30 Mr Pocklington, Prime or Composite Numbers 
3. The method can be applied with exceptional ease to} 
Fermat’s numbers /=2”"+1, where m=2*. Here 2 is always! 
an unsuitable number. We may choose «=3. Then, if in the 
course of the work we find a remainder equal to unity, we know| 
that # must be composite, for 3 is a primitive root of F whenever | 
Fis prime. Also if the mth squaring does not give remainder | 
unity it is clear that Fis composite. If however the mth squaring | 
is the first that gives remainder unity, then F is prime. For) 
our work shows that the exponents to which 3 belongs with/ 
respect to the prime divisors of # all divide 2”, and that one} 
at least does not divide 2". Hence one exponent is 2” and j 
the corresponding prime is at least 2” +1 = F. | 
The method can also be applied very easily to Mersenne’s | 
numbers M=2?—1, where p is prime. Here also 2 is always / 
unsuitable. If Jf is prime we must have 
x 
gM) 2 = (a7) x mod. M, 
where \F 
2 — 1 (the primeness of which is in dispute), we shall have to | 
make 88 multiplications and about the same number of divisions, | 
the multipliers, multiplicands and the divisor having about 27 
digits, and if the number is prime we shall have to doa further | 
amount of work not easy to estimate but not likely to greatly | 
exceed that already done. 
4, This method has the disadvantage that we only (excepting — 
in rare cases) determine whether NV is prime or composite, and 
that we may require to factorize N—1 in part at least. The | 
» advantage lies in the fact that the labour increases approximately | 
as (log V)3, not as ./N, which makes it a much easier method | 
than that of the Idoneals if V is large. It is also well adapted for 
use with the arithmometer. Other methods of proving that a: 
number is prime only give negative evidence of the fact (absence | 
of any divisor or absence of a second representation in a quadratic | 
form), so that a single slip can cause a composite number to be | 
taken as prime. In the present method however it is hardly _ 
possible that we should find #7+=1 mod. NV by making a slip, | 
and the accuracy of such a result as 2-1)/p=% mod. N can be. 
checked by finding what wu? is congruent to. ; 
) is Jacobi’s symbol. We see e.g., that if we wish to test’ 
