1
vote
3answers
56 views

Diophantine equations problem/exercise 3

Find all the pythagorean tripples (x,y,z) with x=40. Well I started with the known formulas for the pythagorean tripples but got me nowhere. Or I was not able continue the thought process required. I ...
1
vote
1answer
27 views

How many n-tuple to genereate zero from some random several variable equation that use constant power and a variable as the base?

Define algebraic number tuple as the aphabetical order sequence of variables that use in the equation. How many algebraic number n-tuple (x,y,z,...) are able to genereate zero to input into a several ...
0
votes
0answers
70 views

System of congruences that do not satisfy CRT assumptions (via algorithm)

Let $x_i,a_i\!\in\!\mathbb{Z}$. The following procedure solves a system of congruences $$x \equiv x_i\pmod{a_i}\;\;\text{ for }i\!=\!1,\ldots,n$$ when $a_i$ are pairwise coprime. Assume that ...
0
votes
1answer
172 views

Do I have this right? Are these conclusions valid in this isomorphic view of $\Bbb{R}$?

Let $F = (\Bbb{R}, \oplus_d, \cdot)$ be the field with usual $\cdot$, and $\oplus_d$ is defined as $a \oplus b = (\sqrt[d]{a} + \sqrt[d]{b})^d$. This field is isomorphic to usual $\Bbb{R}$ structure ...
0
votes
0answers
61 views

What can we say about solutions in fields isomorphic to $\Bbb{R}$?

Let $\phi: \Bbb{R} \to (\Bbb{R}, \oplus, \cdot)$ be a field isomorphism such that $\phi(\Bbb{Z}) \subset \Bbb{Z}. \ $ By FLT, $x^n + y^n - z^n = 0$ has no positive integer solutions for $n \gt 2$. ...
5
votes
1answer
249 views

On Bachet's Duplication Formula and the number $-432$

While reading "Rational Points on Elliptic Curves" by Silverman and Tate, I came across this interesting passage about Bachet's duplication formula: I know how to derive Bachet's duplication ...
4
votes
1answer
193 views

If $x,y,z \in \mathbb{Z}$ such that $x^4+y^4+z^4 \equiv 0 \pmod{29}$, prove that $x^4+y^4+z^4 \equiv 0 \pmod{29^4}$

If $x,y,z \in \mathbb{Z}$ such that $x^4+y^4+z^4 \equiv 0 \pmod{29}$, prove that $x^4+y^4+z^4 \equiv 0 \pmod{29^4}$. I have no idea where to start, but this is my abstract algebra homework, so I ...
0
votes
1answer
65 views

Diophantine equation and cyclicity of $\mathbb{F}_p^*$

I am trying to prove that the diophantine equation $$1998^2x^2+1997x+1995-1998x^{1998}=1998y^4+1993y^3-1991y^{1998}-2001y$$ has no solution in integers (given that $1997$ is a prime). To do so, ...
5
votes
2answers
106 views

Step in a solution of $y^2 = x^3 - 2$

I am reading Algebraic Number Theory notes here by Keith Conrad. In page 9, there is a solution of $y^2=x^3-2$ using unique factorization in $\mathbb{Z}[\sqrt{-2}]$. We start by writing ...
1
vote
0answers
46 views

Is this a fruitless approach to solving diophantine equations?

Let $P(X, Y, Z)$ be a polynomial over $Q$. Let's be concerned with integer solutions. Namely that there are no solutions $(X,Y,Z)$ such that $\gcd(X,Y) = 1$. So let $X,Y$ be coprime and arbitrarily ...
4
votes
3answers
360 views

Infinite solutions of Pell's equation $x^{2} - dy^{2} = 1$

Let $d > 1$ be a squarefree integer. Prove that the equation $x^{2} - dy^{2} = 1$ has infinitely many solutions in $\mathbb{Z} \times \mathbb{Z}$. What I have done: let $ \ \mathbb{K} = ...
1
vote
0answers
72 views

Find integer solutions to $x^2+xy+11y^2=p$ using Ring identities

Let $\theta = (1+\sqrt{-43})/2$ and consider $\mathbb{Z}[\theta]$, a principal ideal domain, with the multiplicative map $\psi (a+b\theta)=a^2+ab+11b^2$. Show there exists an integer solution to ...
4
votes
1answer
61 views

On Selmer's curve

I am trying to prove that the equation $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ has non-trivial solutions for all primes $p$. I divide it into 3 cases: $p \equiv 0,1,2 \pmod{3}$. The cases $p \equiv 0,2 ...
2
votes
2answers
103 views

No rational solutions of a system of equations

Please show that there does not exist $(a,b,c)\in\mathbb{Q}^3$ such that \begin{matrix} a^2b+2b^2c+2ac^2=0\\ a^2c+ab^2+2bc^2=0\\ a^3+2b^3+4c^3+12abc=3. \end{matrix} I'm able to show that this ...
2
votes
1answer
222 views

If $x^p +y^p = z^p$ and $xyz \neq 0$, then $p$ divides $x$ or $y$ or $z$?

I am working on an exercise: If $x^5 +y^5 = z^5$ and $xyz \neq 0$, then $5$ divides at least one of $x$, $y$ or $z$. I am thinking that the answer involves an application of Kummer's theorem, but I'm ...
7
votes
3answers
586 views

Integer Solutions to $x^2+y^2=5z^2$

I'm looking for a formula to generate all solutions $x$, $y$, $z$ for $x^2 + y^2 = 5z^2$. Any advice?
4
votes
3answers
1k views

$\mathbb Z[\sqrt 3]$ contains infinitely many units

I'm asked to show that there are infinitely many units in the ring $\mathbb Z[\sqrt 3]$. But I don't really see a good approach to this one, so far. Some thoughts: The inverse of $a+\sqrt3 b$ ...
5
votes
2answers
362 views

How many positive integer solutions to $a^x+b^x+c^x=abc$?

How many positive integer solutions are there to $a^{x}+b^{x}+c^{x}=abc$? (e.g the solution $x=1$, $a=1$, $b=2$, $c=3$). Are there any solutions with $\gcd(a,b,c)=1$? Any solutions to ...
25
votes
2answers
2k views

Does the equation $x^4+y^4+1 = z^2$ have a non-trivial solution?

The background of this question is this: Fermat proved that the equation, $$x^4+y^4 = z^2$$ has no solution in the positive integers. If we consider the near-miss, $$x^4+y^4-1 = z^2$$ then this has ...
4
votes
10answers
774 views

Polynomial satisfying $p(x)=3^{x}$ for $ x \in \mathbb{N}$

Let $p(x)$ be a polynomial with integer coefficients, which is not constant. Then is this condition possible: $$p(x)=3^{x}$$ whenever $x \in \mathbb{N}$. Motivation: ...