Proving 1/72 Not the Sum of Squares of Two Positive Integers

[icystrike]

Homework Statement

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

Homework Equations

The Attempt at a Solution

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

 

[Дьявол]

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

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

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

Solve the equation and write here the solution.

 

[icystrike]

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

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

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

Solve the equation and write here the solution.

How do i solve that equation?

 

[Дьявол]

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.

 

[icystrike]

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

 

[Дьявол]

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

Ok. Now what "a" is equal to? What can you conclude from the final solution?

 

[icystrike]

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]

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.

 

[icystrike]

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. I am sorry.

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

 

[Mark44]

I got it. I am 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.

 

[icystrike]

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,

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