( 501: ) 
Mathematics. — “Factorisation of large numbers.” (Srl part). 
By F. J. Vars. (Communicated by Prof. P. H. ScHoure). 
XIII. Abbreviated method of § I. 
The determination of the differences of squares allows of a very 
considerable abbreviation. 
According to that method we write G==a,’—,, and to bj we 
add 2a, + 1, 2a,-++ 3 etc. until a square b° is arrived at. 
If to reach this p additions are necessary, then 
b—=b+pX2aq,+(14+3 +54+...... ) 
nd 
p terms 
== bj + p X 2a, + p?, 
so b? = p (2a, + p) + 0}. 
Example: G = 513667 = 717?—422, (See pag. 328). 
We test for p successively the values 1, 2, 3, etc. as follows 
1 X (1434 + 1) + 422 = 1435 4 422 — 1857 
; 2 X (1434 + 2) + 422 = 8294 
3 X (1434 + 3) + 422 = 4733 
4 X (1434 + 4) + 422 = 6174 
etc., 
and we see whether the result is a square. It is immediately 
evident, that in this way the same numbers are obtained as for 
the additions on pages 328 and 329, but at the same time that it 
is not necessary to take all values 1, 2, 3, etc. for p. 
For 6? must terminate in 4 or 9 (pag. 328), so that the product 
(before 422 is added) can have only 2 or 7 as final figure. So we 
have but to calculate: 
2 >< (1434 + 2) + 422 — 2872 + 422 — 3294 
4X 1488 5752 6174 
7X 1441 10087 10509 
9x 1443 12987 13409 
etc., 
33 
Proceedings Royal Acad, Amsterdam. Vol. IV, 
