662 HISTORY OF THE THEORY OF NUMBERS. [CHAP. XXII 



Taking t = 5, we get j/ = 25/4, T=20, z = 39/4. Multiplying the unknowns 

 by 4, we get the solution a; = 39, y = 25, 2 = 20, v = 12. Or we may solve 

 ^ = for v 2 and get v 2 = 2S-x 2 -y 2 -z 2 , S 2 = x 2 y 2 +x 2 z 2 +y 2 z 2 . Set S = xy+tz. 

 Then 



z = 2txyfk, S = xy(x 2 +y 2 +t 2 )/k, k=x*+y*-t 2 . 



Then v 2 is a complicated function of degree 6 and was not treated. ' A 

 solution is said to result from t = 185/153. For = 13/3, x = 5, 2/ = 4, we get 

 the above solution x = 39, etc. 



C. F. Kausler 255 treated the problem to find all rational numbers x, y 

 for which N=(x 2 l)(y 2 1) is an integer. Set y = p/q, where p and q are 

 relatively prime integers. The numerator and denominator of the resulting 

 fraction for x 2 are (N l)q 2 +p 2 = mP 2 and p 2 q 2 = mQ 2 . For ra = l, the 

 latter gives p = (A 2 -\-B 2 }ld, q = (A 2 B 2 )[d, where A, B are relatively prime, 

 one even or both odd according as d= 1 or 2. The first condition then gives 

 N which is an integer for d=l if P2AB is divisible by (A 2 - 2 ) 2 . For 

 m>l, p-\-q = mQ, m or Q 2 , the last two yielding (as far as numbers <100) 

 only the same values of N as above. For p-\-q = mQ, then pq = Q and, 

 dropping the common factor Q/2 in p, q, we have p = m+l, q = ml, m 

 even, ^ = m(P 2 -4)/(m-l) 2 . Then P=F2 = #(m-l) 2 , whence 



G. Eisenstein 256 considered a binary cubic whose coefficients are variables. 

 Its discriminant D is a quartic in these four variables. Given one solution 

 of D = constant, we can find an infinitude of solutions by means of the 

 formulas for the coefficients of the cubic obtained by a linear transformation 

 of determinant unity. 



V. A. Lebesgue 257 noted that 



a 2 t*+ 6 V+ cV - 2&cwV - 2acvH 2 - 2aU 2 u 2 = s 2 

 is satisfied identically by 



t=x(by 2 cz 2 '), u = y(cz 2 ax 2 ), v = z(ax z by 2 ), 

 with s the product of the binomials, and by 



t=x(cy 2 bz 2 ), u = y(az 2 cx 2 }, v=z(bx 2 ay 2 ). 



Several 258 found two numbers whose sum equals the difference of their 

 fourth powers. Let the numbers be (nl}x. Then z = (4n 2 +4)~ 1/3 is 

 rational if n=l. Hence set n = ra+l. Then x = N~ 113 , N=(pm-\-2Y if 

 p = 2/3, m = 9/2. 



E. Lucas 259 stated that the difference of two consecutive cubes is never 

 a biquadrate. Moret-Blanc 75 noted that 3z 2 +3z+l +z 4 since 4z* 1 =t=3 2 . 



D. S. Hart 260 found rational numbers a, b, x for which 



4:X*+4ax z +4bx-\-ab = 0. 



266 Nova Acta Acad. Petrop., 15, ad annos 1799-1802, 1806, 116-45. 

 258 Jour, fur Math., 27, 1844, 76. 



267 Comptes Rendus Paris, 59, 1864, 1069. 



268 Math. Quest. Educ. Times, 2, 1865, 77; cf. (2), 4, 1903, 68-9. 



269 Recherches sur 1'analyse indc'termine'e 73 , 1873, 92; extract in Mathesis, 8, 1888, 21. 

 260 Math. Quest. Educ. Times, 24, 1876, 35-36. 



