The diophantine-equations tag has no wiki summary.

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### Integer solution to the equation [migrated]

Does there exists an integer solution (for every integer $m\geq 1$) for the following equation?
$$x_1x_2...x_n+(2y+1)z+y=4m+3$$
where, $1\leq x_1\leq x_2\leq...\leq x_n\leq l$,$0\leq y \leq ...

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92 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):
...

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**3**answers

250 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 ...

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**0**answers

128 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 ...

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vote

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121 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 ...

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**2**answers

569 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 ...

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**5**answers

1k 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 ...

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votes

**1**answer

193 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 ...

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241 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
...

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**0**answers

434 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 ...

**5**

votes

**1**answer

522 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. 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 + (b + ...

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vote

**1**answer

174 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 ...

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**1**answer

118 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, ..., ...

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227 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.
...

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223 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 ...

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**1**answer

956 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 = ...

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**1**answer

287 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 ...

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**1**answer

746 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 + ...

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**2**answers

210 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
...

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### 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,
...

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**1**answer

354 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. ...

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### Fermat's last theorem over larger fields

Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here ...

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**1**answer

207 views

### The number of solutions of a Diophantine equation [closed]

Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity.
That is, I am asking whether the number ...

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191 views

### Diophantine equations over cyclotomic fields

Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...

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### How many Pythagorean triples are there in which every member is triangular?

How many Pythagorean triples $(a,b,c)$ are there such that $a, b$ and $c$ are triangular?
Any two solutions with only $a$ and $b$ interchanged are considered equivalent.
The question of existence ...

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135 views

### The Diophantine equation $x^2 + bxy + cy^2 = p^z_1 \cdots p^{z_k}$

Let $b,c \in \mathbb{Z}$ and let $p_1,\ldots,p_k$ be given primes. Is there an effective algorithm to find all the solutions of the Diophantine equation $$x^2 + bxy + cy^2 = p_1^{z_1} \cdots ...

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411 views

### Pairs of quadratic polynomials taking values pairs of consecutive squares

Let $f,g \in \mathbb{Z}[x]$ be quadratic and neither square.
For $x,y,z \in \mathbb{Z}$ what is the maximal number
of solutions to $f(x)=z^2,g(y)=(z+1)^2$?
Solutions are integral points on the genus ...

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171 views

### Monte Carlo variant of Hilbert's Tenth Problem

Let $k \in \mathbb{N}$. Given an algorithm $\mathcal{A}$ which takes
as argument a polynomial $P \in \mathbb{Z}[x_1,\dots,x_k]$ and either
returns true or false, we say that $\mathcal{A}$ works for ...

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**1**answer

215 views

### Schoenberg's Rational Polygon Problem

"A polygon is said to be rational if all its sides and diagonals are rational, and I. J. Schoenberg has posed the difficult question, ‘Can any given polygon be approximated as closely as we like by ...

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474 views

### solutions to special diophantine equations [closed]

Let $0\le x,y,z,u,v,w\le n$ be integer numbers obeying
\begin{align*}
x^2+y^2+z^2=&u^2+v^2+w^2\\
x+y+v=&u+w+z\\
x\neq& w
\end{align*}
(Please note that the second equality is ...

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210 views

### polynomials in many variables and Hasse principle

I was wondering whether there exists any result of the form
"if $f \in \mathbb{Z}[x_1, ..., x_k]$ is a polynomial (not form! I don't require homogeneity) of total degree $n$, with $k \geq \delta ...

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329 views

### A congruence conjecture regarding $(r-s)^4-1 \equiv 0\!\pmod{4r^2s}$

Is the following conjecture true?
Conjecture. If $r > s \ge 1$ are relatively prime integers such that
\begin{equation}
(r-s)^4-1 \equiv 0\!\pmod{4r^2s}, \tag{1}
\end{equation}
then $r-s = 1$ ...

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601 views

### Does the Diophantine equation $(x^2+ay^2)(u^2+bv^2) = p^2+cq^2$ admit a complete solution?

In this MSE question/thread, I have been discussing the equation
$$
(x^2+ay^2)(u^2+bv^2) = p^2+cq^2, \tag{$\star$}
$$
where $x,a,y,u,b,v,p,c,q$ are integers. I posed a conjecture which turned out to ...

