2
votes
1answer
72 views

what are the “points” of the scheme $\mathbb{Z}_8[x] /(x^2 + 7)$

I noticed modulo 8 the quadratic $x^2 + 7$ is zero for four separate values $x = 1,3,5,7 \in \mathbb{Z}_8$. The number of zeros exceeds the degree. I would like to define the "variety" ...
1
vote
1answer
55 views

How many natural value of n such that $n^5+2n^4+n-1$ is prime number?

From above polynomial, I can only get one value to make it prime. The value, I guess, is only one. For $n=1$, we got: $$(n^5+2n^4+n-1)= 1+2+1-1= 3 \quad\text{(prime)}$$ But, I cannot find the ...
0
votes
0answers
11 views

Polynomial Roots of Bivariates

I've got a few polynomials that I am trying to get some results for (shown below). They come from the characteristic equation of a matrix. I have two variables in the polynomials, $\eta$ (which is ...
0
votes
0answers
41 views

Gosper summable

I'd like to know why the following is NOT gosper summable: $$\sum_{k\in \Bbb{Z}} \frac{p(k)}{\prod_{j=0}^{m-1}(k+a+j)}$$ where $m>0, m\in\Bbb{Z}$ and $p(k)$ is a polynomial of degree $k=m-1$.
5
votes
3answers
193 views

Positive integer solutions of $a^3 + b^3 = c$

Is there any fast way to solve for positive integer solutions of $$a^3 + b^3 = c$$ knowing $c$? My current method is checking if $c - a^3$ is a perfect cube for a range of numbers for $a$, but this ...
4
votes
3answers
50 views

Integer values of polynomial $a^2+ab-b^2$

Playing with the polynomial $f(a,b)=a^2+ab-b^2=d$ for a given $d \in \mathbb{Z}$ I found that it has integer solutions $(a,b) \in \mathbb{Z}$ for the following values of $d$: ...
4
votes
3answers
295 views

Simply put, what are the similarities between integers and polynomials?

The Princeton Companion to Mathematics mentions that polynomials (for instance, ones with rational coefficients) share similarities with integers, thus leading to the idea of a general structure of ...
1
vote
1answer
24 views

integer solutions to bivariate polynomial of second degree

I am trying to determine if there is a way to quickly determine if an equation of the following type $$0 = axy+x-y-A$$ has integer solutions ($a,A$ are integers). If anyone knows how to do this or ...
2
votes
0answers
30 views

Minimum degree of polynomial assuming exactly k prime values

Dirichlet's theorem states that there are infinitely many primes of the form $an+b$ for coprime integers $a$ and $b$. This implies that The minimum degree of a polynomial $f \in \mathbb{Z}[X]$ ...
0
votes
0answers
41 views

What are some algorithms that can be used to test if a number is transcendental or not?

Well according to the definition of transcendental numbers I find that its any number that doesn't have any polynomial equation of any degree with integer coefficients summing up to 0. So ...
1
vote
1answer
39 views

Comparing coefficients in finite field

We start with the wrong proof of the following theorem: $p| \binom{p}{k}$ for a prime $p$ and $0<k<p.$ Proof: $(1+x)^p \equiv 1+x \equiv 1+x^p \pmod{p}$ by Fermat's little theorem. Comparing ...
0
votes
2answers
80 views

Find three numbers given their sum, product and sum of their squares

Given three unknown positive integers. Is it possible to find the three numbers if we are given their Sum->(a+b+c) = X Product-> (abc) = Y Sum of Squares-> (a^2 + b^2 + c^2) = Z
1
vote
2answers
66 views

number theory of coefficients in an infinite sequence of polynomials

EDIT: equivalent formulation by Hurkyl in comments: if $n$ is odd and $p^\nu \parallel n$ and $n > 2k,$ then $$ p^{(\nu + 2 + 2 k - n)} \; | \; \sum_j \left( \begin{array}{c} n \\ 2j \end{array} ...
1
vote
1answer
114 views

Solution of cubic modulo some prime

Let $f(x)=x^3+3x+12$. Now if we have the relation $$f(x)\equiv0\pmod p$$ for some prime $p$, then what are the values of $p$ for which this equation is solvable for $x$? I know that the cubic ...
2
votes
1answer
62 views

Can this be converted into a polynomial equation for $x$?

