# Why does (p*q+2)-(p+q) always give a prime number?

• John Harris

## Homework Statement

Why does (p*q+2)-(p+q) always give a prime number when p and q are prime? Is there a similar formula that would prove this

## Homework Equations

That's what I'm looking for. It might have something to do with Eulers formula

## The Attempt at a Solution

I tried to find online a formula that would justify this, but was unable to find anything.

Why does (p*q+2)-(p+q) always give a prime number when p and q are prime?
It doesn't.

The first example I picked was a counter example:
(17*47+2)-(17+47)=737=11*67

John Harris
It doesn't.

The first example I picked was a counter example:
(17*47+2)-(17+47)=737=11*67
Oh you're right. I should have tried more examples. Thank you

Oh you're right. I should have tried more examples. Thank you
You couldn't have tried many. It fails whenever p and q differ by 2.

You couldn't have tried many. It fails whenever p and q differ by 2.
I tried 7 and 13

I tried 7 and 13
Only that pair?! Try 3 and 5, 5 and 7, 11 and 13,...

I tried 7 and 13
And you think one example is enough to generalize from ? Probably not a great idea.

Gez no I tried 3 examples 3,7 1,3 7,13, and it would have made sense with the problem I'm doing.

Gez no I tried 3 examples 3,7 1,3 7,13, and it would have made sense with the problem I'm doing.
Well can you see how your statement "I tried 7 and 13" sounds a LOT like "I tried one combination" ? Glad to hear you already realize that just one is not a good idea.

Gez no I tried 3 examples 3,7 1,3 7,13, and it would have made sense with the problem I'm doing.
Your second example isn't valid because 1 isn't a prime number.