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. 
en toeep CBP °[C] [D] (E] FI 
a x pr 22422 2242242 
z=0, 2 0 « dividing by2, 1 O 1 
fT} 4 3 . 
z=2, 6 8 16 “ ome 4° 5 
zie, 8 15 eae: a § 
2=4, 10 24. 26 -« e¥,n 12 18 
2295, 12 35 ov 
2—0, 14 48 50 “7 | «Q4 25 
z=, 16 63 65 
= 18 80 Gee oy Be 
ee9, 20 99 101° 
This table may be extended at pleasure; and since aie set of 
ake may be multiplied by m, it appears, first, that the proportion 
between the sides of rational triangles may be infinitely varied } 
and secondly, that different values may be indefinitely assigned when 
the proportion between the sides remains constant. 
The following properties of the numbers i in the foregoing rable a are 
worthy of notice. 
1. If the numbers in the column [A] be subtracted Jie those in 
_ the column [C], the difference will be a square. Ex. —20=81- 
ve - Demonstration. _(z? +2242) — (2z4+-2)=27 a pets ; 
2. If 1 be added to the numbers in the column Aes the sum wil 
bea asquare. Ex. 3+1=4=0,81+1=9=0 
Demonstration. (z?42z)-+1=2z? pert = 
3. If 1 be taken from the numbers in the column a the remain 
der will be asquare. Ex. 101—1=100=n0, 82— va 0:4 
~ Demonstration. (22 +2242) —1=2?+42z41= 
4. The numbers in the column (F] are all composed ‘of the sump. 
of 2 aa taken, two and two, in the regular series 1, 4; 9, 16, 
&e. Ex. d==1+4,13=4+49, 25=9+16, &e. 
* This is the demonstration of a property stated some weeks since in she. , albany 
Evening Journal, and thenee copied in the National Intelligencer. 
