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31
votes
1answer
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
0answers
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
5answers
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
2answers
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
7answers
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
3answers
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
5answers
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
11answers
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
4answers
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
3answers
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
2answers
800 views

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

Is anybody know a solution of this problem? (Sorry, correct question is here.)
0
votes
1answer
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
3answers
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
1answer
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
1answer
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
2answers
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
2answers
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
3answers
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
0answers
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
2answers
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
3answers
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
3answers
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
0answers
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
1answer
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
3answers
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
2answers
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
3answers
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
1answer
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
3answers
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
3answers
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
3answers
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
2answers
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
3answers
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
1answer
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
2answers
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
3answers
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
2answers
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
2answers
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
3answers
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
0answers
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
4answers
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
0answers
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 $X^2-PY^2=k$ ...