Question

The positive integers p q and r are all primes if p square minus q square is equals to hour then find the set of all possible values of r

Answer 1

Given:

3 primes numbers: $p, q,\ and\ r$.

The condition that:

$p^{2} -q^{2} =r$

To find:

All the possible values of $r$ = ?

Solution:

Property of all prime numbers is that they are always equal to the product of 1 and itself. There are no other factors.

Let us have a look at one more formula:

$a^{2}- b^{2} =(a-b)(a+b)$

Now, the given equation can be written as:

$p^{2} -q^{2} =r\\\Rightarrow (p-q)(p+q)=r$

Using the property of prime numbers:

$\Rightarrow (p-q)(p+q)=1\times r$

Given that $p, q,\ and\ r$ are prime therefore $p+q\neq1$

So, $p-q=1$ and $p+q=r$

Solving $p-q=1$:

Some of the prime numbers are: 2, 3, 5, 7, 11, 13......

Difference 1 is possible only when, $q =2$ and $p=3$.

Therefore, value of $r$ is:

$r=p + q = 3+2=5$