The diophantine-equations tag has no wiki summary.

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### Examples of finiteness of rational points for hypersurfaces in $\mathbb P^3_{\mathbb Q}$ of degree $>4$.

Given an homogeneous polynomial $F(X,Y,Z,T)\in \mathbb Q[X,Y,Z,T]$ of degree $>4$, the surface it defines is well-known to be of general type. Suppose, moreover, that this surface doesn't contain ...

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### seeking an integer parameterization for A^2+B^2=C^2+D^2+1

I'm looking for a complete [integer] parameterization of all integer solutions to the Diophantine equation
$A^2+B^2=C^2+D^2+1$,
analogous to the classical parameterization of the Pythagorean ...

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**2**answers

1k views

### Rational solutions to x^3 + y^3 + z^3 - 3xyz = 1

I can show that there infinitely many solutions to this equation. Is it possible that the set
of rational solutions is dense?

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**1**answer

339 views

### what is the maximum number of rational points of a curve of genus 2 over the rationals

Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.)
We are ...

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631 views

### is there a solution to system of linear Diophantine equations?

I have a matrix A \in Z^{n \by m}, where m > n and a vector b \in Z^n. Then, under what conditions does an integer solution exist to the equation
Ax = b.
Is there a way to bound the norm of the ...

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90 views

### how do you bound exponent of x^2+1=y^p

for p a prime exponent using linear forms in logs?
So far I have (x-i)(x+i)=y^p which are coprime and hence x+i=(a+ib)^p , now how do I get a linear form in logs so that I can find an upper bound on ...

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362 views

### Diophantine equation over Z[i]

I'm trying to generate the set of solutions of a specific diophantine equation over Z[i].
The equation is the following:
$$ z_1^2 + z_2^2 + z_1*z_2 + 39 = 0$$
with $ z_1$ (resp $z_2$) such that ...

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160 views

### Efficient counting of Egyptian fractions with bounded denominators

I was amazed to discover that sequence http://oeis.org/A020473 in the OEIS has almost four hundred terms computed.
I wonder how one can get that far? E.g., how one can compute A020473(100)?
P.S. ...

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**1**answer

342 views

### Symmetric functions on three parameters being perfect squares

Is it possible for $x+y+z, xy+yz+zx$, and $xyz$ to be perfect squares at the same time for positive integer values of $x,y,z$?

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### 0,1 solution to system of linear integer equations.

I have the following problem:
$A x = b$
where $A, b$ - $m \times n$-maxtrix and $m$-vector of nonnegative intgers (respectivelly).
$x \in \{0,1\}^n $ - vector of binary variables, which need to be ...

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304 views

### Growth of $n=n(k)$ for which there's a non-trivial solution to $x_1^k+\cdots+x_n^k=y^k$?

Walter Hayman just asked me the following question. What, if anything, is known about the growth of the function $n(k)$, where $k\geq1$ is an integer, and $n=n(k)\geq2$ is the smallest integer for ...

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**1**answer

214 views

### A problem on cubic Diophatine equations

What is the best algorithm to find all the integer points (X,Y) on this curve
$X^3+aX-bY^3=m,a,b,m\in\mathbb{Z}$(a>0,b>0,b is not a cubic number)?

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751 views

### a family of Pellian equations

I have a question concering the family of Pellian equations
$$x^2 - (k^2+1)y^2 = k^2. \qquad (*)$$
For an integer $k\geq 2$, the equation (*) has at least three classes of solutions
in ...

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**1**answer

346 views

### Infinite solutions of a diophantine equation [closed]

Given the Diophantine equation$$ax^2+bxy+cy^2+dx+ey+f=0$$
if the coefficients $(a,b,c,d,e,f)$ are chosen among all the prime numbers, we have infinite equations. Is it possible to prove that the ...

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206 views

### Hurwitz integers and $F_4$

The Hurwitz integers are
$$
\mathcal H=
\{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}.
$$
I want to know if there is a formula, for $m\in\mathbb Z$, for the number ...

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**1**answer

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### how many consecutive integers $x$ can make $ax^2+bx+c$ square ?

The following problem was raised in a Mathlinks thread:
If $a,b,c\in\mathbb Z$ such that $a\ne0$ and $b^2-4ac\ne 0$, for how many consecutive integers $x$ can $ax^2+bx+c$ ba a perfect square ?
The ...

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### Special arithmetic progressions involving perfect squares

Some time ago the following rather easy problem appeared in an online publication called "Problems in Elementary NT" by Hojoo Lee:
Prove that there are infinitely many positive integers $a$, $b$, $c$ ...

