$x^{a} +1$ and $x^{b} +1$ are the factors of$ 1 + x + x^{2} + ... +x^{255}$ for distinct a and b find maximum value of a+b.
I dont know how to get to solve this problem...
I don't have prior knowledge as well to show my work..
help
Find the maximum value of distinct $a$ and $b$, where $x^a +1$ and $x^b +1$ are the factors of $1 + x + x^2 + \dots+x^{255}$
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elementary-number-theory
polynomials
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0How does this relate to measure-theory ? – 2017-01-04
1 Answers
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Hint:
We know that $$1+x+x^2+\cdots +x^n =\frac {x^{n+1}-1}{x-1}$$ Using this we get our expression as $$\frac {x^{256}-1}{x-1}$$ $$=(x+1)(x^2+1)(x^4+1)(x^8+1)(x^{16}+1)(x^{32}+1)(x^{64}+1)(x^{128}+1) $$ Hope you can take it from here.
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0How do you factor $\frac {x^{256}-1}{x-1}$ ? – 2017-01-04
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0$x^{2a} - 1 = (x^a +1)(x^a-1)$ In this case is $a$ is even (a power of $2,$ in fact) So you factor $(x^a-1)$ the same way again, (and again) – 2017-01-04
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0I hope you know $$x^2-1=(x-1)(x+1)$$ $$x^4-1=(x^2+1)(x+1)(x-1) $$ Proceed similarly. – 2017-01-04
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1I still don't understand the question, but now I am confident that I could answer it if I did ;) – 2017-01-04