145. 



SYSTEMS OF EQUATIONS OF DEGREE THREE IN THREE UNKNOWNS. 



Diophantus, IV, 6, found that # 2 +2 2 is a square and y*-\-z 2 a cube for 

 y = 16/7, x = 3y,z = ty. In IV, 7, 8, he found that x 2 +z 2 is a cube and y*+z z 

 a square for x = 5, y = 5, z = 10 and for x = 40, y = 20, z = 80. 



J. de Billy propos ^d the problem to find three numbers such that if their 

 product is subtracted from any one of the numbers or from the difference 

 of any two or from the product of the second by the first or third or from 

 the square of the second, there results always a square. He expressed his 

 belief that 3/8, 1, 5/8 is the only solution. 



Fermat 373 replied that [if the numbers are denoted by A, 1, 1 A] the 

 problem reduces to the double equal ty 



which has an infinitude of solutions. In addition to de Billy's solution 

 A = 3/8, Fermat gave A = 10416/51865. 



Malezieux 374 proposed the problem to find three rational numbers in 

 A. P. such that one obtains a square by adding to their product either the 

 difference of the s uares of any two of them or the sum of the three differ- 

 ences of the three numbers. 



E. Fauquembergue 375 gave the solution 1/31, 25/589, 1/19. 



J. Ozanam 376 asked for three numbers in G. P. such that one obtains 

 squares by adding to their product the square of each number, and such 

 that if these fractional squares are reduced to their simplest forms the sums 

 by twos of the square roots of the numerators are three cubes in G. P. 



" J. Hob" 377 solved the first part, saying the entire problem is impos- 

 sible. _ 



370 L'intermediaire des math., 10, 1903, 319-20. 



371 Ibid., 11, 1904, 167-70. 



372 Sphinx-Oedipe, 5, 1910, 1-12. 



373 Oeuvres, II, 437, letter to de Billy, Aug. 26, 1659. 



374 Unedited letter to de Billy, Sept. 6, 1675. Cf. P. Tannery, 1'intermediairc des math., 3, 



1896, 37. E16ments de Geometric de M. le Due de Bourgogne, par de Malezieux, 1722. 



375 L'intermediaire des math., 6, 1899, 115-6. 



376 Unedited letter to de Billy, June 25, 1676. Cf. P. Tannery, 1'intermediairc des math., 



3, 1896, 57; C. Henry, Bull. Bibl. Storia Sc. Mat. e Fis., 12, 1879, 517. 



377 L'interm6diairc des math., 4, 1897, 253. 



