**0**

votes

**1**answer

228 views

### A three variable linear diophantine promise problem

Given $a,b,c,s\in\Bbb N$ such that $(a,b,c)=1$ with promise that we have at most one triple $x,y,z\in\Bbb N$ such that $ax+by+cz=s$, what is a good algorithm that runs in $O(\log(abcs))$ time to find ...

**2**

votes

**0**answers

96 views

### Possible argument against Height bound hypothesis

From this paper.
$f(x,y)$ is polynomial with integer coefficients.
$s(f)$ is its size, the sum of the logarithms of the absolute
values of the nonzero coefficients, defined on p. 6. From p. 7.
...

**2**

votes

**1**answer

361 views

### Quadratic Diophantine equation in $\mathbb Z[T]$

I am trying to solve the following quadratic diophantine equation in $\mathbb Z[T]$:
$$((T+1)X+TY-1-Z)((T+1)X+TY-1+Z)=24XY$$
One has the following trivial solutions:
$(X,Y,Z)=(0,Y,\pm(1-TY))$, $(X,0,\...

**2**

votes

**1**answer

191 views

### Finite sequences which happen to be the sequence of orders of elements of a simple group

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation
$$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$
where $\phi$ is the Euler's totient function, $d$ ...

**6**

votes

**4**answers

278 views

### Application and usage of representation of integers as sum of powers?

We know that there are many articles and manuscripts from the ancient to date talking about representation of integers as sum of squares, cubes etc. I would like to know what is it the usage and ...

**2**

votes

**0**answers

133 views

### Egyptian fractions similar to Erdos-Straus conjecture

It is known that the Erdos-Straus conjecture is about writing $4/n$ as three unit fractions.
My question is whether it is known that if $a>4$
$$
\frac an=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k}
$...

**1**

vote

**1**answer

329 views

### Particular case of Beal's Conjecture

Is it known that there exist no coprime positive integers $A$, $B$ and $C$ such that $A^3+B^4=C^3$? This is a particular case of Beal's Conjecture.

**3**

votes

**4**answers

511 views

### Integral points on a particular family of curves

This is a follow-up to this question (and comments thereon). Namely, it follows from Felipe Voloch's comment that for any $n>2$ there is a finite set of integral $(x, y),$ such that
$$
\prod_{i=1}^...

**0**

votes

**1**answer

137 views

### Quadratic diophantine equation in $\mathbb C[T]$

I am trying to solve the following quadratic diophantine equation in $\mathbb C[T]$, but I did not manage. I hope someone could give some hints or solutions to my problem. Here is the equation
$$\...

**2**

votes

**0**answers

186 views

### Can estimate upper bound of $|p_{i}|$ or $|q_{i}|?$

when I Find the diophantine-equation rational points $$2y^2=x^6-x^2+2$$
I using Faltings's theorem showed that there are only finitely many solutions,if we assmue that
$(x_{i},y_{i}),i=1,2,\cdots,N$...

**14**

votes

**4**answers

840 views

### Number of $\mathbb F_p$ points constant mod $p$?

I have some affine varieties $X$ defined over $\mathbb Z$, and associated integers $c(X)$, with the property that $\# X_{\mathbb Z/p} \equiv c(X) \bmod p$ for all $p$. (In particular $c(X)$ is usually ...

**7**

votes

**2**answers

1k views

### How to prove that this equation has only one solution?

I can't find a way to prove that the following equation has only one solution :
$$
X = \frac{2^Q - 1}{2^{P+Q} - 3^P}
$$
with $X,P,Q$ integers $> 0$.
One trivial solution is $X = 1, P = 1, Q = 1$....

**2**

votes

**2**answers

303 views

### Is it possible that $(a,b,c)$, $(x,y,a)$, $(p,q,b)$ are Pythagorean triples simultaneously? [closed]

Do there exist postive integers $a,b,c,x,y,p,q$ such $(a,b,c)$, $(x,y,a)$, $(p,q,b)$ are all Pythagorean triples? That is, does the system
$$\begin{cases}
a^2+b^2=c^2\\
x^2+y^2=a^2\\
p^2+q^2=b^2
\end{...

