58 /. B. Luce on the Theory of Numbers. 



p=am±l = \Q ) and q = m—3 ; and it is 10 2 - 11 x3 2 = 1, with- 



out the necessity of continued extractions. 



2a 



When n is such that we can obtain a whole number for -7-, I 



call n a simple number ; and when such that "T* is a fraction, it 



is called a complex number. Thus the problem is already solved 

 in a simple and easy manner for at least half all numbers that are 

 likely to be needed, since the simple numbers exceed the complex 

 in the table of complex numbers hereafter presented. 



When n is complex, we must find a square factor by which 

 n being multiplied, the product shall be a simple number. For 

 if a 2 be that factor, we shall have^? 2 -n« 2 y 2 = l : but a 2 y 2 ~#*j 

 therefore it will be jd 2 — ?iq 2 = l. 



For instance, 13-9 + 4, is a complex. But if I multiply 13 

 by 5 2 , or 25, the product 325= 18 3 + 1, which is a simple number* 



Hence a = 18, m = 36, and p~am-\- 1 = 18 x 36 +- 1 —649; and 

 y^mSS; wherefore, p 2 - 13 x5 2 xy 2 = 649 2 -13x 180 2 = L 

 The finding of these square factors is a very nice problem, and 

 in some particular cases, rather intricate. The least number that 

 will make 109 a simple number is (S51325) 2 . I shall not pre- 

 sume to lay down any certain and satisfactory rules for the find- 

 ing of them ; I shall only content myself with a few hints on that 

 subject. 



When we have obtained the least values of x and y, in the 

 equation x 2 ~ny 2 ~\, we can readily obtain the least values of 

 p and y, in the equation p 2 — 4fi£*»l. For if y be an even 

 number, y 2 =4q 2 , or ? = £y, andp^x. But if y be an odd num- 

 ber, then will p=2x 2 — 1, andq^xy. For by hypothesis, we 

 have x 2 - ny 2 = 1. Multiply all the terms by 4# 2 , and transpose, 

 it will be 4# 4 — Ax 2 -&nx 2 y 2 — Q. Add one to each side, and it 

 is {2x 2 - l) 2 — Anx 2 y 2 = l; this compared with p 2 — 4wy 2 = l, 

 shows evidently that p = 2x 2 — 1, and q = xy. 



Again, if n be divisible by any square, we can obtain the values 



of X) and y, without the aid of any square factor, if we havejp, 



n 



and 9, in the equation p 2 — -q * = 1. 



For instance, let it be required to find x 3 — 234y 2 = l. As 



234 



-q- = 26 = 5 3 + 1, we say a — 5, m = l0] and placing these val- 

 ues of a, and m, in the fourth converging fraction, we find 

 p=2a*m*+4am+ 1 = 5201, and ? = 1020 = 3 x 340, and it is 

 5201 2 - 234 x340 2 = l. 



If the quotient of n by any square be of the form (? 2 - 1, we 

 place the values of a, and m, in the third converging fraction, since 

 under such form we always have (2a 2 m - 3a) 2 -n{2am — I)* »L 



