High School Mathematics Extensions/Primes/Problem Set

Problem Set
1. Is there a rule to determine whether a 3-digit number is divisible by 11? If so, derive that rule.

2. Show that p, p + 2 and p + 4 cannot all be primes if p is an integer greater than 3.

3. Find x

\begin{matrix} x \equiv 1^7 + 2^7 + 3^7 + 4^7 + 5^7 + 6^7 + 7^7 \ \pmod{7}\\ \end{matrix} $$

4. Show that there are no integers x and y such that
 * $$x^2 - 5y^2 = 3 \!$$

5. In modular arithmetic, if
 * $$x^2 \equiv y \pmod{m} \!$$

for some m, then we can write
 * $$x \equiv \sqrt{y} \pmod{m}$$

we say, x is the square root of y mod m.

Note that if x satisfies x2 &equiv; y, then m - x &equiv; -x when squared is also equivalent to y. We consider both x and -x to be square roots of y.

Let p be a prime number. Show that

(a)

(p-1)! \equiv -1\ \mbox{(mod p)} $$ where

n! = 1 \cdot 2 \cdot 3 \cdots (n-1) \cdot n $$ E.g. 3! = 1*2*3 = 6

(b)

Hence, show that
 * $$\sqrt{-1} \equiv \frac{p - 1}{2}! \pmod{p}$$

for p &equiv; 1 (mod 4), i.e., show that the above when squared gives one.