Questions tagged [diophantine-equations]

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 $\mathbb{Q}$. Topics: Pell equations, quadratic forms, elliptic curves, abelian varieties, hyperelliptic curves, Thue equations, normic forms, K3 surfaces ...

Filter by
Sorted by
Tagged with
69 votes
3 answers
7k views

Can you solve the listed smallest open Diophantine equations?

In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size is substitute 2 instead of all variables, absolute values instead of all coefficients, and ...
Bogdan Grechuk's user avatar
170 votes
3 answers
64k views

Estimating the size of solutions of a diophantine equation

A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ? (I do not know any such numbers). B. Suppose that $\frac{a}{b+c} + \...
alex alexeq's user avatar
  • 1,841
54 votes
6 answers
7k views

What is the smallest unsolved Diophantine equation?

If $P=\sum_{\bf i} a_{\bf i}x^{\bf i}\in {\mathbb Z}[x_1,\dots,x_d]$, let $|P|=\sum_{\bf i}|a_{\bf i}|x^{\bf i}$ and $h(P)=|P|(2,\dots,2)$, so that there is only a finite number of $P$ with $h(P)\leq ...
Zidane's user avatar
  • 917
42 votes
3 answers
19k 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. ...
joro's user avatar
  • 24.2k
20 votes
2 answers
2k 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 ...
José Hdz. Stgo.'s user avatar
37 votes
5 answers
10k 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 ...
András Salamon's user avatar
22 votes
1 answer
1k views

On the smallest open Diophantine equations: beyond Hilbert's 10 problem

In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size of the equation is substitute 2 instead of all variables, absolute values instead of all ...
Bogdan Grechuk's user avatar
21 votes
9 answers
20k 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 ...
amateur's user avatar
  • 213
21 votes
2 answers
2k 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^...
Joseph O'Rourke's user avatar
16 votes
3 answers
1k views

Is Multilinear Hilbert's tenth problem version undecidable?

A multilinear polynomial $f\in\mathbb Z[x_1,\dots,x_t]$ has terms only of form $$b\prod_{i=1}^tx_i^{a_i}$$ where $a_i\in\{0,1\}$ and $b\in\mathbb Z$. Is there no general purpose algorithm for ...
Turbo's user avatar
  • 13.6k
35 votes
3 answers
4k views

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

Does anybody know a solution to this problem? (Sorry, I've missed one summand in the previous post.)
Alexey Ustinov's user avatar
33 votes
4 answers
4k 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 $F_{13}-F_{11}=12^2$...
joro's user avatar
  • 24.2k
29 votes
9 answers
10k 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 ...
Faisal's user avatar
  • 10.2k
20 votes
3 answers
2k 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 ...
Dirk's user avatar
  • 209
18 votes
2 answers
6k 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 ...
Craig Feinstein's user avatar
18 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 ...
Joseph O'Rourke's user avatar
16 votes
2 answers
1k views

Representing $x^3-2$ as a sum of two squares

Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^3-2$ is a sum of two squares of integers. Ideally, I am looking for a proof method that also applies for other $P(x)$, ...
Bogdan Grechuk's user avatar
11 votes
1 answer
2k 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 ...
Max Alekseyev's user avatar
11 votes
2 answers
2k 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?
Luca's user avatar
  • 211
9 votes
2 answers
813 views

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 $...
srs's user avatar
  • 193
7 votes
1 answer
417 views

$y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves

Do there exists rational numbers $x$ and $y$ such that $$ y^3 = x^4 + x + 2 ? $$ Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and ...
Bogdan Grechuk's user avatar
7 votes
1 answer
370 views

A parametric elliptic curve for $x^4+y^4+z^4 = 1$?

