Proving Conjecture: No Real Solutions Greater than 2 for Polynomial Equation

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In summary: Therefore, there are no real solutions greater than 2 for any value of n. In summary, the conjecture that for the polynomial equation -\sum_{n=0}^{\infty}k^{n}=0 there exist no real solutions greater than 2, no matter how large the value of n, is proven to be true using Rouché's theorem. An elementary proof is not known, but it can be shown using a simple example equation.
  • #1
FeDeX_LaTeX
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Hello;

I don't know how to prove this conjecture I've made;

For the polynomial equation

[tex]-\sum_{n=0}^{\infty}k^{n}=0[/tex]

there exist no real solutions greater than 2, no matter how large the value of n.

How do I show that this is true?

If it's a little unclear, what I mean is, for example, if you had the equation

k^30 - k^29 - k^28 ... - k - 1 = 0, k doesn't exceed 2.

So even if you start from 1000, or 100000000000, or any number, you will never find a solution greater than 2.

Thanks.
 
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  • #3
I am assuming your latex equation has something missing, because otherwise the minus sign at the start doesn't have any purpose. So working from your example equation:

[tex]k^n - k^{n-1} - k^{n-2} \cdots - k - 1 = 0[/tex]

Multiply by k:

[tex]k^{n+1} - k^{n} - k^{n-1} \cdots - k^2 - k = 0[/tex]

Subtract the first equation from the second:

[tex]k^{n+1} - 2k^n + 1 = 0[/tex]

[tex]k^n(k-2) + 1 = 0[/tex]

Which has no solution if k > 2
 
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1. How do you prove a conjecture?

There is no one set method for proving a conjecture. It often involves applying mathematical techniques such as induction, contradiction, or direct proof. It also requires a deep understanding of the problem and creative thinking.

2. What if you can't prove a conjecture?

If a conjecture cannot be proven, it is still considered to be an open problem. Mathematicians continue to work on these problems and often make progress towards proving or disproving them over time. Sometimes, a counterexample is found, which shows that the conjecture is not true in all cases.

3. How do you know if a conjecture is true?

A conjecture is a statement that has not yet been proven to be true or false. To determine if a conjecture is true, mathematicians often search for counterexamples or try to find a proof. However, even if a conjecture has been proven to be true, it is always possible that a counterexample may be found in the future.

4. What makes a strong conjecture?

A strong conjecture is one that has been tested extensively and has not been disproven. It is also one that is supported by mathematical evidence, such as numerical data or previous results. A strong conjecture should also be well-defined and not rely on any unproven assumptions.

5. Can a conjecture ever be proven wrong?

Yes, a conjecture can be proven wrong if a counterexample is found or if it is shown to be logically inconsistent. However, even if a conjecture is proven wrong, it can still provide valuable insights and lead to new discoveries in mathematics.

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