I came upon this monster equation while fiddling with the area of pentagons: $$\sqrt{(a+b+c-x)(a+b-c+x)(a+c-b+x)(b+c-a+x)}+\sqrt{(c+d+x)(c+d-x)(c-d+x)(d-c+x)}=4T$$ Where $a,b,c,d,e,T$ are known ...
4
votes
3answers
86 views

Why doesn't this calculation work?

I want to find some closed form for $\gcd(x^3+1,3x^2 + 3x + 1)$ but get $7$ which is not always true.
3
votes
3answers
86 views

Rewrite $\frac{1}{1-\sqrt[3]{2}}$ as a polynomial question

I've been looking for a way to rewrite the following fraction as a polynomial equation in $\sqrt[3]{2}$: $$\frac{1}{1-\sqrt[3]{2}}.$$ Now, to rewrite $1/(1-\sqrt{2})$ as a polynomial equation, it is ...
1
vote
0answers
38 views

factorization of even degree polynomial

i am interested in general criteria about even degree polynomial factorization,maybe number theories rule or some other mathematical properties can be helpful,i am talking rules how to factorize ...
2
votes
0answers
56 views

Irreducibility of a family of univariate polynomials

While doing work on another problem, I came across the following family of polynomials: $$P_e(x) := ex^{2e} - x^{2e-1} - x^{2e-2} - \cdots - x + e\in \mathbb{Z}[x],$$ where $e\geq 1$ is an integer. ...
0
votes
0answers
16 views

Can an arbitrary polynomial be determined to be a permutation polynomial for a finite field with q elements?

For a Dickson polynomial, we have the following result: $D_n(x,α)$ is a permutation polynomial for the field with $q$ elements if and only if $n$ is coprime to $q^2−1$. Suppose I am given an ...
2
votes
2answers
139 views

Does Fermat's Little Theorem work on polynomials?

Let $p$ be a prime number. Then if $ f(x) = (1+x)^p$ and $g(x) = (1+x)$, then is $f \equiv g \mod p$? I'm trying to prove that for integers $a > b > 0$ and a prime integer $p$, ${pa\choose b} ...
3
votes
0answers
54 views

polynomials and functions on $\mathbb{Z}/n\mathbb{Z}$

My general question is How is the set of all polynomial functions on $\mathbb{Z}/n\mathbb{Z}$ structured? What is the number of such functions? How, given a function, one can recognize that it is ...
1
vote
1answer
36 views

n-th power over different algebraic structure

It is a classical result that the group $\mathbb{F}_p^{\times}$ is cyclic and that the equation $x^n \equiv a \pmod{p}$ is solvable iff $a^{(p-1)/gcd (p-1,n)} \equiv 1 \pmod{p}$. Also, we know that if ...
8
votes
1answer
351 views

IMO 1979 problem

The question is $$\text{If }\, p, \ q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad \text{Prove that } 1979\mid p.$$ So my solution went like ...
2
votes
0answers
90 views

How powerful is PA+Con(PA)+Con(PA+Con(PA))… etc?

From what i remember from Godel encoding there was alot of freedom in how one chooses to expresses the statement Con(PA), my question is if one can classify all statements, or some subclass of all ...
6
votes
2answers
127 views

Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?

We know that $f(x)=x^4+1$ is a polynomial irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$. My question is: Is it true that if $f(x)$ has a linear factor over ...
0
votes
2answers
75 views

Intersection of splitting fields of two polynomials

Let $f,g\in \mathbb{Z}[x]$, $(f,g)=h\in \mathbb{Z}[x]$ (if this is not true, can we construct $h$ in another way?). Let $K_1,K_2$ be the splitting fields of $f,g$ over $\mathbb{Q}$ respectively. Is ...
1
vote
1answer
33 views

Factors of $x^n+1$ over $\mathbb{Z}[x]$

Is there any equivalent to $x^n-1 = \prod\limits_{d|n} \phi_d$ where $\phi_d$ is the $d$th cyclotomic polynomial but for $x^n+1$? Even better, can we generalize any further?
0
votes
2answers
39 views

$f(x)\in D[x]$ is irreducible if and only if $f(x)$ is irreducible over $F[x]$.

