CHAP, xxni] PRODUCT OF CONSECUTIVE INTEGERS NOT A POWER. 679 



nomial of degree n 1 in a^ , a n -i. Finally we choose k to make these 

 x's integers. 



If F = can be given the form (3), every set of integral solutions of 

 P; = 0, Qj = (i, j = l, , n l) is evidently a solution of F = 0. Con- 

 versely, if a certain number of integral solutions of P; = Qj = satisfy 

 F = 0, then F = can be given the form (3). In fact, if a polynomial 

 F(XI, - , x n } of degree m always vanishes simultaneously with the products 

 U = Pi---Pp, V = Qi-'-Q q of linear functions of x it , x n , such that 

 not all the values for which any two are zero make a third zero, then 

 F=AU-\-BV, where A and B are polynomials in x\, , x n . 



A. Palmstrom 27 extended the preceding method to the equation 



Pll Pl2 ' ' Pi n-l 



(5) Pal P22 ; p ;-> =o, 







Pnl I Pnl 2 ' ' ' Pnl n-l 



where the P's are linear homogeneous functions of x\, -, x n . For every 

 set of integral x's satisfying (5) there exist n l relatively prime integers 

 oi, , a n -i satisfying 



and conversely. From the latter, Xi/x n = AifA n , so that we may set Xj = kAj 

 (.7 = 1, -, n) and choose k to make the x's integral. Here the a's have 

 any values for which AI, , A n are not all zero. In case the A's are all 

 identically zero, so that only p of the equations (6) are independent, we 

 can assign arbitrary values to n p 1 of the x's and determine the remain- 

 ing x's by p linear equations. He 130 gave a detailed example. 



G. Metrod 27a found the number of ways to decompose a given number 

 into a product of n factors (including unity). 



PRODUCT P n OF n CONSECUTIVE INTEGERS NOT AN EXACT POWER. 



Chr. Goldbach 28 argued that a P 3 is not a square since its root would be 

 a multiple of m and a divisor of (w+l)(m+2), whence m = l or 2. 



J. Liouville 29 proved by use of Bertrand's postulate [Vol. I, Ch. XVIIIJ 

 that m (m +!) (ra+n 1) is not a square or higher power if at least one 

 factor m, - , m-\-n 1 is a prime, or if n>m 5. The latter was proved 

 similarly by E. Mathieu, 30 who verified the theorem for any n when m^lOO. 

 In particular, ml is not an exact power, a fact proved in the same way by 

 W. E. Heal. 31 



Mile. A. D. 32 proved that a P 3 is not an exact power. 



"Skrifter Udgivne af Videnskabsselskabet, Christiania, 1900 (1899), Math.-Naturw. Kl., 



No. 7 (German). 

 270 L'intermediaire des math., 26, 1919, 153-4. Cf. Minetola 192 - 3 of Ch. Ill, and Cesaro 30 of 



Ch. IX; also Index to Vol. I (under "Number," including n=x a y b ). 



28 Corresp. Math. Phys. (ed., Fuss), 2, 1843, 210, letter to D. Bernoulli, July 23, 1724. 



29 Jour, de Math., (2), 2, 1857, 277. Cf . Moreau. 6 " 



30 Nouv. Ann. Math., 17, 1858, 235-6. 

 81 Math. Magazine, 1, 1882-4, 208-9. 



32 Nouv. Ann. Math., 16, 1857, 288-290. Proposed by Faure, p. 183. 



