696 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. XXIII 



is the discriminant of XQ t n +x\t n ~ l -\ +z n = 0, with undetermined coeffi- 

 cients, and roots t\, , t n . By use of Zi = 0, -, z n _ 2 = 0, it is readily 

 proved that D = l has rational solutions. The main theorem is: For 

 n>3, D=l is not solvable in integers; the only equations with integral 

 coefficients and with the discriminant 1 are Q=(ut-}-v)(u 1 t+v l ')=0 and 

 the cubic Q[(u-\-u v )t-\-v-\-v l ~] = Q, where u, u l , v, v l are any integers for 

 which uv l u l v=l. The proof employs the theorem 162 that the dis- 

 criminant of an algebraic domain is always distinct from 1 and the lemma 

 (here proved by use of ideals) : If an equation with integral coefficients is 

 irreducible in the domain of rational numbers, its discriminant is an integer 

 divisible by the discriminant of the domain determined by a root of the 

 equation. 



C. Stormer 163 noted that, if A, B, Mi, Nj are positive integers, 



AM?- - .MZ-BN?' .#*= dbl or 2 



has only a finite number of sets (if any) of integral solutions x i} y } ; and 

 that these can be found by solving a finite number of Pell equations. 



E. Fauquembergue 164 noted that 3x 2 = 4y* z 6 has no solutions with 

 y, z relatively prime, since (x+z*) 3 (x 2 3 ) 3 =(2?/z) 3 gives x = z*, y = z 2 . 

 On z 2 = z 6 -4?/ 3 , see Fuss 11 of Ch. XXI. 



G. B. Mathews 165 noted that xy(x-\-y)=z n has no solution if n = 3m, 

 while if ?i = 3ml the general solution is (A", X"??, X 3 f), where (, 77, f) is 

 the unique solution in which x/y equals a given irreducible fraction, and 

 the g.c.d. of x and y is not divisible by an nth power. 



A. Cunningham 166 solved in integers NiN3 = NzN4, where N r =Xr+4y i r ; 

 also 



N NzN 4 ..-N Zr ' N b ' 

 He solved MiM, = M 2 M 4 , where M r =(x' r +&y?)/(xl+3yD. 



v 



S. 0. Satunovsky 167 discussed the solution in integers of 

 ax mn +aiX mn ~ 1 -\ ----- \-a mn = by n , b = d=a/c m . 



P. F. Teilhet 168 gave, for m = l, recurring series leading to all (an infini- 

 tude of) solutions of x 2m y 2m x m y m l and asked if there are solutions 

 when m>l other than x = y = l. 



* H. Kuhne 169 noted that if the system of n functions a?,-= 0,- ( , , n-i) 

 is equivalent to the system of n functions &=/(#<), , x n -i), the coeffi- 

 cients of the 0's and/'s belonging to the same domain, there exists between 

 the x'a and the ^'s a connection (Verkniipfung) and these connections 

 have the group property. This concept leads to a process of solving all 



182 Minkowski, Geometrie der Zahlen, 1896, 130. 



163 Comptes Rendus Paris, 127, 1898, 752. 



164 L'interm&Iiaire des math., 5, 1898, 106-7. 



166 Math. Quest. Educ. Times, 73, 1900, 37. For 2 = 1, Euler 10 of Ch. XXI. 

 168 Ibid., 75, 1901, 43; (2), 1, 1902, 26-7, 38-9. 



167 Zap. mat. otd. obsc., Odessa, 20, 1902, 1-21 (Russian). 



168 L'interme'diaire des math., 9, 1902, 318. 



169 Math. Naturw. Blatter, 1, 1904, 16-20, 29-33, 45-58. 



