Find All Positive Integer Pairs for (x, y) in sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1

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In summary, the equation <code>sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1</code> can be used to find all positive integer pairs for (x, y). The notation "sqrt[3]" indicates that it is a cube root. This process is significant because it helps us understand relationships between the variables. To solve the equation, we can manipulate it algebraically and use properties of square roots and absolute values. However, there are restrictions on the values of x and y, as they must both be positive integers and the expression inside the square root must be non-negative.
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anemone
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Here is this week's POTW:

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Determine all pairs $(x,\,y)$ of positive integers such that $\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1$.

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No one answered last week's POTW either. However, you can find the official solution below.
WLOG, let $x\ge y$, then we have $7x^2-13xy+7y^2=(x-y+1)^3$.

Now let $x-y=a$ we have

$7a^2+x(x-a)=(a+1)^3 \implies x^2-ax-a^3+4a^2-3a-1=0$.

Now, as $x$ and $y$ are positive integers so the discriminant of the above quadratic in $x$ must be a perfect square.

$4a^3-15a^2+12a+4=(4a+1)(a-2)^2=m^2$ so $4a+1=k^2$ and thus we obtain a family of solution for different values of $k$ for

$x=\dfrac{k^2-1\pm k(k^2-9)}{8}$ and $y=x-\dfrac{k^2-1}{4}=\dfrac{k^2-1\pm k(k^2-9)}{8}-\dfrac{k^2-1}{4}$.
 

FAQ: Find All Positive Integer Pairs for (x, y) in sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1

1. What is the equation for finding all positive integer pairs for (x, y) in the given equation?

The equation is sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.

2. How many positive integer pairs satisfy the given equation?

There are an infinite number of positive integer pairs that satisfy the given equation.

3. How can I find all the positive integer pairs that satisfy the given equation?

To find all the positive integer pairs, you can plug in different values for x and y and solve for the equation. You can also use algebraic manipulation to simplify the equation and find a pattern in the solutions.

4. Can negative integer pairs also satisfy the given equation?

No, the equation only asks for positive integer pairs. However, if you allow negative integers, there are infinitely many solutions.

5. Are there any restrictions on the values of x and y in the given equation?

Yes, x and y must be positive integers. This means they cannot be fractions, decimals, or negative numbers.

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