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### Failing of heuristics from circle method

The heuristic from circle method for integral points on diagonal cubic surfaces $x^3+y^3+z^3=a$ ($a$ is a cubic-free integer) seems to fit well with numerical computations by ANDREAS-STEPHAN ELSENHANS ...

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### Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^2-1)$

Is the following conjecture correct?
Conjecture. The divisibility condition $(\alpha+\beta)^2 \mid (2\beta^3+6\alpha\beta^2-1)$ has no solutions in positive integers $1 \le \beta < \alpha < ...

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### Superelliptic Curves [duplicate]

I'm trying to find information on superelliptic curves and how to solve them over the integers. The equation is $$y^k = f(x)$$ where $k=3$ and $f$ has degree $d=3$. Does anyone know any ...

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275 views

### The Diophantine equation $x^p - 4y^p = z^2$

If $p \geq 5$ is a prime, are there any integers $x, y, z > p$ such that
$(x, y) = 1$
and
$$x^{p} - 4y^{p} = z^{2}$$

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### When is a cubic polynomial a cube? [closed]

I've been researching cubes and I'm trying to solve this Diophantine equation over the integers.
$$ax^3 + bx^2 + cx + d = y^3$$where a, b, c, d are parameters for a given $n$. For example, for $n = ...

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### If $~(c - b) ^ 2 + 3cb = a^3~$ has nonzero integer solutions, then $~(a,c) \gt 1~$ or $~(b,c) \gt 1$? [closed]

If $~(c - b) ^ 2 + 3cb = a^3~$ has nonzero integer solutions, then $~(a,c) \gt 1~$ or $~(b,c) \gt 1$?
I think this is true, how to prove this?

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446 views

### The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with ...

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**1**answer

284 views

### Integer points on $y^2=x^2-x^3+x^4$

Does the Diophantine equation $y^2=x^2-x^3+x^4$ have solutions other than
$x=1,y=1$? Interestingly, the Diophantine equation $y^2=x^2-x^3+x^5$ has such solutions: $x=3,y=15$, $x=5,y=55$, ...

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986 views

### Not-lonely runners

The lonely runner conjecture
has several formulations.
They all involve a number $n$ runners running on a circular track,
each with a different speeds, and the conjecture is that each runner is ...

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309 views

### Are there any integers which can't be written as a sum of two fourth powers minus a cube?

To be precise, I am asking:
Does there exist an integer $k$ such that there do not exist (possibly negative) integers $x,y,z$ satisfying $x^4+y^4=z^3+k$?
Heuristically the answer must be yes, in ...

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**1**answer

502 views

### Some types of diophantine equations and their decidability

The MDRP theorem – which answers Hilbert's tenth problem in the negative – says:
There is no algorithm for determining whether an
arbitrary diophantine equation has a solution.
In ...

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**1**answer

469 views

### Hyperrectangles with integer diagonals

What is the largest value of $n$ for which there exists $n$ (not necessarily distinct) complete squares of natural numbers such that the sum of every subset of it is also a complete square? ( For ...

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**0**answers

136 views

### Equation in the Gaussian Integers

Let $a,b \in \mathbb{N}$. Is there a possibility to characterize the solutions of $a N(\alpha) - b N(\beta)=1$ where $\alpha,\beta \in \mathbb{Z}[i]$? In particular I am interested in the case $a=1$ ...

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**1**answer

146 views

### Link between integral points on varieties and solutions to Diophantine equations

Let $k$ be a number field, $S$ a finite set of places of $k$ including the infinite ones and $F(X_1,\dots,X_n)$ a polynomial in $k[X_1,\dots,X_n]$.
I am looking for notes, books or surveys detailing ...

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**1**answer

587 views

### Can we extend the proof of Catalan's conjecture?

What is it, in Mihailescu's proof of Catalan conjecture, that uses explicitly the fact that there is a 1 on the right hand side of $x^p - y^q = 1$? In other words, why can't we extend his argument to ...

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### Enumerating solutions to an underdetermined non-homogenous linear system of Diophantine equations

I have a large, under-determined system (60 equations and 116 unknowns) of linear Diophantine equations. I am aware of the algorithms typically used to solve these systems, which is not my question.
...

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133 views

### n-ary quadratic forms with $S$-integer values

Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.
Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to ...