The diophantine-equations tag has no wiki summary.

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### 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.

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### Probability of correlated residues

Given $N,c\in\Bbb N$, where $c\ll(\log N)^{1/b}$ for any $b>1$ is fixed, what is the probability that given $A_1,A_2,A_3\in\Bbb N$ with ...

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### 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
$$
...

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### 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
...

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

I want to know the exact number of non-negative integer solutions of $a_1 + 2a_2 + \ldots + k \cdot a_k = n$.
I know that it is the co-efficient of $x^n$ in $(1 - x^{a_1})^{-1} \cdot (1 - ...

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### 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
...

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### 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 ...

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### 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 = ...

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### 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. ...

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### 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 ...

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### 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
...

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### 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 ...

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### 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 ...

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### 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? ...

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### 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 ...

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### 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
$$
P(x_1, \dots, x_k) = 0, \ \ \ P \in \mathbb{Z}[x_1, \dots, x_k]
$$
...

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### 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 ...

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### 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.

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### 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 ...

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### Diophantine equation: Egyptian fraction representations of 1

According to the OEIS (A002966) there are 294314 solutions in positive integers to the equation
$$\sum_{i=1}^7\frac{1}{x_i}=1$$ assuming $x_1\leq x_2\leq\cdots\leq x_7$.
Similarly for 8 summands there ...

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### 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 ...

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### 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 ...

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### 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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### 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.
...

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### 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 ...

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### 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 ...

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### Diophantine: a^n + b^n + c^n = d^n and a^n + b^n = c^n + d^n

Let us consider the equation $a^n+b^n=c^n$ for positive integers $a,b,c$ and $n\ge 2$. The $n=2$ case has a well-known and beautiful parametrization known as Pythagorean triples. Fermat's Last Theorem ...

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### 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 ...

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### 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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### 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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### 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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### Is there an algorithm to solve quadratic Diophantine equations?

I was asked two questions related to Diophantine equations.
Can one find all integer triplets $(x,y,z)$ satisfying $x^2 + x = y^2 + y + z^2 + z$? I mean some kind of parametrization which gives all ...

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### 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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### 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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### 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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### 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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### solution of the equation $a^2+pb^2-2c^2-2kcd+(p+k^2)d^2=0$

i am wondering if there is a complete solution for the equation $a^2+pb^2-2c^2-2kcd+(p+k^2)d^2=0$ in which $a,b,c,d,k$ are integer(not all zero) and $p$ is odd prime.

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### Integer polynomials taking square values

Is there a way to determine a formula giving all integer values of $x$ for which the value of a polynomial $P(x)$ with integer coefficients is a square?
That is, is there a closed formula for:
$X = ...

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### 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 ...

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### 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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### Does the following Diophantine equation have nontrivial rational solutions?

Are there any solutions to the equation $s^{2}(1+t^{2})^{2}+t^{2}(1+s^{2})^{2}=u^2$ where $s,t,u\in \mathbb{Q}$ and $0 < s,t<1$? If so, is there a simple way to parametrize them all?
If I am ...

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### 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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### 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 ...

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### 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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### State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$

As far as I understand, both of the Diophantine equations
$$a^5 + b^5 = c^5 + d^5$$
and
$$a^6 + b^6 = c^6 + d^6$$
have no known nontrivial solutions, but
$$24^5 + 28^5 + 67^5 = 3^5+64^5+62^5$$
and
...

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### $x^4+y^4$ 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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### 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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### 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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### 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 ...