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

THE THEOREMS OF FERMAT AND WILSON 53
where hi is an integer. Therefore
a2=i mod 23.
Squaring (3) we have
22 = I + 24h2;
or 2
o~r a22 _ i mod 24.
It is now easy to see that we shall have in general
a2a- I mod 2a
if > 2. That is,
a30(2a)-I mod 2a if a> 2.
Now in terms of the p-function let us define a new function
X(m) as follows:
X(2a) = (2a) if a =o,,2;
X(2a)-=~(2a) if a>2;
X(pa) = ((pa) if p is an odd prime;
X(2plap2a2... pnan) ==M,
where M is the least common multiple of
X(2a), X(pil), X(P2a2)..., X(p )
2, Pi, p2,..., pn being different primes.
Denote by m the number
m=2cpl"tp2a2. ~. pnrn
Let a be any number prime to m. From our preceding results
we have
aX(2a)I mod 2a
ax(Pl) I mod pli1,
aX(p, 2) _ I d p2%
aX( )I mod pn2a
axv -~ I mod pn.