438 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xiv 



where p, q are relatively prime and one even. Extending the A. P., he 

 proved that the nth term is a square if q = l, 2p = n 2, or q = 2, p = n 2. 



1 Civis ' 24 proved that the common difference of three rational squares 

 in A. P. is never 17. For, if so, 4a6(a 2 -6 2 ) = 17? 2 . Put a = r 2 , 6 = s 2 , 

 r 2 -s 2 = 8y 2 , whence r = 2v-+l, s = 2y 2 -!. Put u = qf(Srsv). Then I7u~-l 

 = 4v 4 , which is impossible in view of the formula for z in the known solu- 

 tion of I7u 2 1=2 2 . A. Martin 25 noted that the theorem is evident for 

 integers since a multiple of 4 cannot equal 17. 



To find 26 three squares in A. P. such that each exceeds its root by a 

 square, employ Frenicle's numbers (say I, m, n), and take Ix, mx, nx as 

 the roots of the required squares. Then I 2 x 2 fo=D, etc., are solved as 

 in Ch. XVIII. 



D. Andre 27 noted that, if three squares are in A.P., 



2y 2 = x 2 +z 2 , y z =( - 7 r- ) +1 ) =a 2 +c 2 , x = a+c, z = a-c 



G. R. Perkins 28 treated the problems 1 [2]: Find three squares in A. P. 

 such that each less [plus] its roots is a square. Take the numbers to be 

 the squares of ^, 7?=b|, fJ, where 4 = +x~ 1 , 4?; = 7/+?/~ 1 , 4f = z+z~ 1 , 

 and the signs are + or according as the problem is 1 or 2. Then each 

 square its root is a square. The squares are in A. P. if 



P p+1' 



These give , 77, f and n=2m(p+l)fN, where A^=2p(2p+l)- The desired 

 numbers are the squares of d=! = ra/cr, m(b, m/c, where 



It remains to make x, y, z rational, using 4 = +#~ 1 , etc. This requires 

 that m^tm be a square for t = a, 6, c. Now 



2 

 if m = - 



Then m 2 T6m=D if fc 2 b(2k a) = D, say (k l) 2 , whence & is a rational 

 function of I Then k 2 -c(2k-a) = D if /=(a+6-c)/2. For Prob. 1, 

 p>2; if p = 3, m/a, -, m/c are quotients of numbers of 14 digits [cf. Hart 5 

 of Ch. XVIII]. Three times as many digits are involved in the answer by 

 D. Kirkwood, 29 who started with x 2 , 25x 2 , 49z 2 . For Prob. 2, the use of 

 p = l gives the answers due to Williams 6 of Ch. XVIII. 



24 The Lady's and Gentleman's Diary, London, 1866, 56-7, Quest. 2041. 

 28 Math. Quest, Educ. Times, 52, 1890, 87. 



26 Ibid., 14, 1871, 54. A Collection of Diophantine Problems by J. Matteson, pub. by A. 



Martin, Washington, D. C., 1888, 10, pp. 14-16. 



27 Nouv. Ann. Math., (2), 10, 1871, 295-7. 



28 The Analyst, 1, 1874, 101-5. 



"Stoddard and Henkle, University Algebra, N. Y., 1861, p. 494. 



