All Questions
Tagged with polynomials elementary-number-theory
668
questions
95
votes
5
answers
4k
views
Reversing an integer's digits is multiplicative for small digits
So my 7 year old son pointed out to me something neat about the number 12: if you multiply it by itself, the result is the same as if you took 12 backwards multiplied by itself, then flipped the ...
- 793
66
votes
9
answers
62k
views
Prove every odd integer is the difference of two squares
I know that I should use the definition of an odd integer ($2k+1$), but that's about it.
Thanks in advance!
- 1,883
51
votes
2
answers
3k
views
Decomposing polynomials with integer coefficients
Can every quadratic with integer coefficients be written as a sum of two polynomials with integer roots? (Any constant $k \in \mathbb{Z}$, including $0$, is also allowed as a term for simplicity's ...
- 2,674
39
votes
2
answers
6k
views
Prove that the polynomial $(x-1)(x-2)\cdots(x-n) + 1$, $ n\ge1 $, $ n\ne4 $ is irreducible over $\mathbb Z$
I try to solve this problem. I seems to come close to the end but I can't get the conclusion. Can someone help me complete my proof. Thanks
Show that the polynomial $h(x) = (x-1)(x-2)\cdots(x-...
- 4,845
23
votes
3
answers
3k
views
Primes dividing a polynomial
Let $g(x)\in \mathbb{Z}[x]$ be a nonconstant polynomial. Show that the set of primes $p$ such that $p\mid g(n)$ for some $n\in \mathbb{Z}$ is infinite.
I don't know how to start. I have tried ...
- 962
20
votes
2
answers
709
views
For what $n$ can we find a degree $\leq n-2$ polynomial such that $P(i) \in \{0 , 1 \}$ for $i \in [n]$, but not all identical.
For what $n$ is the following statement true:
There exists a choice of $ a_1, a_2, \ldots a_n \in \{ 0, 1 \}$, not all identical, such that there is a polynomial $F(x) \in \mathbb{R}[x]$ of degree at ...
- 70.4k
20
votes
4
answers
578
views
When can products of linear terms differ by a constant?
We have
$$ X(X+3) + 2 = (X+1)(X+2)$$
and
$$ X(X+4)(X+5) + 12 = (X+1)(X+2)(X+6)$$
and
$$ X(X+4)(X+7)(X+11) + 180 = (X+1)(X+2)(X+9)(X+10).$$
Do similar polynomial identities exist for each degree?
That ...
- 2,071
19
votes
2
answers
767
views
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
16
votes
3
answers
3k
views
Why is the zero polynomial not assigned a degree? [duplicate]
Yesterday, I read in my textbook,
We assign degree to every polynomial and even a non-zero constant is assigned a degree $0$ but $0$ itself is not assigned a degree.
Why is that? Why we don't ...
- 2,358
16
votes
1
answer
610
views
I would like to determine if exists a polynomial $R$ with integer coefficients such that $P(x)=Q(R(x))$.
Let $P$ and $Q$ be monic polynomials with integer coefficients and degrees $n$ and $d$ respectively, where $d\mid n$. Suppose there are infinitely many pairs of positive integers $(a,b)$ for which $P(...
- 161
16
votes
1
answer
569
views
Roots with equal fractional parts
Question. ¿Does there exist an integer $n>1$ such that there exist positive integers $a,b$ such that $\{\sqrt[n]{a}\}=\{\sqrt[n]{b}\},a\neq b$ and $a$ and $b$ aren't perfect n-th powers? ( $\{x\}$ ...
- 3,493
14
votes
1
answer
420
views
Is $x^n-\sum_{i=0}^{n-1}x^i$ irreducible in $\mathbb{Z}[x]$, for all $n$?
Let the sequence of polynomials $p_n$ from $\mathbb{Z}[x]$ be defined recursively as $$p_n(x)= xp_{n-1}(x)-1$$
with initial term $p_0(x)=1$.
Then $$p_n(x)= x^n-\sum_{i=0}^{n-1}x^i $$
Question 1: is it ...
- 3,716
14
votes
1
answer
343
views
Is there a cubic $Q(x)\in \mathbb{Z}[x]$ so that $|Q(p_1)|=|Q(p_2)|=|Q(p_3)|=|Q(p_4)|=3$, where $p_1, p_2, p_3, p_4$ are distinct primes? [duplicate]
Is there a cubic $Q(x)\in \mathbb{Z}[x]$ so that $|Q(p_1)|=|Q(p_2)|=|Q(p_3)|=|Q(p_4)|=3$, where $p_1, p_2, p_3, p_4$ are distinct primes?
Clearly there must be at least one $Q(p_i)=3$ and at least one ...
- 395
13
votes
14
answers
6k
views
How to prove that $k^3+3k^2+2k$ is always divisible by $3$? [closed]
How can I prove that the following polynomial expression is divisible by 3 for all integers $k$?
$$k^3 + 3k^2 + 2k$$
- 299
13
votes
6
answers
4k
views
If the number $x$ is algebraic, then $x^2$ is also algebraic
Prove that if the number $x$ is algebraic, then $x^2$ is also algebraic. I understand that an algebraic number can be written as a polynomial that is equal to $0$. However, I'm baffled when showing ...
- 131