Tagged Questions

For questions about finding factors of e.g. integers or polynomials

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Theoretical way to prove no positive integer $n$ exists such that $n+3$ and $n^2+3n+3$ are both perfect cubes.

I have to prove that for any positive integer $n$ at least one of $n+3$ and $n^2+3n+3$ is not a perfect cube. Is there a methodical way to solve this problem? I managed to solve it by contradiction, ...
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How to factorise $(x-1)^2 - (x-5)^2$

My attempt: $a = (x-1)$ $c = (x-5)$ $a^2 - c^2$ which is equal to: $$((x-1) - (x-5))((x-1)+(x-5))$$ But the correct answer is : $8(x-3)$ Can you explain, please?
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Solving a characteristic Polynomial of the Hilbert Matrix

I need to find the eigenvalues of the following characteristic polynomial but I can't seem to successfully find the roots of the equation: $P[λ]$ = $λ^5$ - $563/315λ^4$ + $0.3476λ^3$ - $0.0038λ^2$ ...
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Why does factoring eliminate a hole in the limit?

$$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5} = \lim_{x\rightarrow 5} (x+5)$$ I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and ...
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How many positive two-digit integers have exactly 8 positive factors?

I solved this problem by listing all two-digit integers and going through each one. Is there an easy way to solve the problem? How many positive two-digit integers have exactly 8 positive factors?
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Factorization of the polynomial: $m^2+3m^2n^2-30n^2-10$ [closed]

How can we factor the polynomial $m^2+3m^2n^2-30n^2-10$ ?
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Factoring the quintic polynomial $x^5+4x^3+x^2+4=0$

I am trying to factor $$x^5+4x^3+x^2+4=0$$ I've used Ruffini's rule to get $$(x+1)(x^4-x^3+5x^2-4x+4)=0$$ But I don't know what to do next. The solution is $(x+1) (x^2+4) (x^2-x+1) = 0$. I've ...
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A theorem about prime divisors of generalized Fermat numbers?

A theorem of Édouard Lucas related to the Fermat numbers states that : Any prime divisor $p$ of $F_n=2^{2^n}+1$ is of the form $p=k\cdot 2^{n+2}+1$ whenever $n$ is greater than one. Does anyone ...
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Finding 8 co-primes $\le 2^n$

We can find 8 co-prime integers $\le 2^n$ for sufficiently large $n$. I'm looking for asymptotic bounds for the minimum distance away from $2^n$ we have to go before finding 8 co-primes. In other ...
What is the sum of the prime factors of $2^{16}-1$?
I know $2^{10}=1024$ and $2^6=64$, but it seems they are not very helpful in solving this problem. There must be a trick to solve the problem in an easy way. What is the sum of the prime factors ...