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

$\begin{array}{c}x\equiv 1^7+2^7+3^7+4^7+5^7+6^7+7^7(\mathrm{mod}7)\end{array}$

4. Show that there are no integers x and y such that

$x^2-5y^2=3$

5. In modular arithmetic, if

$x^2\equiv y(\mathrm{mod}m)$

for some m, then we can write

$x\equiv \sqrt{y}(\mathrm{mod}m)$

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

Note that if x satisfies x² ≡ y, then m - x ≡ -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)

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

where

$n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n$

E.g. 3! = 1*2*3 = 6

(b)

Hence, show that

$\sqrt{-1}\equiv \frac{p-1}{2}!(\mathrm{mod}p)$

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