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

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THEORY OF NUMBERS

~ 28. EXTENSION OF FERMAT'S THEOREM
The object of this section is to extend Fermat's general
theorem and incidentally to give a new proof of it. We shall
base this proof on the simple Fermat theorem, of which we
have already given a simple independent proof. This theorem
asserts that for every prime p and integer a not divisible by p,
we have the congruence
ap-1 i mod p.
Then let us write
ap-1 =+hp. (I)
Raising each member of this equation to the pth power we
may write the result in- the form
ap(p-l) =I+hlp2 (2)
where hi is an integer. Hence
ap(p- 1) - I mod p2.
By raising each member of (2) to the pth power we can readily
show that
ap(p-1)-I mod p3.
It is now easy to see that we shall have in general
pa-l(p_1)
apt -p — mod pa
where a is a positive integer; that is,
a~(p) -- mod pa.
For the special case when p is 2 this result can be extended.
For this case (i) becomes
a=I+-2h.
Squaring we have
a2 = I +4h(h +I).
Hence,

a2= I +8hl,

(3)