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LSpice
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Raising positive integer to $c,\in\mathbb$c\in\mathbb{R}-\mathbb{N}$ rarely gives an integer!

Problem: Let $c>0$ be a real number, and suppose that for every positive integer $n$, at least one percent of the numbers $1^c,2^c,3^c,\ldots,n^c$$1^c,2^c,3^c,\dotsc,n^c$ are integers. Prove that $c$ is an integer.

My progress: At first we will deal with the case when $c$ is a rational number. Suppose $c=\frac{a}{b}$. It indeed suffices to prove the statement for rationals of the form $\frac{1}{a}$. Observe that there are $\lfloor{M^{\frac{1}{a}}}\rfloor$ integers of the form $n^{\frac{1}{a}}$ between $1$ and $M$. So the percentage of integers of the form $n^{\frac{1}{a}}$ among the first $M$ integers is $$\frac{\lfloor{M^{\frac{1}{a}}}\rfloor}{M}\times 100\le \frac{M^{\frac{1}{a}}}{M}\times 100=\frac{100}{M^{1-\frac{1}{a}}}$$ which will be less than 1 for sufficiently large $M$.

But I am unable to prove the problem for any real $c$. I tried approximating reals with a sequence of rational numbers, but it didn't work well.

I was recently working on an open problem of similar kind, and I stumbled upon this sub-problem. How to solve this one  (preferably not requiring too much heavy tool)? Thanks.

Raising positive integer to $c,\in\mathbb{R}-\mathbb{N}$ rarely gives an integer!

Problem: Let $c>0$ be a real number, and suppose that for every positive integer $n$, at least one percent of the numbers $1^c,2^c,3^c,\ldots,n^c$ are integers. Prove that $c$ is an integer.

My progress: At first we will deal with the case when $c$ is a rational number. Suppose $c=\frac{a}{b}$. It indeed suffices to prove the statement for rationals of the form $\frac{1}{a}$. Observe that there are $\lfloor{M^{\frac{1}{a}}}\rfloor$ integers of the form $n^{\frac{1}{a}}$ between $1$ and $M$. So the percentage of integers of the form $n^{\frac{1}{a}}$ among the first $M$ integers is $$\frac{\lfloor{M^{\frac{1}{a}}}\rfloor}{M}\times 100\le \frac{M^{\frac{1}{a}}}{M}\times 100=\frac{100}{M^{1-\frac{1}{a}}}$$ which will be less than 1 for sufficiently large $M$.

But I am unable to prove the problem for any real $c$. I tried approximating reals with a sequence of rational numbers, but it didn't work well.

I was recently working on an open problem of similar kind, and I stumbled upon this sub-problem. How to solve this one(preferably not requiring too much heavy tool)? Thanks.

Raising positive integer to $c\in\mathbb{R}-\mathbb{N}$ rarely gives an integer!

Problem: Let $c>0$ be a real number, and suppose that for every positive integer $n$, at least one percent of the numbers $1^c,2^c,3^c,\dotsc,n^c$ are integers. Prove that $c$ is an integer.

My progress: At first we will deal with the case when $c$ is a rational number. Suppose $c=\frac{a}{b}$. It indeed suffices to prove the statement for rationals of the form $\frac{1}{a}$. Observe that there are $\lfloor{M^{\frac{1}{a}}}\rfloor$ integers of the form $n^{\frac{1}{a}}$ between $1$ and $M$. So the percentage of integers of the form $n^{\frac{1}{a}}$ among the first $M$ integers is $$\frac{\lfloor{M^{\frac{1}{a}}}\rfloor}{M}\times 100\le \frac{M^{\frac{1}{a}}}{M}\times 100=\frac{100}{M^{1-\frac{1}{a}}}$$ which will be less than 1 for sufficiently large $M$.

But I am unable to prove the problem for any real $c$. I tried approximating reals with a sequence of rational numbers, but it didn't work well.

I was recently working on an open problem of similar kind, and I stumbled upon this sub-problem. How to solve this one  (preferably not requiring too much heavy tool)?

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user141127
user141127
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Raising positive integer to $c,\in\mathbb{R}-\mathbb{N}$ rarely gives an integer!

Problem: Let $c>0$ be a real number, and suppose that for every positive integer $n$, at least one percent of the numbers $1^c,2^c,3^c,\ldots,n^c$ are integers. Prove that $c$ is an integer.

My progress: At first we will deal with the case when $c$ is a rational number. Suppose $c=\frac{a}{b}$. It indeed suffices to prove the statement for rationals of the form $\frac{1}{a}$. Observe that there are $\lfloor{M^{\frac{1}{a}}}\rfloor$ integers of the form $n^{\frac{1}{a}}$ between $1$ and $M$. So the percentage of integers of the form $n^{\frac{1}{a}}$ among the first $M$ integers is $$\frac{\lfloor{M^{\frac{1}{a}}}\rfloor}{M}\times 100\le \frac{M^{\frac{1}{a}}}{M}\times 100=\frac{100}{M^{1-\frac{1}{a}}}$$ which will be less than 1 for sufficiently large $M$.

But I am unable to prove the problem for any real $c$. I tried approximating reals with a sequence of rational numbers, but it didn't work well.

I was recently working on an open problem of similar kind, and I stumbled upon this sub-problem. How to solve this one(preferably not requiring too much heavy tool)? Thanks.

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