Solve Odd Math Puzzle: a^2+b^2=k(ab+1)

  • MHB
  • Thread starter mathpleb
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In summary, the conversation discusses a puzzle involving a formula with variables a, b, and k, where a and b are positive integers. The question is whether k can only be a fraction or a perfect square. The conversation also references a specific problem from a math competition and mentions a proof using a technique called Vieta jumping. The conversation concludes with a comment on the placement of the question in the correct section.
  • #1
mathpleb
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This isn't for me, it's for a friend. I'm still teaching myself stuff before Trigonometry.

Anyways, he has a puzzle. a^2+b^2=k(ab+1).

A and B are given as positive integers.

Q: "Prove that K can only take on the value of fractions or squares."
 
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  • #2
Isn't this clear, since $k = \frac{a^2 + b^2}{ab + 1}$ is always a fraction when $a$ and $b$ are positive integers? Or perhaps I am overlooking something?
Also, could it be you posted this accidentally in the "differential equations" section?

EDIT: I see it was already moved to the right section, thank you (Smile)
 
  • #3
I suspect this is supposed to be problem #6 at IMO 1988, in which case it should read:

Let $\displaystyle k={{a^2+b^2}\over{1+ab}}.
$ Show that if $k$ is an integer then $k$ is a perfect square.​

The proof (along with its history) is given in the Wikipedia article on Vieta jumping.
 

FAQ: Solve Odd Math Puzzle: a^2+b^2=k(ab+1)

What does a^2+b^2=k(ab+1) mean?

This equation is known as an odd math puzzle, where a and b are variables and k is a constant. It is asking for the values of a and b that make the equation true.

How do I solve this equation?

To solve this equation, you can use algebraic techniques such as factoring, substitution, and the quadratic formula.

Are there any restrictions on the values of a, b, and k?

Yes, there are restrictions. The variables a and b must be positive integers, while k can be any positive or negative integer.

Can this equation have more than one solution?

Yes, this equation can have multiple solutions. In fact, there are an infinite number of solutions for a and b that make the equation true.

How is this type of puzzle used in the field of science?

This type of puzzle is often used in mathematics and physics to test problem-solving skills and critical thinking. It also has applications in computer science and engineering.

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