Let $D$ be a principal ideal domain and $F$ be its quotient field. Prove that $f(x)\in D[x]$ is irreducible if and only if $f(x)$ is irreducible over $F[x]$. I only obtained the proof for ...
2
votes
1answer
143 views

Iran Math Olympiad 2012 (perfect power)

Prove that if $t$ is a natural number then there exists a natural number $n > 1$ such that $(n, t) = 1$ and none of the numbers $n + t, n^2 + t, n^3 + t…$ are perfect powers. There is a solution ...
2
votes
1answer
89 views

Find known number of missing natural numbers

Given a set $S$ of distinct natural numbers, we know that a subset $T$ that is $S$ with at most $k$ number of elements missing. Let $M_k := \big\{m_j\big|d_j = \sum_{i\in T}i^j, j\in ...
1
vote
1answer
54 views

A question about cubic roots of rational numbers

I'm trying to understand if, given $K$ a cubic cyclic extension of $\mathbb{Q}(\zeta_3)$, where $\zeta_3$ is a third primitive root of unity, it always exists $b \in \mathbb{Q}$ such that $\sqrt[3]{b} ...
1
vote
2answers
80 views

quadratic equation with two variables

i try to solve the equation below has a solution or not $x^2-97y-40 =0$ if solution exists, $x^2-40$ must be congruent to 0 modulo $97$. if i could show the congruence above implies that ...
3
votes
1answer
66 views

Finding $a_n$ such that $x^n+a_1x^{n-1}+\cdots+a_{n-1}+a_n$ cannot be factored when $a_1,\cdots,a_{n-1}$ given

Let $n\ge 4\in\mathbb N$. Suppose that $a_1,a_2,\cdots,a_{n-1}$ are given integers. Then, here is my question. Question : Is the following true for any $(a_1,a_2,\cdots,a_{n-1})$ ? There ...
0
votes
2answers
93 views

Finding number of positive integral solutions of $x^4-y^4=3879108$

Find the number of positive integral solutions of $$x^4-y^4=3879108$$ $$3879108=36*277*389$$ I tried simplifying factors of $3879108$ to get terms in the form of $x^4-y^4$. However, I am unable to ...
2
votes
0answers
70 views

About the condition such that $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ is the $n$-th power of an integer for every integer $x$

Question : Is the following true for any $n\ge 2\in\mathbb N$? Letting $a_n, a_{n-1}, \cdots, a_1,a_0\in\mathbb R$ be constants, the necessary and sufficient condition for $a_n, a_{n-1}, ...
0
votes
0answers
88 views

Incongruent solutions to $f(x)$ mod p, when $x^n$ is largest poser of $x$ with coefficient not divisible by p

Show that if $p$ is a prime, $f(x)$ is a polynomial with integer coefficients and $x^n$ is the largest power of $x$ with a coefficient not divisible by $p$, then the congruence $f(x)==0$ mod $p$ has ...
4
votes
0answers
65 views

Solve $f(x)\mid g^2(x)+1$ in $\mathbb Z[x]$

We know that if $p\in \mathbb P$ and $p\equiv 1\bmod 4$ then we can find $t\in\mathbb Z$ such that $p\mid t^2+1.$ For what polynomial $f(x)\in \mathbb Z[x]$, we can find $g(x)\in \mathbb Z[x]$ ...
0
votes
0answers
34 views

Lower-Upper bounds on the cardinality of a bounded set

Let $S$ be a finite set which is a subset of $\{(\alpha ,\beta ):\alpha , \beta \in \mathbb{Z}, \alpha\geq 0, \beta \geq 0\}$ and $ T(x,y)=\sum_{(\alpha ,\beta ) \in S} h_{\alpha, \beta} ...
4
votes
0answers
64 views

Can we say anything about the structure of the semigroup of non-coprime pairs after this?

