Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. ...

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Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?

The Diophantine equation $$x^3+y^3+z^3=3$$ has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
14
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3answers
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Which integers can be expressed as a sum of three cubes in infinitely many ways?

For fixed $n \in \mathbb{N}$ consider integer solutions to $$x^3+y^3+z^3=n \qquad (1) $$ If $n$ is a cube or twice a cube, identities exist. Elkies suggests no other polynomial identities are known. ...
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2answers
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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 ...
25
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3answers
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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.)
28
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4answers
3k views

Can the difference of two distinct Fibonacci numbers be a square infinitely often?

Can the difference of two distinct Fibonacci numbers be a square infinitely often? There are few solutions with indices $<10^{4}$ the largest two being $F_{14}-F_{13}=12^2$ and ...
15
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3answers
1k 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 ...
5
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2answers
965 views

The diophantine equation X^2 - Y^2 - Z^2 = +- 1

Hi everybody. I'd like to know if the diophantine equation (1) $$X^2 - Y^2 - Z^2 = \pm 1$$ has been studied and if the set of its solutions $(X,Y,Z)$ is known. I appreciate any reference. Thank you ...
11
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2answers
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On Generalizations of Fermat's Conjecture

We know the following facts: (1) For all $1\leq n\leq 2$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}$ has a solution in $\mathbb{N}$. (2) For all $3\leq n$ the equation ...
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385 views

Solutions of system of diophantine equations

The system of diophantine equations $$\{x^2-y^2+z^2-u^2+q^2-t^2=0,\,xy+zt-uq=0 \}$$ is given. Do the formulas $$x:=(j(p^2-4ps+3s^2)-(p-s)(3p^2-4ps+s^2))k^2+2(j-2(p-s))(p-s)kn+(j-p+s)n^2, $$ ...
9
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1answer
501 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 ...
1
vote
2answers
176 views

Sharply Estimating Pythagorean Triples [closed]

Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$ Is there a way to give a sharp estimate?
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0answers
333 views

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 = ...
38
votes
4answers
3k views

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 ...
25
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2answers
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Does Fermat's last theorem hold in the ordinals?

My question is whether there are no nontrivial solutions in the ordinals of the equations arising in Fermat's last theorem $$x^n+y^n=z^n$$ where $n\gt 2$, and where we use the natural ordinal ...
9
votes
3answers
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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 ...
6
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1answer
2k views

Which types of Diophantine equations are solvable?

Is there a list somewhere of which types of Diophantine equations are solvable, which types are not solvable, and which types are not known to be solvable or not? (When I say solvable, I mean that we ...
36
votes
4answers
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 ...
17
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1answer
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Ramanujan's pi formulas with a twist

Given the binomial function $\binom{n}{k}$. 1. Define the following sequences, $$\begin{aligned} u_1(k) &= \tbinom{2k}{k}\tbinom{3k}{k}\tbinom{6k}{3k} = 1, 120, 83160, 81681600,\dots \\ u_2(k) ...
13
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7answers
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Special arithmetic progressions involving perfect squares

Prove that there are infinitely many positive integers $a$, $b$, $c$ that are consecutive terms of an arithmetic progression and also satisfy the condition that $ab+1$, $bc+1$, $ca+1$ are all perfect ...
10
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1answer
265 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 ...
8
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1answer
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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 ...
30
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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 ...
12
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1answer
743 views

On cubic reciprocity for $x^3+y^3+z^3 = 996$?

I. The Diophantine equation, $$x^3+y^3+z^3 = 3w^3\tag1$$ with $x\geq y \geq z$ and $w=1$ has only two known solutions, namely $1,1,1$ and $4,4,-5$. Are there larger ones? As Noam Elkies points out ...
12
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8answers
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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 ...
10
votes
3answers
574 views

what is the maximum number of rational points of a curve of genus 2 over the rationals

Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.) We are ...
5
votes
5answers
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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 ...
17
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Does the equation $241+2^{2s+1}=m^2$ have a solution?

