# Proving 2n Representable When n is: Converse True?

• cragar
In summary, the conversation discusses proving that 2n is representable when n is, and whether the converse is true. The concept of representability is defined as being able to write a positive integer as the sum of 2 integral squares. An attempt at a solution is made by expressing 2n as 2(x^2+y^2) and considering prime factors. The Brahmagupta-Fibonacci identity is mentioned as a possible proof, and specific examples are suggested. The question of the converse being true is addressed, and a potential method for proving it is discussed.
cragar

## Homework Statement

Prove that 2n is representable when n is. Is the converse true?
Representable is when a positive integer can be written
as the sum of 2 integral squares.

## The Attempt at a Solution

so n can be written as $x^2+y^2$
x and y are positive integers
so then $2n=2(x^2+y^2)$
I am not really sure where to go next, maybe i should look a the prime factors.
Just to make sure that i know what converse is,
Would the converse be if 2n is representable so is n.

cragar said:

## Homework Statement

Prove that 2n is representable when n is. Is the converse true?
Representable is when a positive integer can be written
as the sum of 2 integral squares.

## The Attempt at a Solution

so n can be written as $x^2+y^2$
x and y are positive integers
so then $2n=2(x^2+y^2)$
I am not really sure where to go next, maybe i should look a the prime factors.
Just to make sure that i know what converse is,
Would the converse be if 2n is representable so is n.

Brahmagupta–Fibonacci identity. Look it up. And yes, that would be the statement of the converse.

cragar said:

## Homework Statement

Prove that 2n is representable when n is. Is the converse true?
Try some specific examples, like 25 = 32 + 42

or 29 = 22 + 52 .

Just to make sure that i know what converse is,
Would the converse be if 2n is representable so is n.
Yes, that's the converse.

ok thanks for the responses. Knowing my teacher I would need to prove the
Brahmagupta–Fibonacci identity, but I guess I could multiply it out and show that
the left side equaled the right side. I think the converse is true but ill think about how to prove it.

cragar said:
ok thanks for the responses. Knowing my teacher I would need to prove the
Brahmagupta–Fibonacci identity, but I guess I could multiply it out and show that
the left side equaled the right side. I think the converse is true but ill think about how to prove it.

Yes, you should definitely think much harder about the converse. And sure, it's much easier to prove them than to discover they exist.

Last edited:

## What does "2n representable" mean?

"2n representable" means that a number can be expressed as 2 multiplied by another number, or in other words, it is even.

## Is it possible for a number to be 2n representable and not even?

No, if a number is 2n representable, it must be even because it is divisible by 2.

## How can you prove that a number is 2n representable?

A number can be proved to be 2n representable by showing that it can be expressed as 2 multiplied by another number. This can be done through various methods such as division, multiplication, or algebraic manipulation.

## What is the converse of the statement "2n representable"?

The converse of "2n representable" is "even number". This means that if a number is even, it is 2n representable.

## Why is proving 2n representable important?

Proving 2n representable is important because it helps us understand the properties of even numbers and their relationship to other numbers. It also allows us to solve mathematical problems and equations more efficiently.

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