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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3
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188 views

Logarithmic bound for Diophantine equation

Let $a_1 \geq a_2 \geq a_3$ be given positive integers and let $N(a_1,a_2,a_3)$ be the number of solutions $(x_1,x_2,x_3)$ of the equation $$\dfrac{a_1}{x_1}+\dfrac{a_2}{x_2}+\dfrac{a_3}{x_3} = 1$$...
6
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2answers
250 views

Is there any formula to find number of Pythagorean triplets between two integers 2 and j, j>2?

Given $j \geq 5$, is there a formula for the number of Pythagorean triplets $(a, b, c)$ satisfying the constraint that $a, b, c \leq j$? There exists at least one Pythagorean triplet for $j\geq5$; ...
5
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1answer
524 views

On the Diophantine equation $x^2 = y^p + 2^{r}z^p$ where $p\geq 7$ is an odd prime and $r \geq 2$

It is known that the only nonzero pairwise coprime integer solutions to the above Diophantine equation are for $r=3$, for which $(x, y, z) = (3,1,1)$ and $(-3,1, 1)$. (Cohen, Number Theory Volume 2: ...
0
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0answers
118 views

Counting integer solutions for a system of (in)equalities [migrated]

I wish to enumerate the number of solutions of the system of equations and inequalities for 3 non-negative integer unknowns $x,y,z \ge 0$: ($a$,$b$ specified) \begin{align} x+y+z&=a\\ x+y&>...
2
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1answer
315 views

An elliptic curve for Ramanujan-type cubic identities?

Given the roots $x_i$ of the depressed cubic, $$x^3+px+q=0$$ with rational coefficients. It can be shown that, in general, one can find rational $u,v$ such that, $$(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-...
0
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2answers
83 views

Dependence on parameters of solvability of a non-linear Diophantine system

For which $(k,t)\in\mathbb Z^2\times\mathbb Z$ does there exist $(v,s)\in\mathbb Z^2\times\mathbb Z$ so that $|v|^2=s^2\neq0$ and $v\cdot k+st=0$? I do not care what the solutions $(v,s)$ are but only ...
1
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1answer
103 views

Quadratic (and otherwise) squares, part II

This is a follow-up to this question: Quadratic squares Consider now a polynomial in two variables $f(x, y).$ Are there bounds (upper or lower) for how many $x_0$ of height less than $N$ such that $f(...
31
votes
2answers
1k 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} + \...
4
votes
1answer
245 views

Imprimitive solutions to $x^2+y^3=z^7$

Poonen, Schaefer, & Stoll give the primitive solutions to $x^2+y^3=z^7$: $$ (±1, −1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, −2, 1), (±71, −17, 2),\\ (±2213459, 1414, 65), (±15312283, 9262, 113), (±...
1
vote
1answer
56 views

2nd order Diophantine Equation, when does it have solution(s)? [closed]

I have this problem: Ax^2 + Cy^2 + Dx + Ey + F = 0, (B = 0 => Bxy) I need to know under which circumstances the above has solution(s). Thank you for your time.
0
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1answer
130 views

How do you bound the exponent of x^2+1=y^p?

How do you bound the exponent of $x^2+1=y^p$ for $p$ a prime exponent using linear forms in logs? So far I have $(x-i)(x+i)=y^p$ which are coprime and hence $x+i=(a+ib)^p$. Now how do I get a linear ...
2
votes
1answer
226 views

Non-negative integer solutions of x^2+y^3=n

I have the next equation: $x^2+y^3=n$. Where n is a positive integer constant. I want to know the exact number of non-negative integer solutions. Also I want to know what are those solutions. How ...
0
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0answers
156 views

$z^n-t^m=x^3+y^3$ and Vojta's more general abc conjecture

In A more general abc conjecture, p. 7 Paul Vojta conjectures: If $x_0,\ldots x_{n-1}$ are nonzero coprime integers satisfing $x_0 + \cdots x_{n-1}=0$ $$ \max\{|x_0|,\ldots |x_{n-1}|\} \le C \prod_{...
2
votes
1answer
162 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 ...
6
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2answers
2k views

is there a solution to this linear Diophantine system?