Let $p$ be a prime congruent to $1$ mod. 8. If $p= 17$ one has : $p+ 8 = 5 ^2$. If $p= 41$ one has : $p+ 8 = 7 ^2$. If $p= 73$ one has : $p+ 8 = 9 ^2$. If $p= 89$ one has : $p+ 32 = 11 ^2$. If ...
9
votes
1answer
502 views

Integer Solutions of $x+y^n = y + x^m$ for $n < m$

I found 8 of them and believe there is no more: $$2+3^2=3+2^3$$ $$2+6^2=6+2^5$$ $$6+15^2=15+6^3$$ $$3+16^2=16+3^5$$ $$3+13^3=13+3^7$$ $$2+91^2=91+2^{13}$$ $$5+280^2=280+5^7$$ $$30+4930^2=4930+30^5$$ ...
9
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3answers
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Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$

I'm working on solving the quartic Diophantine equation in the title. Calculations in maxima imply that the only integer solutions are \begin{equation} (r,s) \in \{(-3, -2), (-2, 3), (-1, 0), (0, ...
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2answers
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The Theory of Transfinite Diophantine Equations [closed]

The theory of Diophantine equations is one of the main stream research areas in number theory. There are many known results and unknown conjectures about the existence of non-trivial solutions for ...
6
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3answers
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Reduction from factoring to solving Pell equation

The paper Polynomial-Time Quantum Algorithms for Pell's Equation and the Principal Ideal Problem claims There are reductions from factoring to solving Pell’s equation, and from solving Pell’s ...
3
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1answer
270 views

Generalizing a pattern for the Diophantine $m$-tuples problem?

A set of $m$ non-zero rationals {$a_1, a_2, ... , a_m$} is called a rational Diophantine $m$-tuple if $a_i a_j+1$ is a square. It turns out an $m$-tuple can be extended to $m+2$ if it has certain ...
17
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2answers
711 views

Lower bounds on the easier Waring problem

The easier Waring problem asks for the least number $v=v(k)$ such that every every integer is a sum of $v$ $k$'th powers with signs, i.e. every $n\in \mathbb{N}$ is of the form $$n=x_1^k\pm ...
12
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2answers
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sum of three cubes and parametric solutions

The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$, i.e., there are finite many polynomial triples ...
12
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0answers
920 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 ...
7
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1answer
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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. ...
5
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1answer
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Integer solutions of x^n + y^n = z^{n-1}

This is related to another question I am interested in the non-trivial integer solutions of $$ x^n + y^n = z^{n-1} $$ for $n \ge 4$. A solution is trivial if $xyz=0$ or $x = \pm y$. There are ...
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1answer
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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 ...
12
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1answer
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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 ...
8
votes
1answer
575 views

how many consecutive integers $x$ can make $ax^2+bx+c$ square ?

The following problem was raised in a Mathlinks thread: If $a,b,c\in\mathbb Z$ such that $a\ne0$ and $b^2-4ac\ne 0$, for how many consecutive integers $x$ can $ax^2+bx+c$ ba a perfect square ? The ...
6
votes
2answers
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$3^n - 2^m = \pm 41$ 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?
4
votes
2answers
965 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. ...
2
votes
2answers
843 views

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

Is anybody know a solution of this problem? (Sorry, correct question is here.)
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1answer
322 views

A good introduction to S unit equations

I was looking up some stuff when I stumbled across S unit equations. It seems to me that they are quite helpful in number theory, as given in this paper. ...
10
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1answer
510 views

Can the sum of two non-zero coprime fifth powers be powerful?

I am wondering if the sum of two non-zero coprime fifth powers can be powerful. There are no small solutions. Q1 Can the sum of two non-zero coprime fifth powers be powerful? Got a partial ...
4
votes
0answers
328 views

$a^5+b^5=c^5+d^5$ and polynomial identities

No nontrivial integer solutions to $$ a^5+b^5=c^5+d^5 \qquad (1)$$ are known. (1) has infinitely many solutions in an extension of $\mathbb{Z}$ (root of $9-15x+37x^2 $ ) resulting from genus 0 curve ...
3
votes
0answers
284 views

quasi periodic continued fractions and powers of e, tanh, tan

It is well known that some transcendental numbers like e.g. rational multiples of $e^{2/n}$ with $n\in\mathbb N $, when written as regular continued fractions (R.C.F.), yield what can be called a ...
2
votes
3answers
709 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.
2
votes
0answers
217 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 ...
1
vote
2answers
340 views

For what integer $n$ are there infinitely many $-a+nb+c = -d+ne+f$ where $a^6+b^6+c^6 = d^6+e^6+f^6$?

(Much revised for clarity.) I was considering the system of equations, $$-a+nb+c = -d+ne+f\tag1$$ $$a+b+c = d+e+f\tag2$$ $$a^2+b^2+c^2 = d^2+e^2+f^2\tag3$$ $$a^6+b^6+c^6 = d^6+e^6+f^6\tag4$$ ...