CHAP, xix] PRODUCTS BY Twos PLUS UNITY MADE SQUARES. 517 



PRODUCT OF ANY TWO OF FOUR OR FIVE NUMBERS INCREASED BY UNITY A 



SQUARE. 



Diophantus, IV, 21, required four numbers such that the product of 

 any two increased by unity is a square. He took x, x-{-2, 4x+4 as the 

 first three (by IV, 20), and (3z+l) 2 -l as the product of the first and 

 fourth. Thus the fourth is 9x+6. The product of the second and fourth, 

 increased by unity, is 9z 2 +24x+13; let it equal (3x 4) 2 , whence z = l/16. 

 The remaining conditions are now satisfied. 



Rafael Bombelli 90 treated the problem for four numbers. 



Fermat 91 took 1, 3, 8 as the first three numbers. The conditions on the 

 fourth number x are z+l = D, 3z+l = D, 8x+l = D. His method 

 (Fermat 10 - u of Ch. XV) of solving a "triple equation " gives z = 120. 



L. Euler 92 gave the solution a, b, c = a+b-\-2l, d=4:l(l+a)(l+b), where 

 ab+1 = I 2 , and noted the cases 3, 8, 1, 120 and 3, 8, 21, 2080. He extended the 

 question to five numbers, by seeking z such that l+az, , l-\-dz are all 

 squares. Denote the product of these four sums by P = 1 -}-pz+qz-+rz*-\-sz*, 

 where therefore p = a+b-}-c+d, , s = abcd. Let P be the square of 



^pz+gz 2 , where g = q/2-p 2 {8. Then 



For brevity set a + b + 1 =/, d/4 = k. Then 



c=f+l, 



Now k=f(ab+l)+lab, 4/c 2 = 4A/+4fca6(/+Z). Hence 

 l+<?+s = (2/c+/) 2 =ip 2 , 0=-i(l 

 The denominator g 2 s of z is fortunately the square of (s 1)/2. Thus 



and P is a square. Euler stated that each factor l+az, etc., is then a 

 square. Taking a = 1, 6 = 3, we have 1 = 2, c = 8, d=120, p = 132, # = 1475, 

 r = 4224, s = 2880, z = 777480/2879 2 , and the ten expressions 06+ 1, , 

 dz+1 are the squares of 



9 3 11 Z 1O 31 3011 3250 3809 10070 

 Z, O, 11, O, ly, Ol, 2879) 2879) 2879) 2879 



To obtain smaller (but fractional) numbers, set a = 1/2, 6 = 5/2. Then 



c = 6, d = 48, 2 = 44880/128881. 



A. M. Legendre 93 verified Euler's preceding assertion that \-\-az, etc., 

 are squares by noting that a, b, c, d are the roots of 



90 L'algebra opera, Bologna, 1579, p. 543. 

 91 Oeuvres, III, 251. 



92 Opusc. anal., 1, 1783, 329; Comm. Arith., II, 45. Results stated in a letter to Lagrange, 



Sept. 24, 1773 (Oeuvres, XIV, 235-40); Euler's Opera postuma, 1, 1862, 584-5. 



93 The'orie des nombres, ed. 3, 2, 1830, 142-4; Maser's transl., 2, 1893, 138. 