**4**

votes

**2**answers

984 views

### Find all rational solutions of this diophantine-equation?

Now, today, my friend tell me this problem was posted by American Mathematical Monthly (Vol. 111, No. 2 Feb., 2004), p. 165 by Wu wei Chao ,and It is said that this problem is unsolved, until now. ...

**6**

votes

**2**answers

654 views

### When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$?

When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$ ($n$ being an integer) , i .e., when does $(-1+\sqrt[3]{2})^n$ not have a non-zero term in $\sqrt[3]{4}$. As you might have noticed, I'm ...

**1**

vote

**1**answer

170 views

### How to solve a quadratic diophantine equation [closed]

I'm trying to solve $y^2=3x^2+3x+1$ for integers, which transforms into $(2y)^2-3(2x+1)^2=1$. I know how to solve pell's equation, but how can we extract only (odd,even) pair from the solutions of the ...

**0**

votes

**2**answers

163 views

### Special type Diophantine equations with integer solutions

The following problem on Diophantine equation is still solved or not I don't know. However, I found few solutions by trail and error method.
Problem: $X^2 - X = Y^5 - Y$ has integer solutions or not? ...

**14**

votes

**4**answers

972 views

### Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$

Two years ago, I made a conjecture on stackexchange:
Today, I tried to find all solutions in integers $a,b,c$ to
$$(1-a^2)(1-b^2)(1-c^2)=8abc,\quad a,b,c\in \mathbb{Q}^{+}.$$
I have found some ...

**2**

votes

**1**answer

108 views

### Sufficient condition for solvability of linear diophantine system

I would like to know under what conditions does an integer solution exist to the under-determined linear system:
Ax = b. (without constraints)
Where A is m x n matrix with positive integers entries (...

**15**

votes

**2**answers

880 views

### Origin of the term “Diophantine equation”

It seems that the term "Diophantine equation" has been around at least since the second half of the 19th century, since the historian Hermann Hankel writes (polemically) in the chapter on Diophantus ...

**13**

votes

**1**answer

580 views

### Algorithmic (un-)solvability of diophantine equations of given degree with given number of variables

Question: For which $d, k \in \mathbb{N}$ is there an algorithm to determine
whether a polynomial diophantine equation
$$
P(x_1, \dots, x_k) = 0, \ \ \ P \in \mathbb{Z}[x_1, \dots, x_k]
$$
...

**4**

votes

**1**answer

185 views

### Numbers represented by inhomogeneous forms

I have a family of Diophantine equations that I am trying to solve, and I am trying to figure out what methods could be used to prove existence of solutions. Unfortunately, the equations are ...

**2**

votes

**3**answers

716 views

### A Diophantine equation with prime powers

Let $p$ and $q$ be prime numbers such that $p^2+p+1=3q^a$: is it true that $a=1$?
This specific equation appears when computing order components of finite groups.

**5**

votes

**1**answer

242 views

### Congruence properties of $x_1^6+x_2^6+x_3^6+x_4^6+x_5^6 = z^6$?

(This was posted previously in MSE without getting any answers.)
It is known that given primitive (co-prime) integer solutions to,
$$x_1^4+x_2^4+x_3^4+x_4^4 = z^4$$
then there is one $x_i$ such ...

**11**

votes

**1**answer

602 views

### On a result attributed to W. Ljunggren and T. Nagell

I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation
$$\frac{x^{n}-1}{x-1} = y^{2}$$
doesn't admit solutions in ...

**3**

votes

**0**answers

74 views

### Low height integer points on a rank variety

Let $M_i$ be fixed rectangular matrices with integer coefficients less than $n$. Consider the variety defined by the condition
$$
\mathrm{rank}(\lambda_1M_1 + \lambda_2M_2 + ... + \lambda_kM_k) = 1.
...

