Prove a sum is not the fifth power of any integer

In summary, to prove that a sum is not the fifth power of any integer, one can use the method of contradiction and assume that it is the fifth power of an integer, leading to a contradiction. An example of such a sum is 14 + 27 = 41, which cannot be expressed as the fifth power of any integer. Proving this has significance in understanding number properties and solving complex mathematical problems. Other methods such as modular arithmetic, Fermat's last theorem, and the fundamental theorem of arithmetic can also be used to prove this. While there are some sums that can be the fifth power of an integer, they are rare.
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anemone
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Suppose $X$ is a number of the form $\displaystyle X=\sum_{k=1}^{60} \epsilon_k \cdot k^{k^k}$, where each $\epsilon_k$ is either 1 or -1.

Prove that $X$ is not the fifth power of any integer.
 
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Solution of other:

WLOG, we may assume that $\epsilon_{60}=1$. Let $\large m=60^{\dfrac{1}{5}({60^{60})}}$.

We show that if $\epsilon_{59}=-1$, then $(m-1)^5<X<m^5$---(1) and

if $\epsilon_{59}=1$, then $m^5<X<(m+1)^5$---(2)

From (1), we note that $60^{60}=(59+1)^{60}>59^{60}+60(59)^{59}>2\cdot 59^{60}$---(3)

so that using (3)

$\large m^3=60^{\dfrac{3}{5}({60^{60})}}>60^{\dfrac{6}{5}({59^{60})}}>59^{({59^{60})}}>59^{({59^{59+1})}}=59\cdot59^{59^{59}}$----(4)

Then

$\begin{align*}(m-1)^5&=m^5-5m^3(m-1)-5m(2m-1)-1\\&<m^5-5m^3(m-2)\\&<m^5-5m^3\\&<m^5-m^3\end{align*}$

and from (4) we have

$\begin{align*}m^5-m^3&<m^5-59\cdot59^{59^{59}}\\&<60^{60^{60}}+\sum_{k=1}^{59} (-1)k^{k^k}\\&\le X\\&<m^5-59^{59^{59}}+58\cdot 58^{58^{58}}\\&<m^5\end{align*}$

A similar argument proves (2) and we're done.
 

FAQ: Prove a sum is not the fifth power of any integer

How can you prove that a sum is not the fifth power of any integer?

To prove that a sum is not the fifth power of any integer, we can use the method of contradiction. We assume that the sum is the fifth power of an integer and then show that it leads to a contradiction, thus proving that our assumption was wrong.

Can you provide an example of a sum that is not the fifth power of any integer?

One example is the sum 14 + 27 = 41, which is not the fifth power of any integer. This can be proven by showing that no integer raised to the fifth power can result in 41.

What is the significance of proving that a sum is not the fifth power of any integer?

Proving that a sum is not the fifth power of any integer helps us understand the properties of numbers and how they can be manipulated. It also allows us to solve more complex mathematical problems that involve sums and powers.

Are there any other methods to prove that a sum is not the fifth power of any integer?

Yes, there are other methods such as using modular arithmetic, Fermat's last theorem, and the fundamental theorem of arithmetic. These methods may be more complex but can also provide a stronger proof.

Can a sum ever be the fifth power of any integer?

Yes, there are certain sums that can be the fifth power of an integer. For example, 1 + 1 = 2, which is the fifth power of 2. However, these cases are rare and most sums are not the fifth power of any integer.

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