Number theory: finding integer solution to an equation

[mtayab1994]

Homework Statement

(E): x^2+y^2=6+2xy+3x

The Attempt at a Solution

$$x^{2}+y^{2}=6+2xy+3x\Longleftrightarrow x^{2}-2xy-3x+y^{2}=6\Longleftrightarrow x^{2}+x(-2y-3)+y^{2}=6$$

Any further help to find the answer??

 

[Joffan]

You might try looking at $x^2 - 2xy+y^2=6+3x$

 

[mtayab1994]

You might try looking at $x^2 - 2xy+y^2=6+3x$

That's $$(x-y)^{2}-3x=6$$

 

[Joffan]

That's $$(x-y)^{2}-3x=6$$

Or $(x-y)^2=3(x+2)$ - which should be more interesting.

 

[mtayab1994]

Or $(x-y)^2=3(x+2)$ - which should be more interesting.

Can we use substitution and say that x+2=n?

 

[Joffan]

Can we use substitution and say that x+2=n?

Sure, although we will find a better substitution... what can you tell me about $n$?

 

[mtayab1994]

Sure, although we will find a better substitution... what can you tell me about $n$?

On the question before I proved that x^2 Ξ 0(mod3) and that means that x^2=3n.

 

[Joffan]

On the question before I proved that x^2 Ξ 0(mod3) and that means that x^2=3n.

Hmm, not really. Let's define $m:=(x-y)$ - what can you tell me about $m$?

 

[mtayab1994]

Hmm, not really. Let's define $m:=(x-y)$ - what can you tell me about $m$?

That means that m is the difference of x and y.

 

[Joffan]

That means that m is the difference of x and y.

My question is, what does $m^2=3n$ (using the new definitions) tell you about $m$?

 

[mtayab1994]

My question is, what does $m^2=3n$ (using the new definitions) tell you about $m$?

That means that m^2 is divisible by 3 hence divisible by all multiples of 3.

 

[vela]

Is that what you meant to say? 81 is divisible by 3, but it's not divisible by 6, which is a multiple of 3.

 

[Joffan]

That means that m^2 is divisible by 3

Yes... what does that mean for $m$?

... hence divisible by all multiples of 3.

What? No. For example, 36 is divisible by 3 but not by 15.

 

[mtayab1994]

Is that what you meant to say? 81 is divisible by 3, but it's not divisible by 6, which is a multiple of 3.

If we said that m=x-y then m^2=(x-y)^2=3n .
So then we get that (x-y)^2=3n right?

 

[vela]

Yes, but it doesn't follow that m² is divisible by all multiples of 3, which is what you claimed.

 

[mtayab1994]

Yes, but it doesn't follow that m² is divisible by all multiples of 3, which is what you claimed.

But how are we supposed to find integer solutions out of that? I found that y=n^2-3n-2 and that's wrong I think.

 

[vela]

At this point, you're going to have to think about it a bit on your own. You've got all the pieces. You just need that last little insight, which is what Joffan's been trying to get you to see.

 

[mtayab1994]

My question is, what does $m^2=3n$ (using the new definitions) tell you about $m$?

Which new definitions m=x-y?

 

[Joffan]

Which new definitions m=x-y?

Don't backtrack. We have defined new variables $m$ and $n$; the formula translates into those variables as shown; now you need to understand what $m^2=3n$ tells you about $m$.

 

[mtayab1994]

Don't backtrack. We have defined new variables $m$ and $n$; the formula translates into those variables as shown; now you need to understand what $m^2=3n$ tells you about $m$.

Well there are 3 cases:

Case 1: if m Ξ 0(mod3) then m^2 Ξ 0(mod3)

Case 2: if m Ξ 1(mod3) then m^2 Ξ 1(mod3)

Case 3: if m Ξ 2(mod3) then m^2 Ξ 4(mod3) with is m^2 Ξ 1(mod3)

So that means that m^2=3n . So that means the when m is divided by 3 you get either a remainder of 0,1, or 2 am i right?

 

[vela]

Well there are 3 cases:

Case 1: if m Ξ 0(mod3) then m^2 Ξ 0(mod3)

Case 2: if m Ξ 1(mod3) then m^2 Ξ 1(mod3)

Case 3: if m Ξ 2(mod3) then m^2 Ξ 4(mod3) with is m^2 Ξ 1(mod3)

Looks good.

So that means that m^2=3n . So that means the when m is divided by 3 you get either a remainder of 0,1, or 2 am i right?

How did you come up with that conclusion based on what you wrote above?

 

[Joffan]

What is the value of $3n$ mod 3?

 

[mtayab1994]

What is the value of $3n$ mod 3?

3n mod 3 means that 3n=3k so that means 3k equals the multiples of 3 which are 3n.