All Questions
Tagged with polynomials elementary-number-theory
668
questions
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Prove that the sequence $(a_n)_{n\ge 1}$ is unbounded
Let $f(x)$ be a non-constant polynomial with integer coefficients. For any $n\in\mathbb{N}$, let $a_n$ be the remainder when $f(3^n)$ is divided by $n$. Prove that the sequence $(a_n)_{n\ge 1}$ is ...
- 1,225
2
votes
1
answer
84
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On a weak polynomial version of Erdős conjecture
This question is motivated by Erdős conjecture on arithmetic progressions. It is a weaker conjecture than Erdős' one.
Erdős conjecture on arithmetic progressions states, "If $A$ is a large set, ...
- 20.7k
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68
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Trouble understanding a number theory problem related to polynomials
I am following, titu's book on number theory, the level of examples is escalating at a very quick rate which caused me trouble understanding them.
The problem is as follows:
Let $f$ be a polynomial ...
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Why does a polynomial over a composite modulus have more than $d$ roots, but over a prime modulus it has at most $d$ roots? [duplicate]
Is it true that when considering a degree $d$ polynomial $p(x)$ in a composite modulus $q$, it has more than $d$ roots (i.e., more than $d$ solutions to $p(x) \equiv 0 \pmod q$), and that when we ...
- 3,019
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1
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60
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Find all possible values of $\alpha$
Suppose $n ��� 0$ is an integer and all the roots of $x^3 + αx + 4 − (2 × 2016^n) = 0$ are integers.
Find all possible values of α.
This question is from INMO 2017. The solution goes this way . . .
Let ...
- 2,646
-1
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1
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45
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Existence of Positive Integer Solution for a Polynomial Equation [closed]
Let $f(x)$ be a polynomial of degree n, and it is known that $f(x)$ has no positive integers as its roots. In other words, there are no positive integer values of x for which $f(x)=0$. Consider the ...
7
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For which positive integers $m$ and $n$ do $x^m-x$ and $x^n-x$ being integers imply that $x$ is an integer
This is inspired by
Prove that $x$ is an integer if $x^4-x$ and $x^3-x$ are integers.
For which positive integers $m$ and $n$
do $x^m-x$ and $x^n-x$
being integers
imply that
$x$ is an integer.
$x$ is ...
- 108k
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votes
1
answer
80
views
Parametric solution of quartic diophantine equation in three variables
How can I handle the quartic diophantine equation in three variables $x$, $y$ and $z$ $$x^4-x^2=y^2-z^2$$ in general, i.e, does exists a (three-variable) parametric solution?
What I've tried is ...
3
votes
2
answers
421
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Polynomial system of equations over integers
I want to solve the system of equations:
$$\begin{cases}
x^4+4y^3+6x^2+4y = -137 \\
y^4+4x^3+6y^2+4x = 472
\end{cases}
$$
$x, y \in \Bbb{Z}$.
It most definitely amounts to messing around with algebra ...
- 1,488
1
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1
answer
106
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$f(x)$ has a factor $(x-r)$, if and only if $f(r) = 0$ - why can't $f(r)$ be a factor of $(x-r)$?
It is clear from the polynomial remainder theorem that $(x-r)$ is a factor of $f(x)$ if $f(r) = 0$
Is there no function $f(r)$ which is a factor of $(x-r)$? does this only happen when $f(r) = 0$?
$f(x)...
1
vote
1
answer
86
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Find all integer polynomials $f(x)$ such that $f(n)\mid 2^n-1$ for all $n\in\mathbb{N}^+$. [duplicate]
Find all integer polynomials $f(x)$ such that $f(n)\mid 2^n-1$ for all $n\in\mathbb{N}^+$.
So far, I have tried to plug in values of $n$, and see where that takes me. For example, plugging in $n=1$ ...
user1200609
1
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0
answers
53
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A Theorem Regarding the Factorability of a Quadratic [duplicate]
After noticing some patterns in delta epsilon proofs, I was able to prove a theorem regarding the factorability of quadratics. My question is, if this proof is correct, has it been documented before? ...
- 142
2
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4
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531
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Find the least prime $p$, such that for some positive integers $a$, $b$, $c$, $d$, $(ab + a - b)(bc + b - c)(cd + c - d)(da + d - a) = 2023^2p.$
Problem
Find the least prime $p$, such that for some positive integers $a$, $b$, $c$, $d$, $(ab + a - b)(bc + b - c)(cd + c - d)(da + d - a) = 2023^2p.$
This problem is from the algebra round of a ...
- 167
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1
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314
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Find all integers $n$ such that $(n - 1)^2 + 3$ divides $n^3 + 2023$.
Problem: Find all integers $n$ such that $(n - 1)^2 + 3$ divides $n^3 + 2023$.
My Work:
$(n - 1)^2 + 3 = n^2 - 2n + 4$, which is always greater than 0 for all integers n.
Therefore, if $n^2 - 2n + 4$ ...
- 167
6
votes
1
answer
161
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divisibility of the numerator of a alternating harmonic series
Let $p = 6n+1$ be a prime number. Prove that the numerator of the rational number
$$
\sum_{k=1}^{4n}\frac{(-1)^{k-1}}{k}
$$
is divisible by $p$.
What I've tried: Let
\begin{align}
g(x) &= (x+1)...
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