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**1**answer

231 views

### A good introduction to S unit equations

I was looking up some stuff when I stumbled across S unit equations. It seems to me that they are quite helpful in number theory, as given in this paper.
...

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**1**answer

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### On a remark of Tait on FLT for the exponent 3

This is one of those recreational questions that aren't really about research. I found a curious remark in an old volume of American Mathematical Monthly (1922) which I'll quote below:
In the ...

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915 views

### Impossible Heronian Triangles (Ratio of 2 Sides)

There is no Heronian triangle (or simply consider triangles on an integer lattice
which also have integer side lengths) for which one side is half the length of
another side. What other "side-side ...

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**0**answers

191 views

### is exponential diophantine over Qp

Thanks to Matiyasevic, we all know that exponential is diophantine over the integers. Also, thanks to transcendental number theory, we know that exponential is not diophantine over the rationals. So ...

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### Permission to use Online Notes

Hello,
I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes ...

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### Can the difference of two distinct Fibonacci numbers be a square infinitely often?

Can the difference of two distinct Fibonacci numbers be a square infinitely often?
There are few solutions with indices $<10^{4}$ the largest two being $F_{14}-F_{13}=12^2$ and ...

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**1**answer

342 views

### Explicit solutions of C(n,2)=x^2 ? [closed]

"On a Diophantine Equation" paper of Erdös, at some point it is said that it is well known that $C(n,2)=x^2$ has infinitely many integer solutions. I am just wondering the formula generating all ...

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343 views

### Determining the exceptional set in the theorem of Ax & Kochen

Ax & Kochen [1] proved that for every $d\in\mathbb{N}$ there exists a finite set $A(d)$ such that for every prime $p\not\in A(d),$ every homogeneous polynomial of degree $d$ over $\mathbb{Q}_p$ in ...

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### The “universal” diophantine equation

There is a diophantine equation in some number (I think the minimum is now 9) of variables, that can be used to represent
All other diophantine equations (could be wrong on this)
Any particular set ...

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802 views

### non negative integer solutions : Diophantine Equations [closed]

I want to know the exact number of non-negative integer solutions of a1x1+a2x2+...akxk = n ...
I know that it is the co-efficient of x^n in (1-x^a1)^-1 * (1-x^a2)^-1 * ... (1-x^ak)^-1 ...
but whats ...

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

Let $S\subset \overline{\mathbf{Q}}\subset \mathbf{C}$ 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 ...

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### Which Diophantine equations can be solved using continued fractions?

Pell equations can be solved using continued fractions. I have heard that some elliptic curves can be "solved" using continued fractions. Is this true?
Which Diophantine equations other than Pell ...

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### Existence of a non-trivial zero (in the rational cyclotomic field) of a form

It is well known that if a field K is quasi-algebraically closed (i.e. all forms with coefficients in K of degree d in n > d variables have a non-trivial zero in K) then it has no central divison ...

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489 views

### For any $n$, does there exist a number field with at least $n$ solutions to the unit equation

Let $n$ be a positive integer.
Does there exist a number field $K$ such that the number of solutions of the unit equation $$a+b =1, \quad a,b\in O_{K}^\ast$$ is at least $n$? Can we write down such a ...

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458 views

### Linear diophantine equation in n variables

Let n>3. Is there any way to generate all integer solutions of linear diophantine equation in n variables, or at least to determine number of such solutions?
Thanks in advance.

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### Diophantine $x^p+y^q=(x+y)^r$

Is the equation:
$$x^p+y^q=(x+y)^r$$
in integers $x,y,z,p,q,r$ with $p \geq 2,q \geq 2, r \geq 2$ complete solved?
For $(p,q,r)=(n,n,n+1)$ a parametrization is $t=1-s$ and $ ...

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626 views

### Is there a section disjoint from 0, 1 and infinity on the projective line

Let $K$ be a number field with ring of integers $O_K$. Is there a section of $\mathbf{P}^1_{O_K}$ over $O_K$ whose image is disjoint from $0$, $1$ and $\infty$? If $K=\mathbf{Q}$ this is not possible ...

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401 views

### Is there a two-variable prime-representing polynomial (in the sense of Jones-Sato-Wada-Wiens)?

In the math.se question Proof of no prime-representing polynomial in 2 variables, Alon Amit asks if Ribenboim's claim that a prime-representing polynomial (a Diophantine polynomial in which the ...

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381 views

### Torsion points of abelian varieties in the perfect closure of a function field

The following is a problem, which was recently brought to my attention by H. Esnault and A. Langer.
Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ...

