1
vote
4answers
157 views

Solutions to the diophantine equation: $2a^2 + 2b^2- c^2- d^2 = 0$

As suggested on Mathoverflow (http://mathoverflow.net/questions/168536/solutions-to-the-diophantine-equation-2a2-2b2-c2-d2-0) I am transfering this question to math-stackexchange: I am looking for ...
0
votes
0answers
37 views

Diophantine like philosophy for computing trigonometric functions with approximation around intervals

I noticed that diophantine expressions are great to approximate constants or simple functions, as far as I know, they are not so great when it comes to approximate and compute transcendental functions ...
3
votes
1answer
104 views

solving $x^3-2y^3=1$ using cubic number field

I am trying to solve the diophantine equation $x^3-2y^3=1$ using $\mathbb{Q}(\sqrt[3]{2}).$ I've read this link: Solve $x^3 +1 = 2y^3$ The following is what i have tried: Finding all integer ...
1
vote
0answers
58 views

How to solve diophantine equation $\frac{x^p-y^p}{x-y}=n$

$$\frac{x^p-y^p}{x-y}=n$$ whit $p$ a prime greater than or equal to $3$,for what value to $n$, it's solvable and how to solve,and whether $\frac{x^p-y^p}{x-y}=q_1$ $\frac{x^p-y^p}{x-y}=q_2$ is ...
16
votes
2answers
788 views

Prove that $x^3 + y^3 = z^3$ has no integer solutions as simply as possible

Can someone prove the special case of Fermat's Last Theorem for $n=3$, i.e., that $$x^3 + y^3 = z^3,$$ has no positive integer solutions, as simply as possible? I have seen some good proofs, but ...
4
votes
5answers
175 views

Looking for proof of no solution to 4-variable quadratic diophantine equation

Show there are no integers $a,b,c,d$ such that $$\begin{cases}1=9ac+3ad+3bc-16bd \\ 1=3ad+3bc+2bd \end{cases}$$ Motivation: The ideal $I=(3,1+\sqrt{-17})$ in $R=\mathbb{Z}[\sqrt{-17}]$ has the ...
6
votes
0answers
157 views

Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$

In trying to solve $a^3 - 2b^3 = 1$ over the integers I came across the need to answer the question: when does $(1+ \sqrt[3]{2} + \sqrt[3]{2}^2)^n$ have no $\sqrt[3]{2}^2$ term in it's expansion (in ...
3
votes
1answer
90 views

$\frac{x^5-y^5}{x-y}=p$,give what p ,the diophantine equation is solvable

for$$\frac{x^3-y^3}{x-y}=x^2+xy+y^2=p$$$p=6k+1$give p prime, On what conditions,the diophantine equation $$\frac{x^5-y^5}{x-y}=p$$ is solvable in integers.does it have a linear expression.for ...
0
votes
0answers
36 views

Slight generalization of Pell's equation?

Suppose I want to solve $x^2 - ny^2 = k$, where $n$ is square-free and $$n \equiv2,3 \mod 4.$$ If, moreover, $k$ is a prime such that for all $\mathfrak P_i$ lying above it in the field extension ...
2
votes
0answers
74 views

Find all Integers ($ n$) such that $n\neq 6xy\pm x\pm y$

I am interested in proving that there exist an infinite number of positive integers ($n$) which are not of the form $$ n=6xy\pm x\pm y $$ for $x,y\in\Bbb Z^+$. [Note: The $\pm$ signs above are ...
8
votes
1answer
129 views

Geometric intuition behind the Hasse principle

Let $f(X,Y) \in \mathbb{Q}[X,Y]$ be a quadratic polynomial. The Hasse-Minkowski theorem says that $f(X,Y) = 0$ has a solution $(x,y) \in \mathbb{Q}^2$ iff it has a solution in $\mathbb{R}^2$ and ...
2
votes
2answers
218 views

Finding all solution of a Pell equation

I wanted to solve the equation $x^2-5y^2 = -4$ with $x$ and $y$ integers. Let $\omega=\frac{1+\sqrt5}{2}$ and $A = \mathbb{Z}[\omega]$. One can reduce the Pell equation to finding the elements of $A$ ...
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 ...
4
votes
3answers
355 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} = ...
0
votes
1answer
38 views

Why does a Norm Form have rational coefficients?

