# Squared fractions.

## Homework Statement

Show that $$1/72$$
cannot be written as the sum of the reciprocals
of the squares of two different positive integers.

## The Attempt at a Solution

Available solutions
1/8²-1/24²
1/9²+1/648
Therefore Proven.

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$$\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{72}$$

$$\frac{b^2+a^2}{a^2b^2}=\frac{1}{72}$$

$$72(b^2+a^2)=a^2b^2$$

Solve the equation and write here the solution.

$$\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{72}$$

$$\frac{b^2+a^2}{a^2b^2}=\frac{1}{72}$$

$$72(b^2+a^2)=a^2b^2$$

Solve the equation and write here the solution.
How do i solve that equation?

Move the terms from the left to the right side of the equation:

$$a^2b^2-72b^2-72a^2=0$$

Now factor out b2 or a2 and tell me what you got.

$$a^2=(72b^2)/(b^2-72)$$

$$a^2=(72b^2)/(b^2-72)$$
Ok. Now what "a" is equal to? What can you conclude from the final solution?

Ok. Now what "a" is equal to? What can you conclude from the final solution?
Thanks for your help. But the actual problem is to derive the available solution from the question given.

HallsofIvy
Homework Helper
Yes, that is what he is trying to show you how to do!

However, the problem, as you stated it was

"Show that 1/72 cannot be written as the sum of the reciprocals
of the squares of two different positive integers." (my emphasis)

You can't do that because, as you showed, it is not true.

Yes, that is what he is trying to show you how to do!

However, the problem, as you stated it was

"Show that 1/72 cannot be written as the sum of the reciprocals
of the squares of two different positive integers." (my emphasis)

You can't do that because, as you showed, it is not true.
I got it. Im sorry.

Now for part 2,
How can I write $$1/72$$ with reciprocals of the squares of
three different positve integers.

Last edited:
Mark44
Mentor
I got it. Im sorry.

Now for part 2,
How can I write $$1/72$$ with reciprocals of the squares of
three different positve integers.
Start by writing an equation that expresses this relationship.

Start by writing an equation that expresses this relationship.
okay. $$1/a^2+1/b^2+c^3=1/72$$
By studying the relationship of their factor,
The equation can be translated into :
$$1/x^2+1/(b^2)(x^2)+1/(c^2)(x^2)=1/72$$
Moreover,

$$b^2+c^2+1=x$$