Number theory: finding integer solution to an equation

In summary, the conversation discusses solving a mathematical equation and using substitution to find integer solutions. The participants discuss the importance of understanding what a formula tells them about new variables and conclude that the remainder of m when divided by 3 can be either 0, 1, or 2.
  • #1
mtayab1994
584
0

Homework Statement



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

The Attempt at a Solution



[tex]x^{2}+y^{2}=6+2xy+3x\Longleftrightarrow x^{2}-2xy-3x+y^{2}=6\Longleftrightarrow x^{2}+x(-2y-3)+y^{2}=6[/tex]

Any further help to find the answer??
 
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  • #2
You might try looking at ##x^2 - 2xy+y^2=6+3x##
 
  • #3
Joffan said:
You might try looking at ##x^2 - 2xy+y^2=6+3x##

That's [tex](x-y)^{2}-3x=6[/tex]
 
  • #4
mtayab1994 said:
That's [tex](x-y)^{2}-3x=6[/tex]
Or ##(x-y)^2=3(x+2)## - which should be more interesting.
 
  • #5
Joffan said:
Or ##(x-y)^2=3(x+2)## - which should be more interesting.

Can we use substitution and say that x+2=n?
 
  • #6
mtayab1994 said:
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##?
 
  • #7
Joffan said:
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.
 
  • #8
mtayab1994 said:
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##?
 
  • #9
Joffan said:
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.
 
  • #10
mtayab1994 said:
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##?
 
  • #11
Joffan said:
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.
 
  • #12
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.
 
  • #13
mtayab1994 said:
That means that m^2 is divisible by 3
Yes... what does that mean for ##m##?
mtayab1994 said:
... hence divisible by all multiples of 3.
What? No. For example, 36 is divisible by 3 but not by 15.
 
  • #14
vela said:
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?
 
  • #15
Yes, but it doesn't follow that m2 is divisible by all multiples of 3, which is what you claimed.
 
  • #16
vela said:
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.
 
  • #17
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.
 
  • #18
Joffan said:
My question is, what does ##m^2=3n## (using the new definitions) tell you about ##m##?

Which new definitions m=x-y?
 
  • #19
mtayab1994 said:
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##.
 
  • #20
Joffan said:
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?
 
  • #21
mtayab1994 said:
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?
 
  • #22
What is the value of ##3n## mod 3?
 
Last edited:
  • #23
Joffan said:
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.
 

1. How do you determine if an equation has integer solutions?

The first step is to check if the equation is in the form of a linear Diophantine equation, where all variables are raised to the power of 1. If so, you can use the Extended Euclidean Algorithm to find the greatest common divisor (GCD) of the coefficients. If the GCD divides the constant term, then there are integer solutions. If not, then there are no integer solutions.

2. Can all equations be solved using number theory?

No, not all equations can be solved using number theory. In fact, there is no general algorithm to solve all Diophantine equations. Some equations may require more advanced techniques or may not have any integer solutions at all.

3. How can you find the smallest positive integer solution to an equation?

The simplest method is to use the Extended Euclidean Algorithm to find the GCD of the coefficients, and then use this GCD to find the smallest positive solution. Alternatively, you can use other techniques such as modular arithmetic or linear congruences to find the smallest positive solution.

4. Can negative integers be considered as solutions to a Diophantine equation?

Yes, negative integers can be considered as solutions to a Diophantine equation. In fact, equations with negative solutions are often easier to solve than those without. However, it is important to specify whether you are looking for only positive solutions or all integer solutions.

5. What is the significance of integer solutions in number theory?

Integer solutions play a crucial role in number theory, as they can provide insights into the properties of numbers and their relationships. They can also be used to find patterns and make conjectures about more complex equations. Additionally, integer solutions have practical applications in fields such as cryptography and coding theory.

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