**14**

votes

**4**answers

836 views

### Random Diophantine polynomials: Percent solvable?

Suppose one generates a random polynomial
of degree $d$ with integer coefficients
uniformly distributed within $[-c_\max,c_\max]$.
For example, for
$d=8$, $|c_\max|=100$, here is one random polynomial:...

**12**

votes

**0**answers

569 views

### Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system,
$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$
$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$
and define the function,
$$F(s_m) = x_1+x_2+\dots+x_n$$
For $n\geq3$, using an ...

**9**

votes

**4**answers

482 views

### Nontrivial solutions for $\sum x_i = \sum x_i^3 = 0$

For $x_i \in \mathbb{Z}$, let $\{x_i\}$ be a fundamental solution to the equations:
$$
\sum_{i= 1}^N x_i = \sum_{i=1}^N x_i^3 = 0
$$
if $x \in \{x_i\} \Rightarrow -x \notin \{x_i\}$.
For instance, a ...

**7**

votes

**0**answers

216 views

### Integer solutions of $x^2=4+8y^2+13z^2$

I have been looking for integer solutions of certain Diophantine equations, one of the simplest examples being
$x^2=4+8y^2+13z^2$.
The ideal answer would be a way to parametrize all the integer ...

**8**

votes

**1**answer

305 views

### Integers $d$ for which the Negative Pell equation is soluble for both $d$ and $2d$?

Let $\text{NPE}_d$ denote the negative Pell equation:
$$ x^2-dy^2=-1$$
Where $d$ is a given positive nonsquare integer and integer solutions are sought for x and y.
we know that (in this paper):
...

**8**

votes

**3**answers

370 views

### Deciding a quadratic diophantine equation

Given $a,b\in\Bbb Q_+$, is there an easy way to decide if $$S_{a,b}=\{(x,y)\in\Bbb Z^2:ax^2 + by^2=1\}=\emptyset?$$
I am more interested in seeing if there is a quick way to test for case when ...

**3**

votes

**0**answers

196 views

### Diophantine equations and the numbers $4,7,8$

Consider the diophantine equation
$$
x^n+y^n+z^n=k\cdot xyz,
$$
where $n,x,y,z$ are positive integers. Now consider $k\in\left\{4,7,8\right\}$. It is well-known or easily provable that for $n=1$ and $...

**1**

vote

**3**answers

165 views

### Decidability of sum of powers exponential diophantine equation

I want to ask about decidability of exponential Diophantine equation:
$z_12^{\eta_1} + \ldots + z_n2^{\eta_n} = z$, where $z_i,\eta_i,z \in \mathbb{Z}$, and $\eta_i$ - are variables.
Can we find ...

**8**

votes

**2**answers

649 views

### Sum of consecutive cubes

I'm investigating when the sum of $n$ consecutive cubes equals a cube, i.e., for which $n$ does
$$\sum_{i=0}^{n-1} (k+i)^3 = k^3 + (k+1)^3 + \cdots + (k+n-1)^3 = Y^3 $$
have nontrivial solutions $(k,...

**24**

votes

**5**answers

2k views

### Parametric solutions of Pell's equation

Given a positive integer $n$ which is not a perfect square, it is well-known that
Pell's equation $a^2 - nb^2 = 1$ is always solvable in non-zero integers $a$ and $b$.
Question: Let $n$ be a ...

**-7**

votes

**1**answer

249 views

### Proof of a cubic equation problem [closed]

Well I was doing some questions and i found something. This equation
$x^3+y^3+z^3=w^3$
has only one solution which is
$x=3,y=4,z=5,w=6$.
And what I have have proposed is that there is not other ...

**7**

votes

**2**answers

376 views

### Decidability of diophantine equation in a theory

Given a theory $T \subseteq \operatorname{Th}(\mathbb{N})$, define the decision problem $D_T$ as follows:
Given a polynomial $p$ with integer coefficients and variables $\bar{x}$, decide whether
...