Let $S = \{(a,b) : \ a, b \in \Bbb{Z} \wedge \gcd(a,b) \neq 1 \}$. Then it forms a semigroup under componentwise multiplication and if we add an exception, that even though $\gcd(1,1) = 1$, we ...
3
votes
0answers
69 views

Conditions for polynomial $f$ such that $f(n) \in \mathbb{N}$ for enough $n \in \mathbb{N}^+$ implies $f$ has rational coefficients

This question is suggested by this one: prove: coefficients of $f(x)$ are rational numbers What are the weakest sufficient conditions and strongest necessary conditions on a set of positive integers ...
0
votes
0answers
17 views

What are some techniques for reducing the dimension of an arbitrary Diophantine polynomial?

A set $S \subset \mathbb{N}^k$ is Diophantine if $$(x_1, \dots, x_k) \in S \iff \exists y_1, \dots, y_d \, p(x_1, \dots, x_k, y_1, \dots, y_d) = 0$$ for some Diophantine (integer coefficients) ...
1
vote
0answers
176 views

About polynomials which can represent every prime number

A friend of mine taught me the followings : It has been known that there are some polynomials which can represent every prime number. 1. The $21$st degree polynomial with $21$ variables by ...
3
votes
1answer
111 views

Relationship between divisibility of polynomials and divisibility of its evaluations

Let $f$ and $g$ be primitive polynomials over $\mathbb{Z}$. Decide if the following is true: $f(x) \mid g(x)$ for infinitely many $x\in\mathbb Z$ implies $f\mid g$ as polynomials in ...
3
votes
1answer
257 views

Finding an integer-coefficients quadratic equation which has approximate solutions of $e$ and $\pi$.

Question : Find a quadratic polynomial $f(x)=a_2x^2+a_1x+a_0\ (a_i\in\mathbb Z)$ which has the following three conditions. Suppose that $f(x)=0$ gives two real positive solutions $\alpha, \beta.$ ...
6
votes
1answer
145 views

A low-degree polynomial $g_{a,b}(x)$ which has a zero $x\in\mathbb N$ for any square numbers $a,b$?

Question : Letting $a,b$ be natural numbers, does there exist a $n$-th degree integer-coefficients polynomial $g_{a,b}(x)$ which satisfies the following three conditions? Condition 1: $1\le n\le3.$ ...
0
votes
1answer
63 views

Natural number n-Divisibility

The number of natural number $n$ in the interval $[1005,2010]$ for which the polynomial $$1+x+x^2+x^3\dots +x^{(n-1)}$$ divides the polynomial $$1+x^2+x^4\dots+x^{2010}$$ is: I could realize that ...
3
votes
1answer
46 views

If $x^2+ax+b$ is an integer for every integer $x$ then comment on the coefficients $a$ and $b$ MCQ

Probably a more general category (number theory) multiple choice question but no clue how to get to a clear conclusion . Here's how it goes : Q) If $x^2+ax+b$ is an integer for every integer x ...
5
votes
3answers
107 views

Determine the minimum of $a^2 + b^2$ if $a,b\in\mathbb{R}$ are such that $x^4 + ax^3 + bx^2 + ax + 1 = 0$ has at least one real solution

I just wanted the solution, a hint or a start to the following question. Determine the minimum of $a^2 + b^2$ if $a$ and $b$ are real numbers for which the equation $$x^4 + ax^3 + bx^2 + ax + 1 = 0$$ ...
6
votes
1answer
108 views

Relations between irreducibility on $\mathbb{Q}[x]$, and on $\mathbb{Q}_p[x]$ ($p$-adic numbers)

I'm reading "$p-$adic numbers: An introduction" by Fernando Q.Gouvêa, and I'm currently on page 79 of the book. Problem 121. Show that the equation $(X^2 - 2)(X^2 - 17)(X^2 - 34) = 0$ has a root ...