I have a matrix $A \in \mathbb Z^{n \times m}$, where $m > n$, and a vector $b \in \mathbb Z^n$. Under what conditions does $$Ax = b$$ have an integer solution? Is there a way to bound the norm ...
1
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0answers
47 views

System of diophantine equations with restricted set of solutions [closed]

I'm engineer, not mathematician, so excuse me for wrong terminology, but I hope you'll understand the problem. Example situation: I have N electronic components. Each of them has reactance and ...
11
votes
1answer
607 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 ...
17
votes
2answers
926 views

What is the smallest positive integer for which the congruent number problem is unsolved?

The congruent number problem is the problem of figuring out whether a given positive integer $N$ is the area of a right-angled triangle with all side lengths rational. According to Dickson's "History ...
14
votes
3answers
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. ...
24
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5answers
2k views

Parametric solutions of Pell's equation

Given a positive integer $n$ which is not a perfect square, it is well-known that Pell's equation $a^2 - nb^2 = 1$ is always solvable in non-zero integers $a$ and $b$. Question: Let $n$ be a ...
3
votes
1answer
149 views

4-th order diophantine equation

I met in many places the equation $(a^4-b^4)(c^4-d^4)=\square$ It is well known that this was investigated by Euler. But I was unable to find the general solution of this equation. Could you please ...
7
votes
2answers
383 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 ...
3
votes
2answers
87 views

Complexity of solving systems of linear diophantine equations

It is "well known" that a matrix system $Ax=b$ where $A\in \Bbb Z^{m\times n}$, $x\in \Bbb Z^n,b\in\Bbb Z^m$ for some $m,n \in \Bbb N$, can be solved in polynomial time, using Smith/Hermite Normal ...
1
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0answers
125 views

how to solve this symmetrical equation in number theory

i just have no idea about this equation, i would thank you to you to give me some suggestions on this. $$m_{1}m_{2}m_{3}+2^{\alpha-s-t}m_{1}+2^{\alpha-\gamma-t}m_{2}+2^{\alpha-\gamma -s}m_{3}\\-2^{\...
16
votes
4answers
3k 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 ...
1
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0answers
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 ...
6
votes
2answers
285 views

Integer solutions of (x+1)(xy+1)=z^3

Consider the equation $$(x+1)(xy+1)=z^3,$$ where $x,y$ and $z$ are positive integers with $x$ and $y$ both at least $2$ (and so $z$ is necessarily at least $3$). For every $z\geq 3$, there exists ...
5
votes
2answers
1k views

Algorithm for solving systems of linear Diophantine inequalities

So, I posted on StackOverflow looking for a reasonably fast algorithm to solve systems of linear Diophantine inequalities and was pointed to this article by Cheng-Zhi Gao and Yu-Lin Dong. The problem ...
5
votes
3answers
449 views

Extending rational Diophantine triples to sextuples

(This is a follow-up to a previous post.) A rational Diophantine $m$-tuple is a set of rationals {$a_1,a_2,\dots a_m$} such that (with $i\neq j$), all $a_i a_j+1$ is a square. Problem: Find a class of ...
12
votes
0answers
107 views

Rational inscribed realization of the regular dodecahedron

While it is clear that the regular dodecahedron $D$ cannot be realized with all integer coordinates, it is easy to find a polytope, which is combinatorially equivalent (face lattice isomorphic) to $D$ ...
0
votes
1answer
244 views

On elliptic curves, $\sqrt{x^2-101y^2} ,\sqrt{x^2+101y^2}$, and their ilk

I. Elliptic curves Given integers $a,b,m_k$. Let, $$x^2+a = m_1u_1^2\\x^2+b = m_1u_2^2\tag1$$ If there is a rational point $x_i$, then the pair (after a transformation) is birationally equivalent ...
11
votes
6answers
5k 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) ...
2
votes
0answers
145 views

Bounds for an Egyptian Fraction Inequality

Question: If $A\geq B>0$ are rational and $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ are integers such that $A\geq \sum_{j=1}^{n}\frac{1}{x_{j}}\geq B$, then what is an upper bound on $x_{j}$ in terms ...
3
votes
1answer
276 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 ...
9
votes
4answers
948 views

a family of Pellian equations

I have a question concering the family of Pellian equations $$x^2 - (k^2+1)y^2 = k^2. \qquad (*)$$ For an integer $k\geq 2$, the equation (*) has at least three classes of solutions in ...
37
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 $\...
13
votes
4answers
1k views

Is (n,m)=(18,7) the only positive solution to n^2 + n + 1 = m^3 ?

