How many factors 2 2 the number 3x+1 3^{x} + 1 has given that x x is an even number?

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

32m=(1+2)2m=n=02m(2mn)2n=1+n=12m(2mn)2n3^{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

n=12m(2mn)2n \sum_{n=1}^{2m} \begin{pmatrix} 2m\\ n \end{pmatrix} 2^{n}

is an even number, since the factor 22 happens to appear in all of the terms (and this is only possible because 0<n<2m+1 0 < n < 2m + 1).

Now, this is 3x3^{x}. Adding the 11 we get

32m+1=2+n=12m(2mn)2n=2×[1+n=12m(2mn)2n1] 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+n=12m(2mn)2n1 1 + \sum_{n=1}^{2m} \begin{pmatrix} 2m\\ n \end{pmatrix} 2^{n-1}

is an odd number, since

(2m1) \begin{pmatrix} 2m\\ 1 \end{pmatrix}

is even (for n=1n = 1, where 2n1=12^{n-1} = 1) and for all the other terms 2n12^{n-1} is even. Therefore

1+n=12m(2mn)2n1 1 + \sum_{n=1}^{2m} \begin{pmatrix} 2m\\ n \end{pmatrix} 2^{n-1}

has no factors 22 and

2×[1+n=12m(2mn)2n1] 2 \times \left [ 1 + \sum_{n=1}^{2m} \begin{pmatrix} 2m\\ n \end{pmatrix} 2^{n-1} \right ]

has all the factors 22 in evidence. Then 3x+13^{x} + 1 for an even xx has only one factor 22.

Note by Lucas Tell Marchi
4 years, 4 months ago

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Great!

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

Calvin Lin Staff - 4 years, 4 months ago

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Haven't found it yet :P

Lucas Tell Marchi - 4 years, 4 months ago

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Hint: 9=1+8 9 = 1 + 8

Calvin Lin Staff - 4 years, 4 months ago

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@Calvin Lin (1+8)x=4a+1            (1+8)x+1=4a+2=2(2a+1)(1+8)^{x} = 4a + 1 \;\;\; \Rightarrow \;\;\; (1+8)^{x} + 1 = 4a + 2 = 2(2a+1)

for some natural aa

Lucas Tell Marchi - 4 years, 4 months ago

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@Lucas Tell Marchi Perfecto! Great job!

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

Calvin Lin Staff - 4 years, 4 months ago

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