296 



Solution of a Diophantine Problem. 



which is capable of dividing the same quantities as its compound. 

 If therefore no prime number will divide a given quantity, no num- 

 ber whatever will divide it. 



Now, assigning to z integral values, and dividing by 2, when z is 

 an even number, we form the following table. 



This table may be extended at pleasure ; and since each set of 

 numbers may be multiplied by m, it appears, first, that the proportion 

 between the sides of rational triangles may be infinitely varied ; 

 arid secondly, that different values may be indefinitely assigned when 

 the proportion between the sides remains constant. 



The following properties of the numbers in the foregoing table are 

 worthy of notice. 



1. If the numbers in the column [A] be subtracted from those in 

 the column [C], the difference will be a square. Ex. 101 — 20=81. 



Demonstration, [z^ -\-2z-\-2) — {2z-\-2)=z- a square. 



2. If 1 be added to the numbers in the column [B], the sum will 

 be a square. Ex. 3+1=4= n, 8-|- 1 = 9= n &c. 



Demonstration. {z^-\-2z)-\-l=z^ -\-2z-\-l—U .^' 



3. If 1 be taken from the numbers in the column [C], the remain- 

 der will be a square. Ex. 101-1 = 100=0, 82-l=Sl=n Sic. 



Demonstration, {z^ +2z-{-2) — l—z- -\-2z-\~l=U . 



4. The numbers in tlie column [F] are all composed of the sum 

 of 2 squares, taken, two and two, in the regular series 1, 4, 9, 16, 

 &x. Ex. 5=^1+4,13=44-9,25 = 94-16, &;c. 



* This is the demonstration of a property stated some weeks since in the Albany 

 Evening Journal, and thence copied in the National Intelligencer. 



