Find positive integers a,b,n that satisfy this expression

In summary: That's it.Thank you so much haruspex and SammyS for breaking it down and guiding me through the problem. It helped me a lot :)
  • #1
sunnybrooke
19
0

Homework Statement


Are there any positive integers n, a and b such that

[tex]96n+88=a^2+b^2[/tex]


Homework Equations


The Attempt at a Solution


It resembles the Pythagorean theorem but I'm not sure how that would help me solve it. I factored the LHS

[tex]2^3((2^2)(3)n+11)=a^2+b^2[/tex]

How do I proceed? Thanks.
 
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  • #2
What can you say about perfect squares modulo 4?
 
  • #3
haruspex said:
What can you say about perfect squares modulo 4?

I'm very bad with the whole modulus concept but I can see that a perfect square divided by 4 will always have a remainder of either 0 or 1.
 
  • #4
sunnybrooke said:
a perfect square divided by 4 will always have a remainder of either 0 or 1.
Right, so what can the sum of two squares be mod 4?
 
  • #5
0,1 or 2, right?
 
  • #6
sunnybrooke said:
0,1 or 2, right?
Right. Now which of those could match the LHS (mod 4)?
 
  • #7
haruspex said:
Right. Now which of those could match the LHS (mod 4)?

96 is divisible by 4 (remainder = 0). 88 is also divisible by 4 (remainder = 0). So sum of remainders (0 + 0) = 0.
 
  • #8
sunnybrooke said:
96 is divisible by 4 (remainder = 0). 88 is also divisible by 4 (remainder = 0). So sum of remainders (0 + 0) = 0.
To get the remainder of zero, means that both a and b are even. Correct ?
 
  • #9
SammyS said:
To get the remainder of zero, means that both a and b are even. Correct ?

Yes, they'd both have to be even. Should I substitute, a = 2u, b = 2v ?
 
  • #10
sunnybrooke said:
Yes, they'd both have to be even. Should I substitute, a = 2u, b = 2v ?
Yes, which allows some cancellation. See if you can then repeat the logic.
 
  • #11
haruspex said:
Yes, which allows some cancellation. See if you can then repeat the logic.

96n + 88 = 4u^2 + 4v^2
24n + 22 = u^2 + v^2
(LHS= 2 mod 4 --> both u and v are odd)

24n + 22 = (2w+1)^2 + (2x+1)^2
6n + 5 = w^2 + w + x^2 + x
(LHS = 3 mod 4 --> ??)

If k^2 = 0 mod 4 then either k = 0 mod 4 or k = 2 mod 4
If k^2 = 1 mod 4 then either k = 1 mod 4 or k = 3 mod 4

Is this correct? And please correct me if my notation is wrong. Thanks.
 
  • #12
sunnybrooke said:
6n + 5 = w^2 + w + x^2 + x
w2+w = w(w+1). What does that tell you about w2+w mod 2?
 
  • #13
haruspex said:
w2+w = w(w+1). What does that tell you about w2+w mod 2?

w(w+1)=odd*even=even --> w mod 2 = 0. same goes for x^2 + x. So the RHS is even but the LHS is odd. therefore it's not possible. Is that it?
 
  • #14
sunnybrooke said:
w(w+1)=odd*even=even --> w mod 2 = 0. same goes for x^2 + x. So the RHS is even but the LHS is odd. therefore it's not possible. Is that it?
That's it.
 
  • #15
Thank you so much haruspex and SammyS for breaking it down and guiding me through the problem. It helped me a lot :)
 

1. What is the purpose of finding positive integers a,b,n that satisfy this expression?

The purpose of this task is to explore and understand relationships between different positive integers and how they can satisfy a given mathematical expression.

2. How do you determine which positive integers will satisfy the expression?

To determine which positive integers will satisfy the expression, you need to manipulate and analyze the given expression using mathematical operations and properties. This process may involve trial and error or using specific techniques such as factoring or substitution.

3. Can there be more than one set of positive integers that satisfy the expression?

Yes, there can be more than one set of positive integers that satisfy the expression. This is because there are often multiple ways to manipulate and solve a mathematical expression, resulting in different sets of solutions.

4. Is there a specific method or formula to find positive integers that satisfy the expression?

There is no specific method or formula to find positive integers that satisfy a given expression. The approach may vary depending on the complexity of the expression and the individual's problem-solving strategies. However, having a strong understanding of mathematical concepts and techniques can help in finding solutions.

5. How can finding positive integers that satisfy this expression be applied in real life?

Finding positive integers that satisfy a given expression can have practical applications in fields such as engineering, computer science, and economics. For example, in engineering, it can help in optimizing designs and predicting outcomes. In computer science, it can be used in coding and algorithms. In economics, it can aid in analyzing data and making predictions.

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