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



G. C. Gerono 33 proved that P 4 +n by setting (w+l)(m+4)=2p, 

 whence (m+2)(m+3)=2(p+l), while p(p+l)=|=n. "P. A. G." 34 gave a 

 proof by use of 



ra(ra+l)(w+2)(m+3) + l= {w(ra+3) + l} 2 . 



Gerono 35 proved that P 5 , P 6 or P 7 is not a square. 



V. A. Lebesgue 36 proved that P 5 is not a square or cube. 



A. Guibert 37 proved that, if 8^n^l7, P n =j=D, while P 6 or P 9 or a 

 product of any three integers in arithmetical progression is not a cube. 



A. B. Evans 38 and G. W. Hill 39 proved that P 6 H= D. 



D. Andre 40 proved that, if n>l, P n ^y n or 2/ n d=l. 



A. B. Evans 41 proved that P 5 , P 6 or P 7 is not a square. 



H. Bourget 42 proved that P 5 + D. 



R. Bricard 43 proved that P 8 =}= D by use of a Pell equation. 



L. Aubry 44 proved that P 4 is not a cube by treating the case in which 

 a single one of the four numbers is divisible by 3 and the case in which two 

 are divisible by 3, necessarily the first and fourth, and examining in the 

 second case the residues modulo 9 of the four numbers. 



T. Hayashi 45 proved that P 2 or P 4 is not a square or cube, P 3 =j=z n , n^2. 

 Also (p. 166), y(f/+l)(2y+l) =M, n^2. 



S. Narumi 46 proved that x(x+l] -(x+n) = D 4= is impossible if 



T. Hayashi 47 proved that P 6 =|= D. 



FURTHER PROPERTIES OF PRODUCTS OF CONSECUTIVE INTEGERS. 

 J. Liouville 48 proved that, if p is a prime > 5, 



Berton 49 verified that P=a(a+Ji)(a+2}i)(a+3h) 4=P 4 since 



Hence the area Vp of an inscriptible quadrilateral whose sides are in arith- 

 metical progression is not a square. 



33 Nouv. Ann. Math., 16, 1857, 393-4. 



34 Ibid., 17, 1858, 98. 



36 Ibid., 19, 1860, 38-42. 



36 Ibid., 112-5, 135-6. 



37 Ibid., 213 [400]; (2), 1, 1862, 102-9. 



38 The Lady's and Gentleman's Diary, London, 1870, 88-9, Quest. 2106. 



39 The Analyst, Des Moines, Iowa, 1, 1874, 28-29. 



40 Nouv. Ann. Math., (2), 10, 1871, 207-8. 



41 Math. Quest. Educ. Times, 27, 1877, 30; 44, 1886, 65-9. 



42 Jour, de math. 616m., 1881, 66. 



43 L'interme'diaire des math., 17, 1910, 139-40. 



44 Sphinx-Oedipe, 8, 1913, 136. 



45 Nouv. Ann. Math., (4), 16, 1916, 155-8. 

 46 T6hoku Math. Jour., 11, 1917, 128-142. 



47 Nouv. Ann. Math., (4), 18, 1918, 18-21. 



48 Jour, de Math., (2), 1, 1856, 351. 

 9 Nouv. Ann. Math., 18, 1859, 191. 