**1**

vote

**0**answers

486 views

### Is $ a^2b^2-ab-ap$ a perfect square for suitable $a,b\in \mathbb{Z}^+$?

Consider the expression
$$
a^2b^2-ab-ap\qquad (a,b\in \mathbb{Z}^+),
$$
where $p\equiv1\pmod{4}$.
Question. For every prime $p\equiv1\pmod{4}$ do there exist $a,b\in\mathbb{Z}^+$ such that $b\...

**6**

votes

**1**answer

684 views

### The elliptic curve for $x_1^9+x_2^9+\dots+x_6^9 = y_1^9+y_2^9+\dots+y_6^9$

I. Theorem: "If there are $a,b,c,d,e,f$ such that,
$$a+b+c = d+e+f\tag1$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag2$$
$$3u^3-3uv+w=-def\tag3$$
where $u=a+b+c,\; v = ab+ac+bc,\;w = abc$, then,
$$(a + u)^k +...

**1**

vote

**1**answer

229 views

### Diophantine equation with factorials

Find all natural solutions $m!=a^2n!$ .
It's clear that $m>=n$.
When $m=n$ we have solutions $(1,m,m)$.
When $m=n+1$ we have solutions $(a,a^2,a^2-1)$.
I think that when $m>n+1$ we have no ...

**1**

vote

**1**answer

173 views

### basis of the lattice generated by the integer points inside a subspace of R^L

Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{...

**5**

votes

**2**answers

261 views

### Density of multi-grade solutions to $x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k$ for $k = 5$ or $6$?

Given the Diophantine equation,
$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$
there is the rather curious observation that the smallest positive solutions for $k=5$ or $6$ is multi-grade.
$$24^k+...

**3**

votes

**0**answers

261 views

### On 7th and 8th powers for $x_1^k+x_2^k+x_3^k+x_4^k = y_1^k+y_2^k+y_3^k+y_4^k$

The Diophantine equation,
$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$
for either $k=5$ or $6$ is quite well explored, and it has long been known that it has an infinite number of primitive ...

**24**

votes

**1**answer

1k views

### $x^4+y^4$ powerful for relatively prime $x,y$

I asked this question on the NMBRTHRY mailing list on
17 February 2014, but it remains unsolved as far as I know.
Recall that a "powerful
number" is a positive integer whose prime factorizations
$m = ...

**5**

votes

**1**answer

328 views

### On the elliptic curve $x(x+a^2)(x+b^2) = y^2$

Ajai Choudhry showed that special cases of the elliptic curve,
$$x(x+a^2)(x+b^2)=y^2\tag1$$
can be used to prove that,
$$u_1^7+u_2^7+\dots + u_9^7 = 0\tag2$$
has an infinite number of primitive ...

**13**

votes

**2**answers

1k views

### More elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?

(Note: See also the $a^4+b^4+c^4 = 1$ version in this old MSE post.)
The equation discussed in a paper by Jacobi and Madden,
$$a^4+b^4+c^4+d^4 = (a+b+c+d)^4 = z^4\tag1$$
or equivalently,
$$(p-2q + ...

**3**

votes

**2**answers

219 views

### Sets of squares representing all squares up to $n^2$

Let $S_n=\{1,2,\ldots,n\}$ be natural numbers up to $n$.
Say that a subset $S \subseteq S_n$
square-represents $S_n^2$ if every
square $1^2,2^2,\ldots,n^2$ can be represented by adding or subtracting
...

**37**

votes

**4**answers

2k views

### Why do Pell equations appear in Ramanujan's pi formulas?

While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula.
I. Given the fundamental unit $\...

**7**

votes

**1**answer

597 views

### rational points of a hyperelliptic curve

I have the following hyperelliptic curve of genus $2$:
$$
y^2 = 561 x^6 - 41904 x^5 + 627264 x^4 + 11860992 x^3 - 197074944 x^2 + 124416^2
$$
I need to find all the rational points on this curve. ...