- #1

mathpleb

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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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- 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

- 1

- 0

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."

Mathematics news on Phys.org

- #2

S.G. Janssens

Science Advisor

Education Advisor

- 1,222

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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

MountEvariste

- 87

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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.

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.

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

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

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.

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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