First of all, given that \(x\) is a whole even number, let's write \(x = 2m\) for some natural \(m\). Then observe that we are allowed, according to Newton, to write

\[3^{2m} = (1+2)^{2m} = \sum_{n=0}^{2m} \begin{pmatrix} 2m\\ n \end{pmatrix} 2^{n} = 1 + \sum_{n=1}^{2m} \begin{pmatrix} 2m\\ n \end{pmatrix} 2^{n} \]

And notice that

\[ \sum_{n=1}^{2m} \begin{pmatrix} 2m\\ n \end{pmatrix} 2^{n} \]

is an even number, since the factor \(2\) happens to appear in all of the terms (and this is only possible because \( 0 < n < 2m + 1\)).

Now, this is \(3^{x}\). Adding the \(1\) we get

\[ 3^{2m} + 1 = 2 + \sum_{n=1}^{2m} \begin{pmatrix} 2m\\ n \end{pmatrix} 2^{n} = 2 \times \left [ 1 + \sum_{n=1}^{2m} \begin{pmatrix} 2m\\ n \end{pmatrix} 2^{n-1} \right ] \]

But

\[ 1 + \sum_{n=1}^{2m} \begin{pmatrix} 2m\\ n \end{pmatrix} 2^{n-1} \]

is an odd number, since

\[ \begin{pmatrix} 2m\\ 1 \end{pmatrix} \]

is even (for \(n = 1\), where \(2^{n-1} = 1\)) and for all the other terms \(2^{n-1}\) is even. Therefore

\[ 1 + \sum_{n=1}^{2m} \begin{pmatrix} 2m\\ n \end{pmatrix} 2^{n-1} \]

has no factors \(2\) and

\[ 2 \times \left [ 1 + \sum_{n=1}^{2m} \begin{pmatrix} 2m\\ n \end{pmatrix} 2^{n-1} \right ] \]

has all the factors \(2\) in evidence. Then \(3^{x} + 1\) for an even \(x\) has only one factor \(2\).

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestGreat!

There is a one line solution to this problem, with a similar approach to what you did. Can you figure that out?

Log in to reply

Haven't found it yet :P

Log in to reply

Hint:\( 9 = 1 + 8 \)Log in to reply

for some natural \(a\)

Log in to reply

In fact, it is \( 2 ( 1 + 4b ) \).

Log in to reply