All Questions
21
questions
1
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0
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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
votes
4
answers
208
views
Find natural number $x,y$ satisfy $x^2+7x+4=2^y$
Find natural number $x,y$ satisfy $x^2+7x+4=2^y$
My try: I think $(x;y)=(0;2)$ is only solution. So I try prove $y\ge3$ has no solution, by $(x+1)(x+6)-2=2^y$.
So $2\mid (x+1)(x+6)$, but this is ...
- 727
2
votes
1
answer
64
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Finding roots of polynomial $X^2 - X + 19$ in $\mathbb Z/61 \mathbb Z$
For $p = 61$.
I was given the roots of $X^2 + 3$ in $\mathbb Z/p \mathbb Z$, which are $\pm 27 + p\mathbb Z$.
I then must find the roots of $X^2 - X + 19$ in $\mathbb Z/p\mathbb Z$, which I have found ...
- 35
1
vote
3
answers
63
views
Finding number of integral solutions to an equation.
Find the number of integral solutions to:
$$x^2+y^2-6x-8y=0.$$
My attempt:
The equation can be rewritten as:
$$x^2+y^2-6x-8y+9+16=25,$$
basically adding 25 to both sides, or equivalently,
$$(x-...
- 902
1
vote
5
answers
230
views
Quadratic Diophantine equation $x^2+6y^2-xy=47$ has no solutions.
I am trying to show that $x^2 + 6y^2 - xy = 47$ has no integer solutions. I know that the an efficient way is to look at this equation modulo $n$; other equations can be easily be solved this way. I ...
- 854
-1
votes
3
answers
1k
views
How do you solve a quadratic equation containing a negative power?
I was trying to factor a polynomial with Wolfram and I noticed a quadratic form I've never considered.
$$n^2-4+\frac{6}{n}=0$$
The purpose of it is not important but it made me wonder. How do you ...
- 6,416
3
votes
2
answers
91
views
How many quadratic functions can be simultaneously pairwise coprime?
Is there a limit to how many integer-valued strictly increasing quadratic functions $f(x):=ax^2+bx+c$ can be guaranteed to yield coprime values for a given $x\in\mathbb N$?
For example, the values of
...
- 6,014
1
vote
3
answers
952
views
Let $m$ and $n$ be positive integers such that $m(n-m)=-11n+8$ . Find the sum of all possible values of $m-n$.
Let $m$ and $n$ be positive integers such that $m(n-m)=-11n+8$ . Find the sum of
all possible values of $m-n$.
after manipulation you get the quadratic $0=m^2-mn+(8-11n)$
from that you get $m=\frac{n ...
- 958
0
votes
1
answer
107
views
Determine all pairs $(p,q)$ of integer numbers for which all zeros of $x^2+px+q$ and $x^2+qx+p$ are integers.
Determine all pairs $(p,q)$ of integer numbers for which all zeros of $x^2+px+q$ and $x^2+qx+p$ are integers.
I need help trying to understand the solution of this problem.
This is what it says on ...
- 1,067
3
votes
2
answers
87
views
Find the last three digits of $p$ if the equations $x^6 + px^3 + q = 0$ and $x^2 + 5x - 10^{2013} = 0$ have common roots.
Find the last three digits of $p$ if the equations $x^6 + px^3 + q = 0$ and
$x^2 + 5x - 10^{2013} = 0$ have common roots.
Let $a,b $ be the solutions of second equation, then by Vieta we have $a+...
- 90.7k
5
votes
2
answers
213
views
Is it true that $(a^2-ab+b^2)(c^2-cd+d^2)=h^2-hk+k^2$ for some coprime $h$ and $k$?
Let us consider two numbers of the form $a^2 - ab + b^2$ and $c^2 - cd + d^2$ which are not both divisible by $3$ and such that $(a, b) = 1$ and $(c,d) = 1$. Running some computations it seems that ...
- 51
6
votes
2
answers
148
views
How does one construct general forms that certain variables in an equation must take?
Srinivasa Ramanujan was one of the greatest mathematicians of all time $-$ the greatest in the $20^\text{th}$ century. One day, he stumbled across the equation $$\rm3^3+4^3+5^3=6^3\tag1$$ and only ...
- 9,487
7
votes
1
answer
226
views
Quadratics which produce no primes
There are some famous polynomials which produce a series of consecutive prime numbers, including Euler's $n^2 + n + 41$, which produces primes for $0 \leq n \leq 39$.
What I've been thinking about ...
- 891
2
votes
3
answers
246
views
Find $\frac {\alpha}{\beta} + \frac {\beta}{\alpha}$ if $\alpha^2+3 \alpha+1=\beta^2+3\beta+1=0$
The question:
Let $\alpha$ and $\beta$ be $2$ distinct real numbers which such that $\alpha^2+3 \alpha+1=\beta^2+3\beta+1=0$. Find the value of $\frac {\alpha}{\beta} + \frac {\beta}{\alpha}$.
...
- 2,781
2
votes
3
answers
120
views
Finding integral solutions of $x+y=x^2-xy+y^2$
Find integral solutions of $$x+y=x^2-xy+y^2$$
I simplified the equation down to
$$(x+y)^2 = x^3 + y^3$$
And hence found out solutions $(0,1), (1,0), (1,2), (2,1), (2,2)$ but I dont think my ...
- 67