Prime Numbers

Chapter 3
RECOGNIZING PRIMES AND COMPOSITES
Given a large number, how might one quickly tell whether it is prime or
composite? In this chapter we begin to answer this fundamental question.
3.1 Trial division
3.1.1 Divisibility tests
A divisibility test is a simple procedure to be applied to the decimal digits
of a number n so as to determine whether n is divisible by a particular
small number. For example, if the last digit of n is even, so is n. (In fact,
nonmathematicians sometimes take this criterion as the definition of being
even, rather than being divisible by two.) Similarly, if the last digit is 0 or 5,
then n is a multiple of 5.
The simple nature of the divisibility tests for 2 and 5 are, of course, due
to 2 and 5 being factors of the base 10 of our numeration system. Digital
divisibility tests for other divisors get more complicated. Probably the next
most well-known test is divisibility by 3 or 9: The sum of the digits of n is
congruent to n (mod 9), so by adding up digits themselves and dividing by 3
or 9 respectively reveals divisibility by 3 or 9 for the original n. This follows
from the fact that 10 is one more than 9; if we happened to write numbers
in base 12, for example, then a number would be congruent (mod 11) to the
sum of its base-12 “digits.”
In general, divisibility tests based on digits get more and more complicated
as the multiplicative order of the base modulo the test divisor grows. For
example, the order of 10 (mod 11) is 2, so there is a simple divisibility test
for 11: The alternating sum of the digits of n is congruent to n (mod 11). For
7, the order of 10 is 6, and there is no such neat and tidy divisibility test,
though there are messy ones.
From a computational point of view, there is little difference between a
special divisibility test for the prime p and dividing by p to get the quotient
and the remainder. And with dividing there are no special formulae or rules
peculiar to the trial divisor p. Sowhenworkingonacomputer,orevenfor
extensive hand calculations, trial division by various primes p is simpler and
just as efficient as using various divisibility tests.