60 



/. B. Luce on the Theory of Numbws. 



If in the equation x 



ny 



o 



P 



and 4« + 1, then will always the least value of x be of the form 



4«-|-l, and y of the form 4«. For y 



m 



W; 



P 



rily an even number when n is of the form 4a + 1. If we sub- 



am 



and we shall have d different ways of obtaining the product am ; 

 that is, we shall have so many different square factors of n, each 

 of which will give the least values of x and y. 



For instance, in how many different ways can we obtain the 



least values of x, and y, in the equation x 



Table containing the least square factor by which each com 



obtain a simple number.* 



13y 



l? 



O. N. 



Sq. fact 



C. N. fiq. fact. 



i 



C. N 



94 

 97 

 103 

 106 

 107 

 108 



^q fact 



151 2 



569 2 



472 



389 2 



3 2 



52 



109 I 851325 2 



C. N. 



Ill 



112 

 113 



114 

 115 

 116 



117 



118 

 124 

 125 



126 



2 2 



2 2 



73 2 



3- 



■72 



13 2 



5 2 



51 2 



273 2 



61 



402 I 



127 

 128 

 129 

 130 



131 

 133 

 134 

 135 

 137 

 139 

 149 

 151 

 153 

 154 

 155 

 157 

 158 



Sq tact. 



419775 2 



51 2 

 14 2 



52 

 92 



261 2 



33 2 



32 



149 2 



7472 



9305 2 



3383 2 



8 2 



22 2 



2 2 



315645 2 



72 



Having already found 13x25=18 2 + l, we inferred a =18, 

 w = 36, am =-648, whose divisors are 1, 2, 3, 4, 6, 8, 9, 12, 18, 

 24, 27, 36, 54, 72, 81, 108, 162, 216, 324, 648. Consequently, 

 we ought to obtain 10 different factors of 13 which will, each of 

 them, give the least values of x, and y. These factors are, 10* , 



and 



20 



9 



15% 20 2 , 30 2 , 45 2 , 60% 90% 180 2 , 



I shall say no more upon this new theory, fecund as it is in the 

 resolution of indeterminate problems of all degrees, except the 

 first ; leaving it for others to bring it to its highest degree of per- 

 fection, if they think it worthy of their attention. 



Troy Ladies' Seminary, March 20th, 1849. 



* Each column marked C. N., contains the complex numbers, and the column 

 marked Sq.fact., their square factors. Every number not found in this table, up to 

 1 58, is a simple number. 



