Found problems: 21
Let $p$ be a prime number,let $n_1, n_2, \ldots, n_p$ be positive integer numbers, and let $d$ be the greatest common divisor of the numbers $n_1, n_2, \ldots, n_p$. Prove that the polynomial
\[\dfrac{X^{n_1} + X^{n_2} + \cdots + X^{n_p} - p}{X^d - 1}\]
is irreducible in $\mathbb{Q}[X]$.
[i]Beniamin Bogosel[/i]
Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$
Show that for any positive integer $n$ the polynomial $f(x)=(x^2+x)^{2^n}+1$ cannot be decomposed into the product of two integer non-constant polynomials.
[i]Marius Cavachi[/i]
Let $f(X) = a_nX^n + a_{n-1}X^{n-1} + ...+ a_1X + p$ be a polynomial of integer coefficients where $p$ is a prime number. Assume that $p >\sum_{i=1}^n |a_i|$. Prove that $f(X)$ is irreducible.
Find two smallest natural numbers $n$ for which each of the fractions
$\frac{68}{n+70},\frac{69}{n+71},\frac{70}{n+72},...,\frac{133}{n+135}$ is irreducible.
Show that the fraction $ \frac{n^2+n-1}{n^2+2n}$ is irreducible for every positive integer n.
Let the set $M =\{\frac{1998}{1999},\frac{1999}{2000} \}$.
The set $M$ is completed with new fractions according to the rule:
take two distinct fractions$ \frac{p_1}{q_1}$ and $\frac{p_2}{q_2}$ from $M$ thus there are no other numbers in $M$ located between them and a new fraction is formed, $\frac{p_1+p_2}{q_1+q_2}$ which is included in $M$, etc.
Show that, after each procedure, the newly obtained fraction is irreducible and is different from the fractions previously included in $M$.
Let $f$ be a primitive polynomial with integral coefficients (their highest common factor is $1$ ) such that $f$ is irreducible in $\mathbb{Q}[X]$ , and $f(X^2)$ is reducible in $\mathbb{Q}[X]$ . Show that $f= \pm(u^2-Xv^2)$ for some polynomials $u$ and $v$ with integral coefficients.
Show that for any $n \in N$ the polynomial $f(x) = (x^2 +x)^{2^n}+1$ is irreducible over $Z[x]$.
Find a irreducible fraction with denominator not greater than 2005, that is closest to $\frac{9}{25}$ but is not $\frac{9}{25}$
Find the smallest positive integer $n$ such that the $73$ fractions $\frac{19}{n+21}, \frac{20}{n+22},\frac{21}{n+23},...,\frac{91}{n+93}$ are all irreducible.
Let $f$ be an irreducible (in $Z[x]$) monic polynomial with integer coefficients and of odd degree greater than $1$. Suppose that the modules of the roots of $f$ are greater than $1$ and that $f(0)$ is a square-free number. Prove that
the polynomial $g(x) = f(x^3)$ is also irreducible
a) Prove that the fraction $\frac{3n+5}{2n+3}$ is irreducible for every $n \in N$
b) Let $x,y$ be digits of decimal representation system with $x>0$, and $\frac{\overline{xy}+12}{\overline{xy}-3}\in N$, prove that $x+y=9$. Is the converse true?
Let $P(x)$ be a polynomial with integer coefficients for which there exists a positive integer n such that the real parts of all roots of $P(x)$ are less than $n- \frac{1}{2}$ , polynomial $x-n+1$ does not divide $P(x)$, and $P(n)$ is a prime number. Prove that the polynomial $P(x)$ is irreducible (over $Z[x]$).
Olya wrote fractions of the form $1 / n$ on cards, where $n$ is all possible divisors the numbers $6^{100}$ (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, etc., finally, the sum of the numbers on all the cards. Every amount Olya recorded on the board in the form of irreducible fraction. What is the least different denominators could be on the numbers on the board?
Prove that $p(x)=1+x+x^2 +\ldots+x^n$ is reducible over $\mathbb{F}_{2}$ in case $n+1$ is composite. If $n+1$ is prime, is $p(x)$ irreducible over $\mathbb{F}_{2}$ ?
Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$
Let $n \in N$. Find the maximum number of irreducible fractions a/b (i.e., $gcd(a, b) = 1$) which lie in the interval $(0,1/n)$.
Evaluate the sum of all positive irreducible fractions less than $1$ and having the denominator $1991$.
Let $n$ be a positive integer and $a=2^n\cdot 7^{n+1}+11$ and $b=2^{n+1}\cdot 7^n+3$.
$a)$ Prove that fraction $\frac{a}{b}$ is irreducible
$b)$ Prove that number $a+b-7$ is not a perfect square for any positive integer $n$
For any $k \in \mathbb{Z},$ define $$F_k=X^4+2(1-k)X^2+(1+k)^2.$$ Find all values $k \in \mathbb{Z}$ such that $F_k$ is irreducible over $\mathbb{Z}$ and reducible over $\mathbb{Z}_p,$ for any prime $p.$
[i]Marius Vladoiu[/i]