It's hard to do a Google search on this problem. If I was using Maple correctly, there are no other positive solutions with n at most 10000. I know some of these Diophantine questions succumb to ...
9
votes
1answer
503 views

A congruence conjecture regarding $(r-s)^4-1 \equiv 0\!\pmod{4r^2s}$

Is the following conjecture true? Conjecture. If $r > s \ge 1$ are relatively prime integers such that \begin{equation} (r-s)^4-1 \equiv 0\!\pmod{4r^2s}, \tag{1} \end{equation} then $r-s = 1$ ...
6
votes
3answers
257 views

Uniform bounds on the number of integer points on a family of elliptic curves

Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the ...
1
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0answers
82 views

Simple Diophantine equations for Cartan matrices of Kac-Moody algebras

Let us consider the matrix $A$ defined as follows: $A_{i i} =2$, $A_{i j} = - \frac{2 d_i}{d - 1 - d_{j}}, \qquad i \neq j$; $i, j = 1, \dots, n$. Here $d_1,...,d_n$ are natural numbers, $n > 1$ ...
12
votes
3answers
548 views

How many solutions does $\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}=1$ have?

It is well known that $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$ and this is the only solution to $\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}=1$ with $2\leq x_1<x_2<x_3$. My question is: Let $...
0
votes
1answer
123 views

Find the rational cases where ${t}^{2} - 4$ is a perfect square with height bound $|t| \le N$ for positive integer $N \ge 1$

Find the unique cases when ${t}^{2} - 4$ is a perfect square say, ${n}^{2}$, with height bound $|t| \le N$ for positive integer $N \ge 1$, when $t$ is a rational where $t = p/q$ and integers $p$ an $q$...
13
votes
7answers
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 ...
-1
votes
4answers
199 views

Finding integer zeroes for a particular family of equations [closed]

Given $p,q\in\mathbb Z^+$, and a vector $v=(x_1,\dots,x_{p+q})$ we consider the function $\chi(v)$: $$\chi(v)=x_1^2+\dots+x_p^2-x_{p+1}^2-\dots-x_{p+q}^2$$ We wish to find solutions to $\chi(v)=0$ ...
0
votes
0answers
150 views

A trivial application of Wilson's theorem to Brocard's Problem

Proposition: Let $W(1)$ be the set of all Wilson primes of order $1$ and suppose $n=p-1,$ where $p$ is a prime such that $p\notin W(1)$, then there are no integer solutions to the equation $$n!+1=m^2$$...
2
votes
0answers
146 views

Number of solutions to pentagonal-pentagonal numbers

Continuing the investigation from this question on CGSE about pentagonal-pentagonal numbers: Defining $p(n)$ as the $n$th pentagonal number (a positive integer of the form $n(3n−1)/2,\ n\geq 1$), and ...
1
vote
0answers
142 views

The existence of solution for special equation on integer ring

I have a question which belongs to the field of number theory. Can we prove or disprove the following claim: For all prime number $p=24t+1$ and the natural number $n=6t+1$, there is at least, one ...
12
votes
1answer
762 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 ...
14
votes
4answers
974 views

Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$

Two years ago, I made a conjecture on stackexchange: Today, I tried to find all solutions in integers $a,b,c$ to $$(1-a^2)(1-b^2)(1-c^2)=8abc,\quad a,b,c\in \mathbb{Q}^{+}.$$ I have found some ...
6
votes
1answer
247 views

Why are some solutions of these diophantine equations off the usual patterns?

This is inspired by a recent question about complete multipartite integral graphs. I am wondering if more can be said about tripartite integral graphs with block sizes $a<b<c$. It is easy to see ...