Number Theory

This hypothesis (or theory)

There are no solutions for the following equation,

${ n }^{ p }={ m }^{ p }-1$

${ n }^{ p }={ m }^{ p }-2$

Where $p$ is a prime number,

$n,m$ are positive integers and must be bigger than 1.

#NumberTheory

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Isn't this obvious, for any natural number $p > 2$ ?

The difference of consecutive powers is at least $2 ^ p - 1$ .

Calvin Lin Staff - 6 years, 3 months ago

Wait waaaaa?????????? I just gave a comment on it. Not proved it.

And u must state that $m$ and $n$ are not $0,1,-1$

Julian Poon - 6 years, 3 months ago

No need to state that because u already said that it must be greater than 1

Sudhir Aripirala - 6 years, 3 months ago

I asked whether it is true for n^p=m^p-3 or 4 .............

Sudhir Aripirala - 6 years, 3 months ago

@Sudhir Aripirala – I'll give you a clue, then you can figure out for yourself:

$m^{p}-n^{p}=k$

The difference between $2$ consecutive powers are always more than $k$ .

What must $k$ be?

Julian Poon - 6 years, 3 months ago

@Julian Poon – k=k

Luke Zhang - 6 years, 3 months ago

@Luke Zhang – Wooaaahhhh very insightful! Illuminati!

Julian Poon - 6 years, 3 months ago

@Julian Poon – I understood your theorem. Very good and thanks for making me realize my mistake

Sudhir Aripirala - 6 years, 3 months ago

Can this also be correct instead of 2 if we keep 3,4,5..........

Sudhir Aripirala - 6 years, 3 months ago

im sorry. Im not good

Luke Zhang - 6 years, 3 months ago

I was actually starting to think if this was a troll...

Julian Poon - 6 years, 3 months ago

Dood, im like 10 times worse than u lol.

Luke Zhang - 6 years, 3 months ago

@Luke Zhang – I dun even know calculus

Luke Zhang - 6 years, 3 months ago

@Luke Zhang – K noted.

Julian Poon - 6 years, 3 months ago

@Julian Poon – Glad that u NOTED.

Luke Zhang - 6 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$