The theory of numbers, by Robert D. Carmichael ...

ELEMENTARY PROPERTIES OF INTEGERS

13

III. Among the integers of the arithmetic progression a, a+d,
a+2d, a+3d,.., there is an infinite number of prii;,cs, provided that a and & are relatively prime.
For the special case given in theorem II we have an elementary proof; but for the general theorem the proof is difficult.
We shall not give it here.
EXERCISES
I. Prove that there is an infinite number of primes of the form 4n —I.
2. Show that an odd prime number can be represented as the difference of
two squares in one and in only one way.
3. The expression mlP.-nlP, in which m and n are integers and p is a prime,
is either prime to p or is divisible by p2.
4. Prove that any prime number except 2 and 3 is of one of the forms 6n+i,
6n —.
~ 5. THE FUNDAMENTAL THEOREM OF EUCLID
If a and b are any two positive integers there exist integers
q and r, qo, o <r<b, such that
a=qb +r.
If a is a multiple of b the theorem is at once verified, r being
in this case o. If a is not a multiple of b it must lie between
two consecutive multiples of b; that is, there exists a q such
that
qb<a<(q+i)b.
Hence there is an integer r, o<r<b, such that a=qb+r. In
case b is greater than a it is evident that q=o and r =a. Thus
the proof of the theorem is complete.
~ 6. DIVISIBILITY BY A PRIME NUMBER
I. If p is a prime number and m is any integer, then m either
is divisible by p or is prime to p.
This theorem follows at once from the fact that the only
divisors of p are i and p.
II. The product of two integers each less than a given prime
number p is not divisible by p.