Awesome proof

Let $M$ be a nonempty set of positive integers such that $4x$ and $\left[ \sqrt { x } \right]$ both belong to $M$ whenever $x$ does.Prove that $M$ is the set of all natural numbers.

#NumberTheory #MathematicalInduction #Sets #Proofs

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

(1) If we take the floor of square roots repeatedly we will end up with 1. Hence 1 is in M.

(2) 4 and 2 are in M.

(3) All powers of 2 are in M.

(4) Let's say k is not inside, then all numbers from $k^{2}$ to $(k+1)^{2}-1$ are not inside, so all numbers from $k^{4}$ to $(k+1)^{4}-1$ are not inside, ...

Hence for all natural $n$ the numbers from $k^{2^{n}}$ to $(k+1)^{2^{n}}-1$ are not inside.

(5) Choose $n$ large enough such that the ratio between these two values is way greater than 2. Contradiction!

Joel Tan - 6 years, 1 month ago

Joel Tan good one.I have similar thing.For your 5th step I have something..

$f\left( x \right) =\log _{ 2 }{ x }$ defined for ${ R }_{ + }\rightarrow R$ is increasing and hence we have

$\log _{ 2 }{ (n+1) } -\log _{ 2 }{ n } >0$

Since $g\left( x \right) ={ 2 }^{ -x }$ is decreasing,for a sufficiently large positive integer $h$ we will have

${ 2 }^{ -h }<\log _{ 2 }{ (n+1) } -\log _{ 2 }{ n } \quad$

Which implies ${ (n+1) }^{ { 2 }^{ h } }>{ 2n }^{ { 2 }^{ h } }$

Therefore interval $[{ n }^{ { 2 }^{ h } },{ 2n }^{ { 2 }^{ h } }]\quad$ is totally contained in $[{ n }^{ { 2 }^{ h } },{ (n+1) }^{ { 2 }^{ h } })\quad$ .

But $[{ n }^{ { 2 }^{ h } },{ 2n }^{ { 2 }^{ h } }]\quad$ contains a power of $2$ .A contradiction.

shivamani patil - 6 years, 1 month ago

Nihar Mahajan,Sharky Kesa,Satvik any one?

shivamani patil - 6 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$