( 328 ) 
The easiest way to obtain such an ~ consists in increasing a, several 
times by unity and continuing this operation 
N= a — bi 
= (ai + 1)? — (bj + 2a, + 1) or = ag? — by, 
= (aq + 1)? — (bg + 2a, +1) or = ag” — bz, ete. 
until the number 5 in the last line has become a square. 
Example WV = 57. 
We write N= 8?—7 
= 9 (741284 1)=9?— 24 
= 102 — (24 2% 9+41)=—109— 43 
— 112 — (43 +2 10 4 1)=—112— 64 
= 11? — 62 (11 + 8)(11 — 8) = 19 X3. 
For shortness’ sake we make use of the algorithm 
57 = 8? — 7 
2x8 -l- L=17 
EN 
19 
ES 
21 
 64= 82, 
so 57=(8-+ number of additions)? — 8°, where only the first 
additional number 17 has to be calculated. 
An important abbreviation can be obtained by paying attention 
to the terminal figures, as is shown by the following example. Here 
the reckoning was to be: 
However as a? can terminate in 0, 1, 4, 
Mesen 5, 6 or 9 only and the last figure of is 7, 
ee 1 eer a®2—N can terminate in 3, 4, 7, 8, 9 or 2 
1857 only. But a?—J being also a square (67), it 
1437 can only terminate in 4 of 9. So it is un- 
3294 necessary to perform all the additions and - 
etc. the abbreviated algorithm comes to this: 
