All Questions
Tagged with polynomials elementary-number-theory
666
questions
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Factoring integer polynomials by factoring their values then interpolating
Let $f\in\mathbb Z[x]$ be a non-constant integer polynomial, and let $n$ be a positive integer which is at least $\deg f$. Choose positive integers $a_1<\cdots<a_n$ such that $f(a_i)\neq 0$ for ...
- 1,079
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3
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100
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Does there exists an integer-coefficient polynomial that extracts the highest digit of an integer in base p? [closed]
Given an odd prime $p$, a positive integer $1 \lt n$, and an integer $x \in \mathbb{Z}/p^n\mathbb{Z}$, does there exist an an integer-coefficient polynomial that extracts the highest digit of $x$?
The ...
- 9
3
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Find non-zero polynomials $f(n)$ and $g(n)$ with integer coefficients such that $(n^2+3)f(n)^2+1=g(n)^2$
Find non-zero polynomials $f(n)$ and $g(n)$ with integer coefficients such that $(n^2+3)f(n)^2+1=g(n)^2$.
This problem comes from Pell's equation, which states that for $\forall m\in\mathbb{N}^*$, ...
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-4
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Does this proof of $f(X)\not\equiv0\pmod{n}$ (for this specific $f$) imply there is an integer $k$ such that $f(k)\not\equiv0\pmod{n}$? [closed]
A generalization of the following statement is proven in "Primes is in P":
$(X+1)^n-X^n-1\equiv0\pmod{n}$ iff $n$ is prime.
Their proof of the only if part goes something like: if $q^k\Vert ...
- 133
3
votes
2
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142
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A conjecture about the sum of Faulhaber polynomials mod 2.
Context: In this question I asked about "Finding a polynomial function for alternating $m$ numbers of odds and even numbers in sequences" I noticed that all Faulhaber polynomials are ...
- 6,352
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Finding a polynomial function for alternating m numbers of odds and even numbers in sequences
Context: The function $(-1)^n$ alternates between $1,-1$ because $n$ alternates between odd and even, I was trying to find a function $f(n)$ that will give $m$ positives then $m$ negatives and so on ...
- 6,352
2
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1
answer
67
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Polynomial Divisibility by Factorial Plus One [closed]
Find all polynomials with integer coefficients $f$ such that $f(p) \mid (p-1)! + 1$ for all primes $p$.
Using Wilson’s theorem we find that $f(p)=p;-p$ satisfy the problem. I also thought about using ...
- 637
1
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157
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Prove $a^n+\frac{1}{a^n}$ is rational if $a+\frac{1}{a}$ is rational [duplicate]
Assume
$$a+\frac{1}{a}$$
is a rational number.
Prove that $a^n+\frac{1}{a^n}$ is a rational number.
I can easily prove it using induction. I would like to know if is it possible to prove it without ...
- 392
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2
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170
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Sufficient condition for a polynomial $f$ over $\mathbb Q$ to be linear is $f(\mathbb Q)=\mathbb Q$.
I am reading a book (Polynomial Mappings by W. Narkiewicz) on invariance of polynomial maps where I found the following result along with a proof:
Theorem: Let $f\in\mathbb Q[x]$ be a polynomial such ...
- 1,079
1
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0
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55
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Do polynomials taken modulo integers compose?
Intuitive Idea:
Say we have polynomials modulo positive integers; for instance, $f(x) = 3x + 1\pmod 5$ and $g(x) = 5x^2 - x + 3 \pmod 7$. (Yes, there is some abuse of notation here.) We may "...
- 181
19
votes
2
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756
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Is the smallest root of a polynomial always complex if the coefficients is the sequence of prime numbers?
The smallest root of a polynomial is defined as the root which has the smallest absolute value. Consider the polynomial $a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$. I observed that if $a_n$, is the ...
- 20.8k
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45
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Proof of existence of composite number P(x) if P is a non-constant polynomial [duplicate]
Prove that if $P$ is a polynomial with integer coefficients, which is not a constant polynomial, then there is a natural number $x$ such that $P(x)$ is a composite number.
My first thought was to ...
- 11
1
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69
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Divisibility of sum of powers of $2$ and $3$
I am looking at the question, whether for some positive integers $h,k,l\in\mathbb{N}$ we have that
$$\left(3^{h+l}-2^{2(k+h)+l}\right)\,\big| \left( 2^{2k+l+1}-2^2 3^{l-1}-2^{2k+1}3^l\right).$$
I ...
- 1,271
3
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40
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Calculating $P(x) \bmod x^{k}$ at target value by sampling $P(x)$
Motivation
The motivation behind this question is from a computational mathematics and combinatorics background. It is often convenient to express a problem as a product of polynomials so that the ...
- 294
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0
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24
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Difference between polynomials with integer and non integer coefficiants regarding number of roots of polynomial [duplicate]
There is a prior question to show that polynomial of degree 7 with all integer coefficiants has 7 integer values $ P(x1,x2...x7)=+-1$ cant be expressed as product of two polynomials with integer ...
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