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Irreducibility of family of polynomials

Consider the following family of polynomials over $\mathbb{Q}$: $$f_n = x^n - x^{n-1} - \dots - 1$$ Notice that these polynomials satisfy the recurrence $$f_{n+1} = x f_n - 1$$ I would like to ...
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Factorisation in $\mathbb{N}[X]$?

Do we know an efficient algorithm to factorise in $\mathbb{N}[X]$ ? One way to do factorisation in $\mathbb{N}[X]$ is to use an algorithm to factorise in $\mathbb{Z}[X]$ and to combine some factor to ...
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Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...
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More generalized RSA construction

Is there a way to construct RSA type cryptosystem over general number rings? Can Number Field Sieve technique be applied here?
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Solution of $\left(\Gamma(x+c)/ \Gamma(x+d)\right)y^d/y^c = {\rm const}$

When I try to solve $F(x,y)= \Gamma(x+c)/\Gamma(x+d) y^d/y^c = {\rm const}$, I find that $y = p x + q$ satisfies the above equation, whith specific $p$ and $q$ constants for the given constants $c$ ...
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Reference Request on the existence of $k$ satisfying $P(\Phi_d(2))^k \gt \Phi_d(2)$ for all $d$

I am working my way through the literature regarding the following conjecture: There is a positive integer $k$ such that for all positive integers $d$, $$P(\Phi_d(2))^k \gt \Phi_d(2).$$ I am ...
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Expressing Numbers with a Minimal Sum in Powers of 2 [closed]

The first 64 bits of pi are: 11.00100100001111110110101010001000100001011010001100001000110100 Computer multiplication can be sped up by looking for patterns and ...
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Does Coppersmith's method always finds non-trivial factor of integers of the form $n=a(2^k b+1)$ assuming $1 < a<2^k b +1$ and $b < n^{1/4-0.05}$?

Got an argument and numeric evidence that pari's implementation of Coppersmith's method finds non trivial factor of integers of certain form under some assumptions very efficiently. Three $5000$ bit ...
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Finding integer representation as difference of two triangular numbers

Since $n = \frac{n(n+1)}{2}-\frac{n(n-1)}{2}$, every natural number can be represented as the difference of two triangular numbers: $n = \frac{a(a+1)}{2}-\frac{b(b-1)}{2}$. Finding such a ...
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A Different 2-factor in a graph

We know that a k-factor of G is a k-regular spanning subgraph of G. And if G is 4-regular (or 2k-regular), it can be partitioned into 2 (k) edge-disjoint 2-factors (Petersen 1891). My question is in ...
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Why going to number fields in number field sieve help beat quadratic sieve?

To factor an $n$ bit integer number field sieve roughly takes $$e^{c{(\ln\ln n)^{\frac23}}({\ln n})^{\frac13}}$$ time while quadratic sieve takes $$e^{c{(\ln\ln n)^{\frac12}}({\ln n})^{\frac12}}$$ ...
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Simple decomposition of $K_{2n}-I$ into hamiltonian cycles

http://mathworld.wolfram.com/HamiltonDecomposition.html In the 1890s, Walecki showed that complete graphs K_n admit a Hamilton decomposition for odd n, and decompositions into Hamiltonian cycles ...
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Dynamics of the distribution of prime factorization types in increasing intervals

I've tagged this as reference request as surely this question must be very well investigated, I just don't know how to look for it. Most likely the perfect answer will be in form of a keyword for ...
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Is this factoring algorithm sufficiently efficient for some integers of special kind?

Basically the question is if $m$ is factored over the integers, can it be relatively efficiently factored over $\mathbb{Z}[\sqrt{n}]$ where $n$ is not factored, but might be of special form? Suppose ...
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How did Cole factor $2^{67}-1$ in 1903

I just heard a This American Life episode which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193707721 \times 761838257287$. There doesn't seem to be a historical ...
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On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$, Part II

I posted this question on MSE two days ago, but did not receive any responses. I have cross-posted it on MO, hoping it gets more attention here and that it is appropriate for this site. A positive ...
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Algorithm for fast factorization of polynomial over $\mathbb Z$ or over $\mathbb F_p$

I want to fast decompose polynomial over ring of integers (original polynomial has integer coefficients and all of factors have integer coefficients) and also over ring of integers modulo prime number....
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$2^n$-1 consisting only of small factors

I've checked the factorization of $2^N - 1$ up through N = 120 for the largest prime factor, and it looks like the largest value of N where $2^N-1$ has a largest prime factor under 2500 is N = 60 (...
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Factorization when a factor is partially known

Let's say that I have a very large number of the order ($10^{250+}$) which is composite. I have been given one of its factor partially to a significant amount of digits (say 75+). Then, how can I ...
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On a reciprocal of Ostrowski theorem on Newton polytopes and factorization

$\newcommand\KK{\mathbb{K}}$Let $\KK$ be any field and $f\in\KK[x_1,\dotsc,x_n]$ be a polynomial. Its support $S_f$ is the set $\{(e_1,\dotsc, e_n) : x_1^{e_1}\dotsb x_n^{e_n}$ has a nonzero ...
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counting irreducible factors

In How hard is it to compute the number of prime factors of a given integer? a question was asked on computing number of prime factors of an integer. Suppose we have a polynomial $f(X)\in \Bbb Z[X]$ ...
Let $f(x),g(x)\in\Bbb Z[x]$ with $deg(f)>deg(g)$. Given an integer $B$, is there any algorithm that runs in $\log^c |B|$ for some fixed $c\in \Bbb R$ to find a $h(x)\in\Bbb Z[x]$ (if one exists) ...