Three twin primes

Number Theory Level 3

We define a twin prime as a pair of prime numbers whose absolute difference is precisely 2.

Let $p,q$ and $r$ be odd prime numbers such that $p,q$ are twin primes and $r$ is a twin prime with some prime.

Given that $\dfrac{p^2+q}2 = r$ and $p<r-p< q < r < p^2$ , find the product $pqr$ .

2 solutions

Kushal Bose
Sep 24, 2016

As $p$ and $q$ are twin primes and $p<q$ .It can be concluded that $q=p+2$ .

There exist only one integer between $p,q$ .So $r-p=q-1=p+1 \\ =>r=2 p+1$

Putting all primes in terms of $p$ we get:

$p^2+p+2=2 (2 p+1) \\ =>p^2-3 p=0 \\ => p=3$ .

Then $q=5,r=7$

Sswag SSwagf
Sep 24, 2016

We can develop all the statements to get

$r=p+4$

$q=p+2$

$(r-p)=p+1$

$(p+4)-p=p+1$

$p=3$

so we have primes with the fllowing difference

$3, (3+2), (3+4)$

$pqr=3.5.7=105$