How Many Solutions Satisfy This Diophantine Equation?

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In summary, the purpose of finding the maximum value of (x+y) and minimum value of x is to determine the largest and smallest possible values of the given expression, which can be useful in mathematical and scientific applications. To find these values, one can take the derivatives of the expression and solve for the coordinates of the maximum and minimum points. It is possible for the maximum value and minimum value to be equal, and there is not a specific range for the values of x and y. These calculations can be applied in various scientific fields, such as engineering, physics, and economics.
  • #1
Albert1
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$x,y\in N$
$\dfrac {1}{x}+\dfrac{1}{y}=\dfrac{1}{1987}---(1)$
find :$max(x+y)$ and $min(x)$
How many solutions of (x,y) will satisfy (1) ?
 
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  • #2
I will answer your second question first : If $$\frac1x + \frac1y = \frac1{n}$$ For some nonzero positive integer $n$ then it's clear by rearranging that $n(x + y) = xy$. If $(x, y) = k$, i.e., $x = a \cdot k$ and $y = b \cdot k$ where $(a, b) = 1$, then $n \cdot (a + b) = k \cdot ab$. Thus, $a | n$. But then $n/a \cdot ( a + b) = k \cdot b$, and $b$ must divide $n/a$ (which is an integer). Thus, the number of solutions $(a, b)$ are

$$\sum_{a | n} \sum_{b | n/a} 1 = \tau(n)$$

Where $\tau(n)$ is the number of divisors of $n$. In this case, however $1987$ is a prime, thus $\tau(1987) = 2$. This makes answering the former question a lot easier : one clear solution is $(3974, 3974)$ as $1/2 + 1/2 = 1$. Another solution is $(1988, 3950156)$, coming from a little nonobvious identity $(a + 1)^{-1} + \left ( a \cdot (a + 1) \right )^{-1} = a^{-1}$. By the calculations above, these are all the possible solutions. It is thus clear that $\text{max}(x + y) = 1988 + 3950156 = 3952144$ and $\text{min}(x) = 1988$.
 
  • #3
A terribly elementary solution inspired by http://mathhelpboards.com/challenge-questions-puzzles-28/find-ab-bc-ca-11916.html#post56296 :

$$\frac1{x} + \frac1{y} = \frac1{n} \Rightarrow n( x+ y) = xy \Rightarrow xy - n(x + y) + n^2 = n^2 \Rightarrow (x - n)(y - n) = n^2$$

In this case, $n = 1987$ is a prime, so WLOG either $(x - n, y- n) = (n^2, 1)$ or $(n, n)$. If the former, then $x = n^2 + n = 3950156$ and $y = n + 1 = 1988$ and if otherwise then $x = y = 2n = 3974$. Thus, $\max(x + y) = 3952144$ and $\min(x) = 1988$
 
  • #4
mathbalarka said:
A terribly elementary solution inspired by http://mathhelpboards.com/challenge-questions-puzzles-28/find-ab-bc-ca-11916.html#post56296 :

$$\frac1{x} + \frac1{y} = \frac1{n} \Rightarrow n( x+ y) = xy \Rightarrow xy - n(x + y) + n^2 = n^2 \Rightarrow (x - n)(y - n) = n^2$$

In this case, $n = 1987$ is a prime, so WLOG either $(x - n, y- n) = (n^2, 1)$ or $(n, n)$. If the former, then $x = n^2 + n = 3950156$ and $y = n + 1 = 1988$ and if otherwise then $x = y = 2n = 3974$. Thus, $\max(x + y) = 3952144$ and $\min(x) = 1988$
perfect !
 
  • #5


As a scientist, my response to this content would be as follows:

To find the maximum and minimum values of x+y and x, we need to first solve for the variables in equation (1). Multiplying both sides by xy, we get:

xy * (1/x) + xy * (1/y) = xy * (1/1987)
y + x = 1987

Now, to find the maximum value of x+y, we can use the fact that the sum of two numbers is maximum when they are equal. Therefore, the maximum value of x+y would be 1987.

To find the minimum value of x, we can use the fact that the minimum value of 1/x occurs when x is maximum. Since x can only take positive integer values, the minimum value of x would be 1.

To find the solutions of (x,y) that satisfy equation (1), we can use the fact that x and y must be positive integers. Therefore, the solutions would be all pairs of positive integers (x,y) that satisfy the equation y + x = 1987. There are multiple solutions for this, such as (1,1986), (2,1985), (3,1984), etc.

In conclusion, the maximum value of x+y is 1987 and the minimum value of x is 1. There are multiple solutions for (x,y) that satisfy equation (1).
 

FAQ: How Many Solutions Satisfy This Diophantine Equation?

What is the purpose of finding the maximum value of (x+y) and minimum value of x?

The purpose of finding the maximum value of (x+y) and minimum value of x is to determine the largest and smallest possible values of the given expression. This can be useful in various mathematical and scientific applications, such as optimization problems and data analysis.

How do you find the maximum value of (x+y) and minimum value of x?

To find the maximum value of (x+y), you can take the derivative of the expression with respect to both x and y, set them equal to 0, and solve for x and y. The resulting values will give you the coordinates of the maximum point. To find the minimum value of x, you can take the derivative of the expression with respect to x, set it equal to 0, and solve for x. The resulting value will give you the x-coordinate of the minimum point.

Can the maximum value of (x+y) and minimum value of x be equal?

Yes, it is possible for the maximum value of (x+y) and minimum value of x to be equal. This would occur when the expression is constant, meaning it does not change with different values of x and y. In this case, the maximum value and minimum value will both be the same constant value.

Is there a specific range for the values of x and y when finding the maximum value of (x+y) and minimum value of x?

No, there is not a specific range for the values of x and y when finding the maximum value of (x+y) and minimum value of x. The range will depend on the given expression and the constraints of the problem. It is important to carefully consider the problem and its constraints when finding the maximum and minimum values.

How can finding the maximum value of (x+y) and minimum value of x be applied in the scientific field?

Finding the maximum value of (x+y) and minimum value of x can be applied in various scientific fields, such as engineering, physics, and economics. For example, in engineering, these values can be used to optimize the design and performance of a system. In physics, they can be used to analyze and predict the behavior of a physical phenomenon. In economics, they can be used to determine the most profitable values for certain variables.

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