Noam Elkies found that $x^4+y^4+z^4 = 1$ has infinitely many rational points $xyz \neq 0$ using an elliptic curve. We use a different approach that will produce pairs of solutions and a parametric ...
Tito Piezas III's user avatar
7 votes
3 answers
756 views

Something interesting about the quintic $x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0$ and its cousins

(Update): Courtesy of Myerson's and Elkies' answers, we find a second cyclic quintic for $\cos\frac{2\pi}{p}$ with $p=\text{1 mod 10}$ as, $$\frac{z^5}{\beta} = 10 z^3 - 20 n^2 z^2 + 5 (3 n^4 - 25 n^...
Tito Piezas III's user avatar
4 votes
2 answers
669 views

On the product $\prod_{k=1}^{(p-1)/2}(x-e^{2\pi i k^2/p})$ with $x$ a root of unity

Let $p$ be an odd prime. Dirichlet's class number formula for quadratic fields essentially determines the value of the product $\prod_{k=1}^{(p-1)/2}(1-e^{2\pi ik^2/p})$. I think it is interesting to ...
Zhi-Wei Sun's user avatar
  • 14.4k
48 votes
4 answers
4k 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 $\...
Pablo's user avatar
  • 11.2k
39 votes
2 answers
3k 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 ...
Noam D. Elkies's user avatar
35 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 ...
Wadim Zudilin's user avatar
32 votes
2 answers
2k 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 ...
Joel David Hamkins's user avatar
27 votes
3 answers
1k views

When does $axy+byz+czx$ represent all integers?

For which $a,b,c$ does $axy+byz+czx$ represent all integers? In a recent answer, I conjectured that this holds whenever $\gcd(a,b,c)=1$, and I hope someone will know. I also conjectured that $axy+byz+...
user avatar
26 votes
3 answers
3k views

The diophantine eq. $x^4 +y^4 +1=z^2$

This question is an exact duplicate of the question Does the equation $x^4+y^4+1=z^2$ have a non-trivial solution? posted by Tito Piezas III on math.stackexchange.com. The background of ...
user avatar
23 votes
1 answer
2k 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) &...
Tito Piezas III's user avatar
20 votes
1 answer
1k views

Is "almost-solvability" of Diophantine equations decidable?

Say that a Diophantine equation is almost-satisfiable iff for each $n\in\mathbb{N}$ it has a solution mod $n$. Trivially genuine satisfiability over $\mathbb{N}$ implies almost-satisfiability, but the ...
Noah Schweber's user avatar
18 votes
1 answer
1k views

When is $(q^k-1)/(q-1)$ a perfect square?

Let $q$ be a prime power and $k>1$ a positive integer. For what values of $k$ and $q$ is the number $(q^k-1)/(q-1)$ a perfect square, that is the square of another integer? Is the number of such ...
Huangjun Zhu's user avatar
16 votes
6 answers
3k views

Advances and difficulties in effective version of Thue-Roth-Siegel Theorem

A fundamental result in Diophantine approximation, which was largely responsible for Klaus Roth being awarded the Fields Medal in 1958, is the following simple-to-state result: If $\alpha$ is a real ...
Stanley Yao Xiao's user avatar
13 votes
2 answers
933 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}...
user avatar
13 votes
2 answers
1k views

On Euler's elliptic curve for $A^4+B^4 = C^4+D^4$?

To solve, $$A^4+B^4 = C^4+D^4$$ we use Euler's method. Let, $$(p+q)^4+(r-s)^4=(p-q)^4+(r+s)^4$$ and define $p = (a^3 - b),\, q = a y,\, r = b (a^3 - b),\, s = y.\,$ The equation above transforms to ...
Tito Piezas III's user avatar
10 votes
2 answers
3k 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 ...
Richard Bonne's user avatar
10 votes
3 answers
734 views

Solve in integers: $y(x^2+1)=z^2+1$

Find all integer solutions to the equation $$ y(x^2+1)=z^2+1. $$ There is, for example, an infinite family of solutions $x=u$, $y=(uv\pm1)^2+v^2$, $z=(u^2+1)v \pm u$, $u,v \in {\mathbb Z}$, but there ...
Bogdan Grechuk's user avatar
10 votes
1 answer
540 views

Formula for "cointersection" of three circles?