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**3**answers

1k views

### Reduction from factoring to solving Pell equation

The paper Polynomial-Time Quantum Algorithms for Pell's Equation and the Principal Ideal Problem claims
There are reductions from factoring to solving Pell’s equation, and from solving Pell’s
...

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318 views

### Bilinear system of Diophantine Equations

$\forall i,j \in$ $\{$$1, \cdots,n$$\},$ let $x_{i},y_{i}$ be unknowns and $n_{ij} \in \mathbb{Z}$ with $i \le j$ be the knowns.
Consider the following $\frac{n(n+1)}{2}$ with $n > 2$ ...

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367 views

### System of Diophantine equations

$p + p' = m$
$q - q' = n$
$pp' = qq'$
$(m^{2} + n^{2})\equiv1\pmod 4$ and $n^{2}\equiv0\pmod 4$.
Only $m,n$ are known in the above. Are there any known techniques to guess the values of $p$ and ...

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543 views

### algorithm for solving systems of linear Diophantine inequalities

So, I posted on stack overflow looking for a reasonably fast algorithm to solve systems of linear Diophantine inequalities and was pointed to this article by Cheng-Zhi Gao and Yu-Lin Dong. The problem ...

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**1**answer

273 views

### Coprime integer solutions to $ \frac{x^n \pm y^n}{x \pm y}=z^m $ with $n>5 , m>1$

Are there coprime integer solutions to:
$$ \frac{x^n \pm y^n}{x \pm y}=z^m $$
with $n>5 , m>1$ and excluding $z=0$?
I suppose the abc conjecture implies finitely many solutions.

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618 views

### Are there any solutions to $\frac{3^n - 2^n}{2^k-3^n} = N$

Are there any solutions to $\frac{3^n - 2^n}{2^k-3^n} = N$ for $n$, $k$, $N$ $\in\mathbb{N}$, greater than 2.
This is related to a previous answered question: Are there any solutions to $2^n-3^m=1$
...

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### Integral roots to degree $d$-forms in four variables inside a box

Hi,
After the response to the following question, Rational roots to quadratic forms in 4 variables, I am now considering the following question.
Let $d \geq 2$ be a positive integer, and suppose ...

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613 views

### Rational roots to quadratic forms in 4 variables

Hi,
I am interested in the following question. Let $F(x_1, x_2, x_3, x_4)$ be a quadratic form in four variables with integer coefficients. Let $B > 0$ be a parameter. Define $N_1(F,B)$ to be the ...

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289 views

### Quartic case of a theorem of Bombieri and Pila

I am interested in the ternary case of a theorem of Bombieri and Pila, in E. Bombieri and J. Pila, "The number of integral points on arcs and ovals", Duke Mathematical Journal., 59 (1989), 337-357. ...

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833 views

### A heuristic for the density of solutions to Diophantine equations

Let $f\in\mathbb{Z}[X_1,\ldots,X_n]$ be a Diophantine equation which, for the purposes of this question, I will assume is homogeneous and nonsingular on $\mathbb{R}^n\setminus\{0\}$ (so that $\nabla ...

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262 views

### Exponential Diophantine $\prod x_i^{e_{x_i}} + \prod y_i^{e_{y_i}} = \prod z_i^{e_{z_i}}$ ,$e_{x_i}>3,e_{y_i}>3,e_{z_i}>3$

Does the exponential diophantine equation
$$ \prod x_i^{e_{x_i}} + \prod y_i^{e_{y_i}} = \prod z_i^{e_{z_i}}$$
with $x_i, y_i,z_i$ coprime and $e_{x_i}>3,e_{y_i}>3,e_{z_i}>3$ have ...

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266 views

### A question on M. Mignotte's Paper: “Petho's Cubics”

I have been reading the paper "Petho's Cubics" by M. Mignotte (appeared in Publ. Math. Debrecen, 56/3-4 (2000)) unfortunately I don't understand section 4 of the paper. I am hoping someone could help ...

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### Why can Diophantine equations represent exponential growth?

The wikipedia page on Matiyasevich's theorem challenges:
Unfortunately there seems to be as yet no short intuitive explanation as to why Diophantine equations can represent exponential growth only ...

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2k views

### General integer solution for $x^2+y^2-z^2=\pm 1$

How to find general solution (in terms of parameters) for diophantine equations
$x^2+y^2-z^2=1$ and $x^2+y^2-z^2=-1$?
It's easy to find such solutions for $x^2+y^2-z^2=0$ or $x^2+y^2-z^2-w^2=0$ or ...