Number Theory Proof of the day--11/21

[Steven G]

Prove that if p>q>5, where p and q are both prime, then 24| p²-q²

 

[blamocur]

Spoiler

Both (p-q) and (p+q) are divisible by 2, and one of the is divisible by 4, thus [imath]p^2-q^2[/imath] is divisible by 8. Among three numbers p-q, p, p+q one is divisible by 3, and it cannot be 'p', thus [imath]p^2-q^2[/imath] is divisible by 3.

 

[khansaheb]

Prove that if p>q>5, where p and q are both prime, then 24| p²-q²

24|p²- q² ...... should be written as ......... 24|(p² - q²)

Otherwise the required conclusion is false!

 

[blamocur]

24|p²- q² ...... should be written as ......... 24|(p² - q²)

Otherwise the required conclusion is false!

Python would return [imath]-q^2[/imath], which it would consider true

 

[Steven G]

Spoiler

Both (p-q) and (p+q) are divisible by 2, and one of the is divisible by 4, thus [imath]p^2-q^2[/imath] is divisible by 8. Among three numbers p-q, p, p+q one is divisible by 3, and it cannot be 'p', thus [imath]p^2-q^2[/imath] is divisible by 3.

Number theory is fascinating! I never knew that one of p-q, p, p+q must be a multiple of 3.
Let's see if I can see the proof. If we set A=p-q, the the three numbers are A, A+q, A+2q. Their sum is certainly a multiple of 3. I need more time for this one.
OK, I see the proof! Thanks for that one!

 

[Steven G]

Number theory is fascinating! I never knew that one of p-q, p, p+q must be a multiple of 3.
Let's see if I can see the proof. If we set A=p-q, the the three numbers are A, A+q, A+2q. Their sum is certainly a multiple of 3. I need more time for this one.
OK, I see the proof! Thanks for that one!

I should of said that I see a proof.

 

[khansaheb]

Python would return [imath]-q^2[/imath], which it would consider true

Garbage in → Garbage out