Numbers with a quadratic property

In summary: So the smallest such pair is $(22,18)$ and the next smallest is $(2180,1780)$.In summary, the conversation discusses the challenge of finding pairs of positive integers $x$ and $y$ that satisfy the equation $2x^2+x = 3y^2+y$. The key to solving this problem is realizing that $x-y$ is a perfect square and using this to find an equation for $x$ in terms of $u$, where $u$ must satisfy a condition to ensure that $x$ is an integer. By using continued fraction convergents, an infinite sequence of positive integer solutions for the equation is found, with the smallest being $(22,18)$ and the next smallest being
  • #1
Opalg
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A recent https://mathhelpboards.com/potw-secondary-school-high-school-students-35/problem-week-411-apr-5th-2020-a-27196.html#post119308 asked about properties of a pair of positive integers $x$, $y$ such that $2x^2+x = 3y^2+y$. But it is not obvious that any such pairs exist. So the challenge is, are there any such pairs of positive integers? If so, what is the smallest such pair? After that, what is the next smallest pair?
 
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  • #2
Opalg said:
A recent https://mathhelpboards.com/potw-secondary-school-high-school-students-35/problem-week-411-apr-5th-2020-a-27196.html#post119308 asked about properties of a pair of positive integers $x$, $y$ such that $2x^2+x = 3y^2+y$. But it is not obvious that any such pairs exist. So the challenge is, are there any such pairs of positive integers? If so, what is the smallest such pair? After that, what is the next smallest pair?
Hint:
[sp]As shown in the https://mathhelpboards.com/potw-secondary-school-high-school-students-35/problem-week-411-apr-5th-2020-a-27196.html#post119308, $x-y$ is a perfect square, say $x-y = u^2$. Then $y = x-u^2$. Use that to find and solve an equation for $x$ is terms of $u$. What condition must $u$ satisfy to ensure that $x$ is an integer?[/sp]
 
  • #3
[sp]
As a matter of fact, I was interested in that very question. The question is about finding points with integer coordinates on a hyperbola, and this is a classical problem on representation by quadratic forms. I wrote something about it here. Sorry, it's in French, but ‶the equations speak for themselves″ :)
[/sp]
 
  • #4
Congratulations to castor28 for his solution. Mine is quite similar:

[sp]Substituting $y=x-u^2$, the equation $2x^2+x = 3y^2+y$ becomes $$2x^2+x = 3(x-u^2)^2 + x - u^2 = 3x^2 - 6u^2x + 3u^4 + x - u^2,$$ $$x^2 - 6u^2x + u^2(3u^2-1) = 0,$$ $$x = 3u^2 \pm\sqrt{u^2(6u^2 + 1)}.$$ We want $x$ to be positive, so take the positive square root to get $x = 3u^2 + uv$, where $v = \sqrt{6u^2+1}$. We also want $v$ to be an integer, so we want integer solutions to the equation $v^2 = 6u^2+1$. That is a https://mathhelpboards.com/showthread.php?2905-The-Pell-Sequence-type equation. To see how to solve it, divide by $u^2$ to get $\left(\frac vu\right)^2 = 6 + \frac1{u^2}$. If $u$ is large, then $\frac1{u^2}$ is very small and so $\frac vu$ will be close to $\sqrt6$. The best rational approximations to $\sqrt6$ come from its continued fraction convergents. The helpful continued fraction calculator here gives this table:

https://www.physicsforums.com/attachments/9693._xfImport

The convergents $\frac vu$ are alternately slightly larger and slightly smaller than $\sqrt6$, corresponding to solutions of $v^2 = 6u^2+1$ and $v^2 = 6u^2-1$. So from alternate rows of that table we get

$$\begin{array}{r|r|r|r}v&u&x=u(3u+v)&y=u(2u+v) \\ \hline 5&2&22&18 \\ 49&20&2180&1780 \\ 485&198&213642&174438 \\ 4801&1960&20934760&17093160\end{array}$$

After that, the numbers increase rapidly, giving an infinite sequence of positive integer solutions of $2x^2+x = 3y^2+y$. (I'm pleased to see that my numbers tally exactly with castor28's!)[/sp]
 

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FAQ: Numbers with a quadratic property

1. What is a number with a quadratic property?

A number with a quadratic property is a number that can be expressed as the product of two equal factors. In other words, it is a number that can be written in the form of x2, where x is a whole number.

2. How do you identify if a number has a quadratic property?

To identify if a number has a quadratic property, you can take its square root. If the square root is a whole number, then the original number has a quadratic property. For example, the number 9 has a quadratic property because its square root is 3, a whole number.

3. What is the significance of numbers with a quadratic property?

Numbers with a quadratic property have several applications in mathematics, including in algebra, geometry, and number theory. They also have real-world applications, such as in physics and engineering.

4. Can all numbers have a quadratic property?

No, not all numbers have a quadratic property. Only numbers that can be expressed as the product of two equal factors have a quadratic property. For example, the number 5 does not have a quadratic property because it cannot be written as x2 for any whole number x.

5. How are numbers with a quadratic property used in equations and formulas?

Numbers with a quadratic property are often used in equations and formulas involving squares and square roots. For example, the Pythagorean theorem, which relates the sides of a right triangle, uses numbers with a quadratic property. They are also used in solving quadratic equations, which have the form ax2 + bx + c = 0.

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