CHAP, xxill] PRODUCT OP FACTORS (x+l)fx. 687 



T. Suzuki 105 noted that there are at least (p 2)(p l) n ~ 2 solutions of 



a?+--+al* = Q (modp), 



if two of the a's are primitive roots of the prime p. Also there are solutions 

 if ai is a primitive root and if not every a t =l (mod p) for i = 2, , n. 



RATIONAL SOLUTIONS OF x y = y x . 



L. Euler 106 set y = tx and deduced x t ~ 1 = t. The graph is composed of 

 y = x, a branch asymptotic to the positive x and y axes, and an infinity of 

 isolated points. Among the rational solutions are (x, y) = (2, 4), (3 2 /2 2 , 

 3 3 /2 3 ), (4 8 /3 3 , 4 4 /3 4 ). 



D. Bernoulli 107 noted that, for x=^y } the only integral solution is 2, 4; 

 but that there is an infinitude of rational solutions. 



J. van Hengel 108 remarked that r r+n >(r+n) r if r and n are positive 

 integers either one =^3. Thus if a b = b a , it remains to treat the cases a = l 

 or 2. If a = 2, 6>4, whence b = 2+n, we apply the above remark. 



* C. Herbst 109 noted that 2, 4 give the only solution in integers. 



* A. Flechsenhaar 110 and R. Schimmack 111 discussed the rational solu- 

 tions. 



A. M. Nesbitt 112 and E. J. Moulton 113 discussed the graph of x y = y x . 

 A. Tanturri 114 proved that 2, 4 give the only solution in integers. 



PRODUCT OF FACTORS (x+l)/x EQUAL TO SUCH A FRACTION. 



Fermat 115 proposed the problem to find in how many ways (n+l)/n can 

 be expressed as a product of k such fractions, citing the case n = 8, k = 10, as 

 suitable to be proposed to all mathematicians of his time. Tannery noted 

 that of the decompositions of 9/8 the difference of the factors is least and 

 greatest in respectively 



908988878685 84 838281 9+1 9 2 +l 9 4 +l 9 256 +l 9 512 



89'88'87'86'85"84"83*82'8l'80 } 9 9 2 9 4 9 256 > 12 -1' 



V. Bouniakowksy 116 noted that an irreducible fraction a/b less than 

 unity can be expressed in an infinitude of ways as a product of fractions 

 of the form xf(x+l). We may often find fewer than the b a fractions 



105 Tohoku Math. Jour., 5, 1914, 48-53. Cf . papers 265-6 of Ch. XXVI. 



106 Introductio in analysin infin., lib. 2, cap. 21, 519; French transl. by J. B. Labey, 2, 



1797 and 1835, 297. 



107 Corresp. Math. Phys. (ed., Fuss), 2, 1843, 262; letter to Goldbach, June 29, 1728. 



101 Beweis des Satzes, das unter alien reellen positiven ganzen Zahlen nur das Zahlen Paar 4 



und 2 fur a und b der Gleichung a a = b b gentigt, Progr. Emmerich, 1888. 

 109 Unterrichtsbl. fur Math., 15, 1909, 62-3. 

 uo Ibid., 17, 1911, 70-3. 



111 Ibid., 18, 1912, 34-5. 



112 Math. Quest. Educ. Times, (2), 23, 1913, 77-8. 



113 Amer. Math. Monthly, 23, 1916, 233. 



114 Periodico di Mat., 30, 1915, 186-7. 



115 Oeuvres, I, 397. Quoted by Tannery, I'interme'diaire des math., 9, 1902, 170-1. 



116 Mem. Acad. Sc. St. P<tersbourg (Sc. Math. Phys.), (6), 3, 1844, 1-16. 



