Prime or Composite

[Fahim Shahriar]

Is the number $\frac{2^{62} + 1}5$ prime or composite?
Easy.. What's your method?

[Fahim Shahriar]

I did...--

$2^{62}=2² {2^4}^{15} ≡4 (mod 5) $
So, $2^{62}+1≡0 (mod 5)$
Therefore, $\frac{2^{62} + 1}5$ is an integer.

Now, $2^{31}+1)² = (2^{62}+1)+2^{32}$
$=> (2^{62}+1) = (2^{31}+1)² - (2^{16})²$
$=> (2^{62}+1) = (2^{31}+2^{16}+1)*(2^{31}-2^{16}+1)$

The factors are greater than 5. So it can be written as $2^{62}+1 = 5xy$ ; where $x,y$ are integers and both greater than 1.
Hence, $\frac{2^{62} + 1}5$ is composite.

Any other way

[sakibtanvir]

I solved the same problem a month ago and i did like this....