694 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxni 



Then X, Y, Z are positive integers for which XYZ = X+ Y+Z. If X is the 

 largestofX, Y, Z, then XYZ <3X, YZ = 2orl. Wemay take 7=2, Z = l. 

 Then 2 = 5, y = , x = 3. See the next two papers, and 341 of Ch. XXI. 



Housel 151 proved that the sum of n distinct positive integers equals their 

 product only when the integers are 1, 2, 3. 



J. Murent 152 discussed the positive integral solutions (i, , o n ) of 



"X n ( 



One solution is (n, 2, 1, ,!). Always at least two a's exceed unity. If 

 n>2, at least one a is unity; call i the index of a solution (01, , a,-, 1, 

 , 1) with ai>l, , <Zi>l. Then 2 i i^=n', if =n, theni= = a; = 2. 

 If n = 5 = 2 3 3, there is a single solution (2, 2, 2, 1, 1) of index 3, while the 

 only remaining solutions are (3, 3, 1, 1, 1) and (5, 2, 1, 1, 1) of index 2. 



P. di San Robert 153 noted that F(x, y,z)=Q can be solved by use of the 

 slide rule only if reducible to X(x) + Y(y) =Z(z), a necessary and sufficient 

 condition for which is 



dxdy dx By' 



S. Re"alis 154 noted that 



<2=^ 



m 1 



is not an mth power, being between a m and (a+l) OT , and that mQ is not 

 divisible by (a-fl) m a m . 



E. Lucas 155 noted that x k +x+k = y* is impossible if k is odd. 



S. Realis 156 noted that, if xy^O, 6xy(3x 4 -\-y 4 ) 4=2 3 or 4^ 3 . The impossi- 

 bility (p. 524-5) of 



7 



or 7z+5, 



is easily verified by use of remainders modulo 9 or 7. M. Rochetti 1560 

 expressed 



as a sum of three cubes. 



A. Markoff 157 gave complicated formulas for all positive integral 

 solutions of x z -\-y 2 -\-z 2 = 3xyz. 



E. Fauquembergue 158 proved that l+3+3 2 H ----- f-3 n = i/ 2 only when 



= 0, 1, 4, by using the powers of a+frV 2 to treat 3 n+1 = l+2i/ 2 . 



161 Nouv. Ann. Math., (2), 1, 1862, 67-69. 

 lbid., (2), 4, 1865, 116-20. 



163 Atti della R. Accad. Sc. Torino, 2, 1866-7, 454-5. 

 1M Nouv. Ann. Math., (2), 12, 1873, 450-1. 



165 Nouv. Corresp. Math., 4, 1878, 122, 224. 



166 Nouv. Ann. Math., (2), 17, 1878, 468. 

 a lbid., (2), 19, 1880,459. 



167 Math. Annalen, 17, 1880, 396. Cf. Hurwitz. 171 



168 Mathesis, (2), 4, 1894, 169-170. 



