All Questions
10
questions
1
vote
1
answer
73
views
Prove or disprove that $\sum_{k=1}^p G(\lambda^k) = ps(p)$
Prove or disprove the following: if $\lambda$ is a pth root of unity not equal to one, $G(x) = (1+x)(1+x^2)\cdots (1+x^p),$ and $s(p)$ is the sum of the coefficients of $x^n$ for $n$ divisible by $p$ ...
- 1,225
2
votes
1
answer
161
views
Integer solution to $x^4 + x^3 = y^4 +7$
Find an integer solution or show there are none of $$x^4+x^3=y^4+7$$
I have found using python that there are no solutions for x, y less than 100
but also that the equation has at least one solution ...
- 96
1
vote
2
answers
133
views
Proving a cubic polynomial has no rational roots [duplicate]
This is an exercise in polynomials/algebra/discrete mathematics I have just met:
For odd integers $a,b \in \mathbb{Z}$ we are asked to show the polynomial $ x^3+ax+b$ has no rational roots.
The ...
- 3,847
1
vote
3
answers
422
views
Prove that there exists a polynomial p(x) with coefficients belonging to the set {-1, 0, 1} such that p(3) = n, for some positive integer n
Prove that there exists a polynomial p(x) with coefficients belonging to the set {-1, 0, 1} such that p(3) = n, for some positive integer n.
I started off my proof by noticing that n = either 3k or ...
- 93
3
votes
3
answers
94
views
Solve $(y^2 + xy)(x^2 - x + 1) = 3x - 1$ over the integers.
Solve $$(y^2 + xy)(x^2 - x + 1) = 3x - 1$$ over the integers.
There are many solutions to this problem, and perhaps I chose the worst one possible. I hope that someone could come up with a better ...
- 4,246
3
votes
2
answers
253
views
Solve $x^2y^2 - 4x^2y + y^3 + 4x^2 - 3y^2 + 1 = 0$ over the integers.
Solve $$x^2y^2 - 4x^2y + y^3 + 4x^2 - 3y^2 + 1 = 0$$ over the integers.
You can probably guess by now... This problem is adapted from a recent competition.
If there are any other solutions, please ...
- 4,246
5
votes
1
answer
105
views
there are at least $\frac{p+1}{2}$ integers $d$, with $0 \leq d < p$, so that the equation $x^3+x=d \pmod p$ has a root$\pmod p$
Prove that for infinitely many prime numbers p, there are at least $\frac{p+1}{2}$ integers $d$ with $0 \leq d<p$ such that the equation $x^3+x \equiv d \pmod p$ has a root $\pmod p$
My first ...
- 65
1
vote
2
answers
84
views
Show that $7 |(n^6 + 6)$ if $7 ∤ n$, $∀ n ∈ ℤ$
Show that $7 |(n^6 + 6)$ if $7 ∤ n$, $∀ n ∈ ℤ$
I need to prove this by the little Fermat's theorem.
My attempt
$n^6 \equiv -6 \pmod 7$
To show $7 ∤ n$ I need to show that $N$ is not congruent to $...
- 637
1
vote
1
answer
710
views
How do you determine where a polynomial evaluates to a perfect square?
How do you determine where a polynomial evaluates to a perfect square?
One example would be $f(x)=x^2+148x-288$. $f(68) = 14400 = 120^2$.
Another one would be $f(x)=x^2+204x-88$. $f(2) = 324 = 18^2$...
- 13
10
votes
2
answers
14k
views
Proof that no polynomial with integer coefficients can only produce primes [duplicate]
Doing a discrete math review and am trying to solve problem 1.6 in the text found here: http://courses.csail.mit.edu/6.042/fall13/ch1-to-3.pdf - I believe I've gotten parts (a) and (b) correctly, but (...
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