412 TRANSACTIONS OF SECTION A. 
Department oF MaTuematics. 
The following Papers and Report were read :— 
1. Arithmetical Factors of the Pellian Terms. 
By Lieut.-Colonel ALLAN Cunninauam, R.E. 
Let 7’, v’,), (7,, v,) be the zth solutions of the Pellian equations 
q'2 — Dy’? = — J, 7 — Qu? = +1, 
and let p, w, « denote an (odd) prime, an odd nwmber, an even number respectively. 
Ed. Lucas has shown that 
1. Every p divides into some v,. 
2. Every p = 8w+ 1 divides into one of 7’, v',, 7 
bd 
3. Every p = 8a+ 3, 5, 7 divides into 7,, v’,, 7', respectively. 
And, also that, in general, v, = 0 (mod p), 
where x 
1 
(p—1), when p = 8w+ 1 
D 
and = 2m. 
x 1), when p = 8a 3] 
ra cite ao ji + 
New criteria are now given, showing when n = 4m, 8m, 16m, 32m, and 3:2m; 
depending only on the linear and quadratic forms of p, and showing a close analogy 
between the theories of 
v, = 0 (mod p), and (27 — 1) = 0 (mod p). 
i. As to 2 = 2w; this happens when p = 8w + 3, 5, 7. 
p = 8m+t 3 gives v, = 0, 7, = 0, where 2 = 3(p + 1), y = ia. 
p= 8w+t Sgivest,=0,v,=0, , w=3(pt+1)y=4 (e+ 1). 
p= 8w+ Tgivesv,=0,7=0, , w© =3(p—1),y=3(@ + 1). 
ii. As to m = 4m, 8m, &e., these require p = 8w + 1 = a? + b? = c? + 2d?, with 
b = 48, d = 25; and then give n as below— 
Pp d n | Pp b | ad n 
/ 8a 41 2Qw 4o’ | 320+1 8e 8w | 16w’ 
| 7 4m |" 8a? || M 8a | 8e | 16’ 
| 8%+1 de 8u' | 55 8w 8w | lbe 
| 166 +1 4e | 160’ 
iii. As to 2 = 2°3m; p 
p 
8a +1 
8H + 3 
3m’ + 1 involves p = G*? + 6H”. 
3m’ — linvolves p = 2G? + 3H? 
Wu 
Then » = 2°3m when—(and only when)—H = 3h. 
iv. As to m = 25m, 27m, &c., the criteria seem not to depend solely on the linear 
and quadratic forms of p. 
v. As to the suffixes of 7’, v', 7; these are derivable from the suffix of v (which is 
fundamental). 
Let n', 7, — be the least suffixes of 7’, u', 7, v giving 
7’ = 0, wv’ =0, 7, = 0, ve = 0, (mod p). 
Then ¢ = ¢ gives 7 = 3€ always. 
i 