[From Wolfgang Schmidt's "Diophantine Approximation and Diophantine Equations"] "Let $K$ be a number field with $[K:\mathbb{Q}] = d$ and $L(X)$ be a linear form, say $L(X) = L(x_1,\dots,x_n) = ...
6
votes
0answers
204 views

Ramanujan-Nagell Theorem Proof Question

I'm currently working through Stewart and Tall's Algebraic Number Theory. In particular, section 4.9 of this book provides a proof of the Ramanujan-Nagell Theorem, which states that the only integer ...
7
votes
2answers
208 views

Solve: $x^2-py^2=q$

Solve $$x^2-py^2=q$$ for integers $x,y$, here $p,q$ are both given prime numbers. It's obvious that $p,q$ should satisfy $(\frac{p}{q})=(\frac{q}{p})=1,$ here $(\frac{p}{q})$ is the Jacobi symbol. ...
2
votes
3answers
246 views

Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$

For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$ where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
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 ...
1
vote
0answers
94 views

Solve the equation $x^4+y^4=d*z^2$

Solve the equation:$$x^4+y^4=d*z^2,$$ where $x,y,z$ are positive integers,and $d>1$ is a given square-free integer. I know if $p$ is an odd prime and $p|d,$ then $t^4\equiv -1 \pmod p$ is ...
1
vote
2answers
158 views

How to prove that there exist infinitely many integer solutions to the equation $x^2-ny^2=1$ without Algebraic Number Theory

I learned in my Intro Algebraic Number Theory class that there exist infinitely many integer pairs $(x,y)$ that satisfy the hyperbola $x^2-ny^2=1$; just consider that there are infinitely many units ...
2
votes
1answer
221 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 ...
4
votes
1answer
168 views

Solving $x^2+19=y^5$

I was given several exercises and there is a particular one, I am not able to solve. Let it be given that $Pic(\mathbb{Z}[\sqrt{−19}])$ is a finite group of order $3$. Use this to find all integral ...
5
votes
1answer
107 views

Quadratic Diophantic equation

Hello :) i want to give a answer op the following question: For which prime number $p$ can we give a solution of the diophantic equation given by $x^2-65y^2=p$. I want to solve the question without ...
4
votes
2answers
175 views

Does this equation have integer solutions

Let $g\geq 2$ be an integer. (It will be the genus of some curve.) Are there positive integers $d$ and $e$ such that the equality $$ (e-2)(e-1) = 2d(g-1)+2$$ holds?
1
vote
0answers
24 views

Can one determine in finite time whether a point is $S$-integral

Let $x$ be a $\mathbf{Q}$-rational point of $\mathbf{P}^1-\{0,1,\infty\}$. Let $S$ be a finite set of primes. How do I check in finite time whether $x$ is $S$-integral or not? I know how to do this ...
2
votes
2answers
106 views

Does the equation have infinite number of solutions in integers?

Does either of the equations ${a^2} - 10{b^2} = \pm 1$ have infinite number of solutions in integers? If the answers is yes, a hint about how to reduce this problem to the problem of Pythagorean ...
4
votes
1answer
234 views

For which $(n,m,l)$ does $L_n^2+L_m^2=L_l^2$ hold?

Let $L_n = 2 \sin(\frac{\pi}{n})$ be the length of a side of the regular $n$-gon inscribed in a circle. For which $(n,m,l)$ does $$L_n^2+L_m^2=L_l^2$$ hold? I found by computer search that (3,6,2), ...
1
vote
1answer
100 views

Clarifying a step in the solution of $y^2 + 2 = x^3$

In Ian Stewart's book on Algebraic Number Theory, he gives the following proof that the solutions to $y^2 + 2 = x^3$ is $x=3, y = \pm 5$: I'm confused about the step "since $c^2 + 2d^2 | 4y^2$ and ...
4
votes
2answers
348 views

Diophantine equation (use class ideal group to solve)

Use ideal class group to find all integer solutions to the equation $$x^3=y^2+200$$ My approach: Observe that $\mathbb{Z}[\sqrt-2]$ is the field of integers in the ring $\mathbb{Q}(\sqrt -2).$ ...
4
votes
1answer
294 views

Solving a Diophantine Equation using factorisation of ideals

I am stuck on the following question which is given as follows: Prove that the only integer solutions to the equation \begin{equation} x^2 + 13 = y^3 \end{equation} are $(70,17)$ and $(-70, 17)$. ...
3
votes
0answers
82 views

Do the solutions to the unit equation lie dense in the complex numbers

Let $S\subset \overline{\mathbf{Q}}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of algebraic integers. ...
40
votes
5answers
1k views

Golden Number Theory

The Gaussian $\mathbb{Z}[i]$ and Eisenstein $\mathbb{Z}[\omega]$ integers have been used to solve some diophantine equations. I have never seen any examples of the golden integers ...
6
votes
3answers
1k views

Integral solutions to $y^{2}=x^{3}-1$

How to prove that the only integral solutions to the equation $$y^{2}=x^{3}-1$$ is $x=1, y=0$. I rewrote the equation as $y^{2}+1=x^{3}$ and then we can factorize $y^{2}+1$ as $$y^{2}+1 = (y+i) \cdot ...