I am working on the problem of finding "rational" dodecahedrons, and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point? ...
Thomas Blok's user avatar
8 votes
1 answer
604 views

On Markoff-type diophantine equation

Do there exist integers $x,y,z$ such that $$ x^2+y^2-z^2 = xyz -2 \quad ? $$ Why this is interesting? First, this equation arose in an answer to the previous Mathoverflow question What is the smallest ...
Bogdan's user avatar
  • 781
8 votes
4 answers
836 views

A cubic equation, and integers of the form $a^2+32b^2$

I am trying to determine whether there are any integers $x,y,z$ such that $$ 1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1) $$ It is clear that $x$ is odd. We can consider this equation as quadratic ...
Bogdan Grechuk's user avatar
8 votes
1 answer
580 views

Hilbert 10th problem for cubic equations

Hilbert 10th problem, asking for algorithm for determining whether a polynomial Diopantine equation has an integer solution, is undecidable in general, but decidable or open in some restricted ...
Bogdan Grechuk's user avatar
6 votes
2 answers
686 views

Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$

Let $P(z) = z^4 +z^3 +z^2 +z+1$ where $z$ is a positive integer. While working with the diophantine equation $(xz+1)(yz+1)=P(z)$, I was able to construct a seemingly infinite and complete solution set ...
ASP's user avatar
  • 319
6 votes
2 answers
441 views

Difficult elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?

Similar to the case $x^4+y^4+z^4 = 1$ discussed in this MO post, define the system, $$x^4+y^4+z^4+1 = (x+y+z+1)^4\tag1$$ $$\frac{x^2+x+1}{(x+y+1)(x+z+1)}=u\tag2$$ $$\frac{y^2+y+1}{(y+z+1)(y+x+1)}=v\...
Tito Piezas III's user avatar
4 votes
0 answers
231 views

On the values of $\prod_{k=1}^{(p-1)/2}(e^{2\pi i/12}-e^{2\pi i k^2/p})$ for primes $p>3$

In a recent preprint, I investigated $$S_p(x):=\prod_{k=1}^{(p-1)/2}(x-e^{2\pi ik^2/p}),$$ where $p$ is an odd prime and $x$ is a root of unity. Motivated by Question 337879 and Question 338325, ...
Zhi-Wei Sun's user avatar
  • 14.4k
4 votes
1 answer
188 views

Small linear relations between primitive Pythagorean triples $\mathsf{II}$

WillJagy answered a linear relation question on Pythagorean Triples in Small linear relations between primitive Pythagorean triples $\mathsf I$. Now let $a^2+b^2=c^2$ be a primitive Pythagorean ...
VS.'s user avatar
  • 1,816
4 votes
1 answer
492 views

Large radical of an integer and three AB conjectures

In this Note, We propose a new definition called "large radical of an integer". Using this definition, three very useful $AB$ conjecture are given. 1. Large counter examples of the ABC conjecture ...
Đào Thanh Oai's user avatar
4 votes
0 answers
297 views

What does it mean to solve an equation?

Assume that we want to find all integer (or rational) solutions to the polynomial Diophantine equation $$ P(x_1,\dots,x_n) = 0 $$ where $P$ is a polynomial with integer coefficients. Do we have a ...
Bogdan Grechuk's user avatar
3 votes
0 answers
254 views

Are all odd integers greater than $599$ of the form $x^2+y^2+z^4+t^4$?

For $a \le 10^7$, the equation over integers $4a+1=x^2+y^2+z^4+t^4$ has solutions. Q1 Is it true that all integers of the form $4a+1$ are also of the form $x^2+y^2+z^4+t^4$? Heuristic argument: ...
joro's user avatar
  • 24.2k
3 votes
0 answers
559 views

On Choudhry's $x_1^k+x_2^k+x_3^k+x_4^k = y_1^k+y_2^k+y_3^k+y_4^k$ for $k=7$?

I. Fifth Powers The Diophantine equation, $$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$ for $k=5$ is quite well-explored. It has an infinite number of primitive solutions (as points on an elliptic ...
Tito Piezas III's user avatar