All Questions
Tagged with factorization reference-request
25 questions
12
votes
5
answers
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Polynomials all of whose roots are rational
I have two questions about the class of integer-coefficient polynomials all of whose roots are rational.
I asked this at MSE, but it attracted little interest (perhaps because it is not interesting!)
...
- 151k
12
votes
2
answers
4k
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How can one construct a sparse null space basis using recursive LU decomposition?
Given an $m$ by $n$ matrix $A$ I'm familiar with the standard method to compute a basis for the null space of $A$ by computing a QR factorization of $A^T$. If $A$ is large and sparse, we can use ...
- 723
9
votes
2
answers
1k
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runs of consecutive non squarefree integers
This question gained no attention at Math SE.
Call a sequence of $k$ consecutive naturals squary if each one of them is divided by a square > 1. The Chinese Remainder theorem trivially guarantees us ...
- 13.4k
9
votes
3
answers
980
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$\omega(p^n - 1)$ as $n \rightarrow \infty$
Although I am also interested in the number of distinct prime factors (not counting
multiplicity), today I use $\omega(m)$ to denote the number of (positive) prime
factors (with multiplicity) of the ...
- 2,158
9
votes
0
answers
275
views
What is known about vector subspaces of polynomial rings closed under factors?
Let $R$ be a commutative ring. Call a nonempty subset $F$ of $R$ a factroid if it is closed under sums and factors. That is:
If $a,b \in F$, then $a+b \in F$, and
If $a,b \in R$ with $a\in R$ ...
- 1,802
7
votes
1
answer
382
views
$\log \log p / \log \log n$, where $p|n$, gets equidistributed in [0,1] (for almost all $n$)
According to Hardy-Ramanujan/Erdős-Kac we know that usually there are $\sim\log\log n$ prime numbers in a factorization. But if you pick up a natural number at random, and you factor it, what is the ...
- 1,384
6
votes
2
answers
804
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Must Mersenne numbers be divisible by arbitrary large primes with exponent one?
Let $M_n$ denote the Mersenne numbers $M_n=2^n-1$.
As $n$ varies, must $M_n$ be divisible by arbitrary large prime $p$
with exponent one, i.e. $p \mid M_n, p^2 \nmid M_n$?
In other words, must the ...
- 25.4k
6
votes
1
answer
1k
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Using the decomposition $641 = 5^4 + 2^4$ to factor $F_5$
The question in the title arises from a problem in Stewart's "Galois Theory, Third Edition" (and possibly elsewhere) which has been bugging me for a few days since reading it:
Problem 19.5 (p. 224) ...
- 5,191
5
votes
1
answer
214
views
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 ...
- 18.4k
5
votes
0
answers
66
views
weak factorization systems (co)generated by an arbitrary class of morphisms
Under what assumptions can one prove existence of weak factorization systems (co)generated by a class of morphisms ?
Are there counterexamples ? I am interested both in assumptions on the class of ...
- 513
5
votes
0
answers
682
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Is integer factorization harder than RSA ($n=pq$) factorization? [closed]
This is a repost. I could not get a precise answer on math.SE and cstheory.SE
Let FACT denote the integer factoring problem: given $n \in \mathbb{N},$ find primes $p_i \in \mathbb{N},$ and integers $...
- 236
4
votes
1
answer
414
views
A Handbook of Matrix Factorizations
I am looking for a good collection of facts regarding the various types of matrix factorizations, something like a "Handbook of Matrix Factorizations" or a very-thorough review paper. I am hoping for ...
- 143
4
votes
1
answer
138
views
Irreducible skew polynomials over an algebraically closed field
Let $\mathbb{F}$ be a field, and denote with $\mathbb{F}[t,\sigma]$ the skew-polynomial ring, where $\sigma$ is an automorphim of $\mathbb{F}$. Recall that the multiplication of this ring is defined ...
4
votes
1
answer
288
views
Is total degree version and $x,y$ degree version of Coppersmith's theorem correct?
The notes here https://web.eecs.umich.edu/~cpeikert/lic13/lec04.pdf have the note 'Small decryption exponent $d$: so far the best known attack recovers $d$ if it is less than $N^{.292}$. This uses a ...
- 13.9k
4
votes
2
answers
199
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Counting integers with k large prime divisors
If $x \ge y \ge 1$ are real numbers and if $k$ is a positive integer, take $\Phi_k(x, y)$ to be the number of integers $\le x$ with exactly $k$ prime factors and no prime factor $\le y$. If $y$ is ...
- 443
4
votes
0
answers
119
views
Adjoining new factors for primes in UFDs
It is well-known that if we pass from a UFD to a new ring where we have factored one of the primes, it does not need to stay a UFD. The classic example is passing from $\mathbb{Z}$ to $\mathbb{Z}[\...
- 18.7k
4
votes
0
answers
216
views
Characterizing atomicity in a commutative domain
In Proposition 1.1 of [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ...
- 10.5k
3
votes
3
answers
382
views
Closed formula for number of ones in a proper factor tree
Edit [2023 Dec 7]: One of my specific wonders, along with that of students, is around when a recursive formula might have – or be expected to have – an explicit or closed formula. What is the ...
- 7,831
3
votes
1
answer
313
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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 ...
- 173
3
votes
1
answer
137
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Subexponential algorithms that apply only one of factoring and discrete logarithm?
Shor (quantum polynomial), Number Field Sieve (subexponential), Pollard rho (square root) all have both factoring and discrete logarithm over $\mathbb F_p^*$ variants.
What are the subexponential ...
- 13.9k
2
votes
1
answer
450
views
Is there a "small $\omega$" number theorem?
In my studies of how primes jump (search this forum for a link), a question has been raised which may have been studied. Can anyone jump-start my literature search with references regarding the ...
- 13k
2
votes
0
answers
140
views
Integers with exactly three factor pairs whose successors are relatively prime
I am interested in the following problem, and will appreciate pointers around how it can be solved – partially or fully – and/or indicators around whether it is even tractable:
Characterize $N \in \...
- 7,831
1
vote
1
answer
178
views
+50
On a probabilistic integer factorization algorithm given bounds for one prime factor
We got a probabilistic integer factorization algorithm and experimental evidence with large
integers given bounds for one factor.
Let $D \ge 2$ be real number and let $p,q$ be primes and $N=pq$.
...
- 25.4k
0
votes
0
answers
65
views
Reference Request: Factorization method for polynomials whose maximum absolute value of coefficient is 1
So, today I came up with a method for factoring polynomials whose coefficients are either $-1$ or $1.$
Let me explain with examples.
Example No. 1. Factorize $P(x)=x^8+x^7+1$
Solution. It is known ...
- 161
0
votes
0
answers
138
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A diophantine equation involving partial sums of exponentials similar than the equation in Fermat's Last Theorem
I'm curious about the following diophantine equation from my invention: I don't know if this is in the literature, I wrote it using creativity in an attempt to write a variant of the equation in ...
