Question

8) (Problem 17 (a) on page 49) Let p and q be two distinct primes. Show that for any integer a, pq|(a p+q − a p+1 − a q+1 + a 2 ). Hint: Find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo p, and then find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo q. After that, use the following result: Suppose x, y, and z are integers. If x|z, y|z, and (x, y) = 1, then xy|z. (Fermat's or Wilson's Theorem Number Theory)

Answer 1

Let p and q be two distinct primes.

According to Fermat's little theorem, if p is a prime then for any integer a we have

aP a (mod p)

Therefore

ala ala (mod p)

aptq aq+1 (mod p)

And also we have

a a a a (mod p)

+ 1 ap a2 (mod p)

Then, subtracting (2) from (1) we have

$a^{p+q}- a^{p+1} \equiv a^{q+1}-a^2 \text{ (mod p)}$

Then

$a^{p+q}- a^{p+1} -a^{q+1}+a^2 \equiv0 \text{ (mod p)}$

Thus,

plaPtg aP+1

Analogously, due to Fermat's little theorem we have that, for any a

a a (mod q)

Therefore

aPaaPa (mod q)

(mod q)

And also we have

a aa a (mod q)

+ 1 aq a2 (mod q)

Then, subtracting (4) from (3) we have

$a^{p+q}- a^{q+1} \equiv a^{p+1}-a^2 \text{ (mod q)}$

Then

$a^{p+q}- a^{p+1} -a^{q+1}+a^2 \equiv0 \text{ (mod q)}$

Thus

aP+y aP+1 a941 Cl

Now, since p and q are primes then (p,q)= 1

Therefore