CHAP, xiv] SQUARES IN GEOMETRICAL PROGRESSION. 441 



be squares, the first three being v 2 , w 2 , x 2 . As known, v, x = t 2 u z =F2tu, 

 W= =t 2 +u 2 . Since the common difference of these squares is 8 = 4:tu(t 2 u 2 ~), 

 the fifth number is iv 2 +3d = z 2 . This has an infinitude of solutions t, u, z 

 derivable in succession from the minimum solution. From the solution 

 7 2 , 13 2 , 17 2 , 409, 23 2 , there are deduced two solutions in much larger integers. 



SQUARES IN GEOMETRICAL PROGRESSION. 



Beha-Eddin BO (1547-1622) included (as Prob. 6) among the 7 problems 

 remaining unsolved from former times: Find 3 squares in G. P. whose sum 

 is a square. Nesselmann noted that the problem is impossible since 

 x 2 +x 2 y 2 +x 2 y*= D has no rational solution [Adrain, 113 Anderson, 114 Genoc- 

 chi, 119 Pocklington 138 of Ch. XXII]. 



To find three squares in G. P. and three numbers in A. P. such that the 

 three sums of corresponding terms are squares, W. Saint 51 took a 2 , a 2 x 2 , 

 a 2 x 4 as the squares in G. P. and 2a+l, ax*+a+l, 2ax 2 +l as the numbers 

 in A. P. It suffices to make a 2 x 2 +ax 2 +a+l = D = (az+z/2) 2 , say, whence 

 = I X 2_ ! Others took x 2 , 4z 2 , IQx 2 and either l,4z+l,&c+lor 2ax+a*, 

 8ax+4a 2 , 14ax+7a 2 . 



W. Wright 52 found three squares x 2 , a 2 x 2 , a 4 x 2 in G. P. each plus its root 

 being a square. Thus z 2 +z=D, x 2 +xla=E3, x 2 +xfa 2 =U, which are 

 satisfied in the usual way (Ch. XVIII). 



To find three squares in G. P. each less its root being a square, J. Ander- 

 son 53 took x 2 , xy, y 2 as the roots and x 2 l = (p-x) 2 , y 2 -l = (q-y} 2 , which 

 give x,y. Then x 2 y 2 xy=d leads to a quartic in p which is solved as usual. 

 Isaac Newton (I. c.) took {r 2 /(2r-l)j 2 , r 2 , (2r-l) 2 as the numbers. The 

 first of the three conditions is satisfied identically. Take r 2 r = n 2 r 2 , 

 whence r = l/(l-n 2 }. Then (2r-l) 2 -(2r-l) = D if 2n 2 +2=D. Set 

 n = m+l. Then 2n 2 +2 = (sm+2) 2 determines m. 



S. Ward 54 found three squares x 2 , 4x 2 , IQx 2 in G. P., such that if any one 

 of them is increased by its root, the sum is a square. Take x 2 +x = p 2 x 2 . 

 The remaining two conditions become 2p 2 +2= D, p 2 +3= D, which hold 16 

 if p = 23/7. 



60 Essenz der Rechenkunst von Mohammed Beha-eddin ben Alhossain aus Amul, arabisch 

 u. deutsch von G. H. F. Nesselmann, Berlin, 1843, p. 56. French transl. by A. Marre, 

 Nouv. Ann. Math., 5, 1846, 313. 



51 The Diary Companion, Suppl. to Ladies' Diary, London, 1806, 36-37. 



52 The Gentleman's Math. Companion, 5, No. 24, 1821, 41-44. 

 Ibid., 5, No. 27, 1824, 274-7. 



" J. R. Young's Algebra, Amer. ed., 1832, 341. 



