The diophantine-equations tag has no wiki summary.

**31**

votes

**1**answer

2k views

### Infinitely many solutions of a diophantine equation

If $P(x,y,...,z)$ is a polynomial with integer coefficients then every integer solution of $P=0$ corresponds to a homomorphism from $\mathbb{Z}[x,y,...,z]/(P)$ to $\mathbb{Z}$. So there are infinitely ...

**3**

votes

**0**answers

320 views

### The surface $ x^2 y^2 + 1 = (x^2 + y^2) z^2 $

Hi, I'm trying to find all rational points on the surface of the title, in connection with the Euler Brick (AKA Rational Box) problem.
This surface is equivalent to $ x^2 z^2 - 1 = (x^2 - z^2) y^2 $, ...

**7**

votes

**5**answers

3k views

### Methods for solving Pell's equation?

It is known that the minimum solution of Pell's equation $x^2-dy^2=\pm1$ can be found from the continued fraction expansion of $\sqrt d$. Are there other methods for finding the minimum (or any other) ...

**3**

votes

**2**answers

350 views

### Decision Procedure for Inequalities between Homogeneous Polynomials

Given two polynomials $p_1$ and $p2$ each of which is a multi-variate polynomial with positive integer coefficients, we want to decide if $p_1 \leq p_2$ over all integral values of the variables.
The ...

**14**

votes

**7**answers

3k views

### Diophantine equation with no integer solutions, but with solutions modulo every integer

It's probably common knowledge that there are Diophantine equations which do not admit any solutions in the integers, but which admit solutions modulo $n$ for every $n$. This fact is stated, for ...

**6**

votes

**3**answers

705 views

### English translation of Voronoi's dissertation

I am looking for an English translation of Voronoi's doctoral dissertation, "On a generalization of the Algorithm of Continued Fractions." I can only find it in the original Russian.

**11**

votes

**5**answers

1k views

### Analysis of a quadratic diophantine equation

Hi! This is my first post on Math Overflow. I have two equations: $a(3a-1) + b(3b-1) = c(3c-1)$ and $a(3a-1) - b(3b-1) = d(3d-1)$. I'm trying to find properties of $a$ and $b$ that lead to solutions, ...

**51**

votes

**11**answers

4k views

### Why certain diophantine equations are interesting (and others are not) ?

It is quite clear why certain differential equations, among the jungle of possible diff equations that is possible to conceive, are studied: some come from physical problems, or from "spontaneous" ...

**12**

votes

**4**answers

2k views

### Is there an elementary way to find the integer solutions to $x^2-y^3=1$?

I gave this problem to my undergraduate assistant, as I saw that Euler had originally solved it (although I am having trouble finding his proof). After working on it for two weeks, we boiled the hard ...

**23**

votes

**3**answers

3k views

### Solve in positive integers $n!=m(m+1)$

Is anybody know a solution of this problem? (Sorry, I've missed one summand in the previous post.)

**1**

vote

**2**answers

800 views

### Solve in positive integers $n!=m^2$

Is anybody know a solution of this problem?
(Sorry, correct question is here.)

**0**

votes

**1**answer

349 views

### An asymptotic expression for the solution to the squares problem suggested by statistical mechanics

The $s$ squares problem is to count the number $r_s (n)$ of integer solutions $(x_1,x_2,...,x_s)$ of the Diophantine equation $x_{1}^{2}+x_{2}^{2}+...+x_{s}^{2}=n$ in which changing the sign or ...

**2**

votes

**3**answers

396 views

### 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.

**0**

votes

**1**answer

450 views

### Rational solutions of homogeneous equations

Can every solution of a homogeneous linear system be approximated by a solution in rational numbers?
In mathematical terms: Let $$Ax=0$$ be a homogeneous linear system in $n$ determinates for an ...

**11**

votes

**1**answer

852 views

### Fermat's Bachet-Mordell Equation

Fermat once claimed that the only integral solutions to $y^2 = x^3 - 2$ are $(3, \pm 5)$.
Fermat knew Bachet's duplication formulas (more precisely, Bachet had a formula for computing what we call ...

**24**

votes

**2**answers

834 views

### Trost's Discriminant Trick

The following little trick was introduced by E. Trost
(Eine Bemerkung zur Diophantischen Analysis,
Elem. Math. 26 (1971), 60-61). For showing that a diophantine equation
such as $x^4 - 2y^2 = 1$ ...

**3**

votes

**2**answers

368 views

### Is there any solution to the diophantine equation $1/x^m+1/y^n=1/z^t$ in which x, y, z are not coprime

If $x,y,z$ are coprime, this equation has not any solution with an elementary method. I want to know a solution of this equation when $x,y,z$ are not coprime.

**13**

votes

**3**answers

1k views

### Striking applications of Baker's theorem

I saw that there are many "applications" questions in Mathoverflow; so hopefully this is an appropriate question. I was rather surprised that there were only five questions at Mathoverflow so far with ...

**10**

votes

**0**answers

813 views

### Effective proofs of Siegel's theorem using arithmetic geometry

This is a speculation and perhaps naive. The theorem of Siegel that
There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is ...

**5**

votes

**2**answers

859 views

### Are there consecutive integers of the form $a^2b^3$ where $a$, $b$ > 1?

Let $S$ = { $a^2b^3$ : $a, b \in \mathbb{Z}_{>1}$ }.
Does there exist $n$ such that $n$, $n+1 \in S$?
Motivation: I was thinking about Question on consecutive integers with similar prime ...

**2**

votes

**3**answers

447 views

### Hexagonal Triangular Squares

Is there a hexagonal, triangular, square (apart from 0 and 1)?
In other words, is there a positive integer that is simultaneously
(1) a perfect square, $n^2$, $n \ge 2$,
(2) a triangular number, ...

