CHAP, xxni] PROPERTIES OF PRODUCTS OF CONSECUTIVE INTEGERS 681 



C. Moreau 50 repeated the first remark by Liouville. 29 



H. Brocard 51 asked for values of x making 1+z! a square. He 52 sugges- 

 ted that the only solutions are 4, 5, 7. 



E. Lucas 53 noted that the product P of the first n primes is not of the 

 form a p b p , where a and b are positive integers and p>l, P>2. 



E. Lionnet 54 stated that no product 1 -3 -5 of consecutive odd numbers 

 is a square or higher power. Moret-Blanc 55 proved the last statement by 

 Bertrand's postulate. 



Moret-Blanc 56 solved y(y+l')(y+2)=x(x+l), proposed by Lionnet. 

 Adding 1 to the product by 4, we are to make 4?/ 3 + 127/ 2 +8i/+l = D, say 

 (my I) 2 . The discriminant of the quadratic in y is to be rational. Thus 

 m = 2n, ft 4 6ft 2 4n+l = D, which holds for ft = 3. Thus solutions are 

 l-2.3 = 2-3, 5 -6 -7 = 14 -15. G. C. Gerono (p. 432) noted that, since 

 2x+l = 2ny 1, the initial equation becomes y 2 (n 2 3)y+n+2 = Q and 

 proved that ft = 3. 



E. Lionnet proposed and Moret-Blanc 57 solved the problem to find N 

 such that both N and N/2 are products of two consecutive integers, the 

 smaller factor of N/2 being a product x(x-\-l) of two consecutive integers. 

 Thus 



2(x 2 +x)(x 2 +x+l}=y 2 +y, 8z 4 +16:E 3 +16:c 2 +8z+l = (2?/+l) 2 . 



Euler's process to deduce new solutions from x = l leads only to z = or 

 fractional values. 



E. Lemoine 58 asked if the product of three consecutive numbers (besides 

 2, 3, 4) is of the form px 3 , where p is a prime. H. Brocard (p. 304) noted 

 that the problem reduces to y*y = px z , took y = p and concluded that 

 x = 2, y = 3. Several replies (p. 369) show readily that 2, 3, 4 is the only 

 solution. 



E. B. Escott 59 proved that x(x+4)(x+6) + D. 



G. de Rocquigny 60 proposed for solution 



E. B. Escott 61 noted the solutions x = l or 6, y = &, besides the evident 

 solutions z = 0, 1, - -, 5. P. F. Teilhet 62 proved that these are the 

 only solutions by noting that the left member becomes (2 4)2(2+2) for 



60 Nouv. Ann. Math., (2), 11, 1872, 172. 



61 Nouv. Corresp. Math., 2, 1876, 287; Nouv. Ann. Math., (3), 4, 1885, 391. 



62 Mathesis, 7, 1887, 280. 



63 Nouv. Corresp. Math., 4, 1878, 123; ThSorie des nombres, 1891, 351, Ex. 4. Proof by P. 



Bachmann, Niedere Zahlentheorie, I, 1902, 44-6. 

 84 Nouv. Ann. Math., (2), 20, 1881, 515. 

 56 Ibid., (3), 1, 1882, 362. Invalid objection by G. C. Gerono, p. 520. 



66 Nouv. Ann. Math., (2), 20, 1881, 431-2. Same, Zeitschr. Math. Naturw. Unterricht, 13, 



1882, 451. 



67 Nouv. Ann. Math., (2), 20, 1881, 375. 



68 L'intermediaire des math., 2, 1895, 15. 



69 Ibid., 7, 1900, 211-3. 



60 Ibid., 9, 1902, 203. 



61 Ibid., 10, 1903, 132. 

 *Ibid., 12, 1905, 116-8. 



