**20**

votes

**2**answers

1k views

### 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
$$3^...

**28**

votes

**4**answers

5k views

### 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**

votes

**3**answers

4k views

### 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.
...

**25**

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.)

**29**

votes

**4**answers

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 $F_{13}-F_{11}=12^...

**15**

votes

**3**answers

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 ...

**11**

votes

**2**answers

833 views

### 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 $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}...

**11**

votes

**1**answer

602 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

**2**answers

178 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?

**1**

vote

**0**answers

335 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 $n = 5$...

**37**

votes

**4**answers

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 $\...

**24**

votes

**1**answer

1k 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 = ...

**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 ...

**37**

votes

**4**answers

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 $\...

**25**

votes

**2**answers

1k views

### 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 ...

**6**

votes

**1**answer

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 ...

**17**

votes

**1**answer

1k views

### 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**

votes

**7**answers

2k views

### 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**

votes

**1**answer

277 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 a ...

**8**

votes

**1**answer

1k 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 ...

**30**

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 ...

**5**

votes

**2**answers

999 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 ...

**12**

votes

**8**answers

6k views

### 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

**3**answers

580 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 ...

**17**

votes

**2**answers

3k views

### 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

**1**answer

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**

votes

**3**answers

2k views

### 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, -1),...

**7**

votes

**2**answers

500 views

### 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**

votes

**3**answers

2k views

### 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**

votes

**1**answer

275 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**

votes

**2**answers

717 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 x_2^k\pm\...

**12**

votes

**2**answers

1k views

### 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 $(x(t),y(t),z(...

**12**

votes

**0**answers

939 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**

votes

**1**answer

603 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. ...

**5**

votes

**1**answer

2k views

### 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 ...

**4**

votes

**1**answer

1k 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 ...

**13**

votes

**1**answer

1k 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 $-...

**8**

votes

**1**answer

581 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

**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?

**4**

votes

**2**answers

984 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

**2**answers

849 views

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

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

**1**

vote

**0**answers

178 views

### System of diophantine equations related to Ozanam's problem

Could you please help with finding of general solution of diophantine system for rational a, b, c, d
$(a^2+b^2)(c^2+d^2)=A^2$
$(a^2-b^2)(c^2-d^2)=B^2$
for some rational A and B.
This is related ...

**1**

vote

**1**answer

326 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.
http://faculty.nps.edu/pstanica/research/...

**10**

votes

**1**answer

517 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

**0**answers

329 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

**0**answers

288 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

**0**answers

219 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 ...

**2**

votes

**3**answers

717 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

**1**answer

160 views

### The exponential Diophantine equation $a^n-b^m=x^3+y^3$ for arbitrary large $n,m$

Let $a,b$ be coprime integers, neither a perfect power, $n,m$ naturals and $x,y$ integers.
Consider the exponential Diophantine equation
$$ a^n-b^m=x^3+y^3 \qquad (*) $$
Nontrivial solution ...

**1**

vote

**2**answers

345 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,
$$-a+nb+c = -d+ne+f\tag1$$
$$a+b+c = d+e+f\tag2$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag3$$
$$a^6+b^6+c^6 = d^6+e^6+f^6\tag4$$
...