**3**

votes

**3**answers

647 views

### 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 = ...

**1**

vote

**0**answers

171 views

### Checking local solubility of varieties at “bad” primes

Let $X$ be smooth variety defined over $\mathbb{Q}$. If we want to check that $X$ is locally soluble at a prime $p$, then it suffices to find a non-singular $\mathbb{F}_p$ point, which can be lifted ...

**7**

votes

**1**answer

989 views

### Integer values of a rational function

Suppose we are given a rational function with numerator and denominator being polynomials with integer coefficients. Is there an efficient algorithm for finding all integers arguments at which the ...

**21**

votes

**3**answers

1k views

### Proving non-existence of solutions to $3^n-2^m=t$ without using congruences

I made a passing comment under Max Alekseyev's cute answer to this question and Pete Clark suggested I raise it explicitly as a different question. I cannot give any motivation for it however---it was ...

**6**

votes

**2**answers

1k views

### $3^n - 2^m = \pm 41$ is not possible. How to prove it?

$3^n - 2^m = \pm 41$ is not possible for integers $n$ and $m$. How to prove it?

**6**

votes

**3**answers

784 views

### Is there a solution for the equation x^m-y^n=k in which k > 1?

The Catalan conjecture state that $x^m-y^n=1$ has only the solution $x=3, m=2, y=2, n=3$. This conjecture was proved by Preda Mihailescu in 2004, but I want to know about the equation mentioned above. ...

**4**

votes

**1**answer

984 views

### Solving a Diophantine equation related to Algebraic Geometry, Steiner systems and $q$-binomials?

The short version of my question is:
1)For which positive integers $k, n$ is there a solution to the equation $$k(6k+1)=1+q+q^2+\cdots+q^n$$ with $q$ a prime power?
2) For which positive ...

**9**

votes

**3**answers

6k views

### Status of Beal, Granville, Tijdeman-Zagier Conjecture [closed]

The Beal, Granville, Tijdeman-Zagier Conjecture, i.e.
If $A^x+B^y=C^z$ , where $A, B, C, x, y,z$ are positive integers and $x,
y$ and $z$ are all greater than $2$, then $A, B$ and $C$ must have a ...

**10**

votes

**3**answers

1k views

### 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 ...

**4**

votes

**3**answers

2k views

### Non-negative integer solutions of a single Linear Diophantine Equation

Consider the following linear Diophantine Equation::
ax + by + cz = d ------------ (1)
for all, a,b,c and d positive integers, and relatively prime, and ...

**4**

votes

**2**answers

2k views

### Quadratic Diophantine equations solver

Is there software that helps list small solutions of the Diophantine equation
$$
x_0^2=1+x_1^2+x_2^2+\cdots+ x_n^2
$$
where "small" is negotiable, but e.g. we could fix $x_0$ and and ask for the list ...

**28**

votes

**3**answers

2k views

### A binomial generalization of the FLT: Bombieri's Napkin Problem

This is an extract from Apéry's biography
(which some of the people have already enjoyed in
this answer).
During a mathematician's dinner in
Kingston, Canada, in 1979, the
conversation turned ...

**20**

votes

**1**answer

2k views

### Polynomials with rational coefficients

Long time ago there was a question
on whether a polynomial bijection $\mathbb Q^2\to\mathbb Q$ exists. Only one attempt
of answering it has been given, highly downvoted by the way. But this answer ...

**12**

votes

**2**answers

1k views

### Transforming a Diophantine equation to an elliptic curve

I heard that the following problem lead to determine the rational points of an elliptic curve:
For which integers $n$ there are integers $x,y,z$ such that $x/y+y/z+z/x=n$. Could anyone show me why ...

**7**

votes

**3**answers

721 views

### Is there a solution to the a+b^m=b+c^n=c+a^l for l,m,n >1 and a, b, c distinct odd primes?

Is there a solution to:
$a+b^m=b+c^n=c+a^l$ for l,m,n >1 and a, b, c distinct odd primes?
I've had a play around with specific possible solutions and there are lots of possibilities that may be ...

**5**

votes

**2**answers

1k views

### 4900, a particularly square number

I read in "Letters to a young mathematician" that 4900 is the only square integer that is the sum of consecutive squares (I believe he meant by that "starting from 1", but maybe that's not even ...

**1**

vote

**2**answers

519 views

### Diophantine equation problem

How many positive integer solutions does the equation x^2+y^2+z^2-xz-yz = 1 have? My guess is (1,0,1), (0,1,1) and (1,1,1). What is proof of that fact that there are none other?

**1**

vote

**3**answers

2k views

### Integer points of an elliptic curve

I would like to find those integers $x,y$ that satisfies $y^2=x^3+1$. Is there some elementary way to find those?

**3**

votes

**0**answers

354 views

### Asymptotics related to the Erdos--Moser diophantine equation

I share the authorship of this question with Pieter Moree.
In our recent joint work with Y. Gallot (arXiv:0907.1356 [math.NT]) we attack
the Erdős--Moser diophantine equation
$$
...

**11**

votes

**4**answers

2k views

### hard diophantine equation: $x^3 + y^5 = z^7$

Does the equation $x^3+y^5=z^7$ have a solution $(x,y,z)$ with $x,y,z$ positive integers and $(x,y)=1$? In his book H. Cohen (Number theory,2007) said "[...] seems presently out of reach".
I couldn't ...

**0**

votes

**0**answers

274 views

### When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, a square?

It is easy to show that the following problems are equivalent.
a. When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, and $n$ any integer, a square?
and
b. When is $X^2-PY^2=k$ ...