APR. 19,1923 HEYL: REMARKABLE FORMULA FOR PRIME NUMBERS 151 
Substituting for 2*4 its remainder when divided by 483, namely 211, 
we obtain: . 
211° x 4 = (2112) x 4 
Taking remainder of 211? 85° x 4 = (852)5 x 4 
4635 X 4 = 4634 x 463 x 4. 
Remainder of 463? = 400 and of 463 «x 4 = 403 
Hence we obtain 400? x 403 
127 x 403 = 51181 
the remainder of which is 466, not unity; hence 483 is composite. 
Also, remainder of Bia Tint 
483 
the original formula, will in about 75 per cent of the cases examined 
(1000 in number) contain a factor of the original number, which may 
be found by the usual arithmetical process. For example: 
= 465. This remainder, resulting from 
465)483(1 
465 
18) 465 (25 
36 
105 
90 
15)18(1 
15 
3=H.C.F. 
The time required for such an operation is, for large numbers, very 
much less than that required for trying prime divisors up to vn. 
For example, a number of the order 108 might require the trial of 1228 
primes, which, at the rate of one division per minute, would take 20 
hours. By the method of powers of 2 the reduction can be made in 
one hour. } 
This method lends itself admirably to checking, as the original 
numerator can be expressed in several different ways, for example: 
9482 — (224)20 4 92 = (216) 30 x 92 = (250) 9 4 932 etc., 
each leading, on reduction, to a different series of numbers, but all 
ending in the same way. 
