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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76 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 ...
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 ...
1
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1answer
83 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 ...
3
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1answer
123 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 ...
-1
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0answers
43 views

General Solution of Quadratic Diophantine Equation System

I am looking for the general solution of: $x^2-y^2=\square$ $x^2-z^2=\square$ $y^2-z^2=\square$ I have found a solution in tpiezas site. But not really sure if this is the general solution.
7
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2answers
331 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
71 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
121 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 ...
15
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 ...
1
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0answers
169 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 ...
3
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1answer
361 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: ...
6
votes
2answers
274 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
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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
420 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
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0answers
99 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
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1answer
220 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 ...
12
votes
6answers
4k 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
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0answers
135 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
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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 ...
9
votes
4answers
934 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 ...
36
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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 ...
2
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0answers
235 views

A problem on Hecke operators

I have a problem that is related to page 11 in http://www.cims.nyu.edu/~venkatesh/research/ml.pdf Fix $m\in\Bbb N$. Pick $(a,b)\in\Bbb Z_p^2$ randomly such that $m<a,b<2m$ with $a,b$ ...
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
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1answer
498 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
245 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
81 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
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3answers
503 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
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1answer
121 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 ...
13
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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 ...
2
votes
0answers
105 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} = ...
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votes
4answers
192 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 $\chi(v)=0$ ...
30
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} + ...
0
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0answers
127 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 ...
2
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0answers
144 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 ...
0
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0answers
141 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
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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 ...
14
votes
4answers
956 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
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1answer
245 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 ...
4
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1answer
256 views

Is this problem of Schinzel and Tijdeman misquoted? It appears easy with Pell equations

In Diophantine equations over the twentieth century: a (very) brief overview , p. 5 Problem Let $f(x) \in \mathbf{Z}[x]$ be an irreducible polynomial of degree at least 2. Do the Diophantine ...
1
vote
1answer
263 views

How can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution for every prime $p>3$?

Without applying Fermat's Last Theorem, how can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution $(x,y) = (0, \frac{1}{2})$ for ever prime $p ...
6
votes
2answers
751 views

A new result on the Diophantine equation $x^3 + y^3 +z^3 = 3$ [closed]

The above Diophantine equation is unknown to have any further integer solutions other than $(x, y, z) = (1, 1, 1)$ and $(4, 4, -5)$. I am a prospective undergraduate mathematics student in Zimbabwe ...
3
votes
1answer
242 views

rational numbers and triangular numbers

This question is an offshoot of Ratio of triangular numbers. Suppose $ka(a+1)=nb(b+1)$, where $k,n >1$ are relative prime integers, and $a,b \geq 0$ are integers. Which $k,n$ pairs have no solution ...
0
votes
2answers
218 views

Minimal solution of simultaneous congruences

I would to determine the set of values $\lbrace a_1,a_2,a_3,\ldots,a_n \rbrace$ that minimizes the value of $x$ such that: $$x\equiv a_1\mod p_1$$ $$\vdots$$ $$x\equiv a_n\mod p_n$$ where every ...
19
votes
2answers
708 views

Rational points on the “quintic circle” $x^5 + y^5 = 7$

I suspect that the curve $x^5 + y^5=7$ has no $\mathbb Q$ points, and a brief computer search verifies this hypothesis for denominators up to $10^4$. What techniques can be used to show that there are ...
35
votes
1answer
3k 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 ...
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
1answer
213 views

The number of integral solutions to $x^2+y^2-az^2=0$

I think this must be well-known (and probably not hard to prove either), but I cannot find a reference: for a (positive) rational number $a$, the number of integral solutions to the equation $$ ...
10
votes
2answers
493 views

Equation $x^2=y^p + 1$

can you help me please for solving this diophantine equation : $x^2=y^p+1$ and if you can give me a general method to studying such equation : $x^2=y^p+t$ Thanks
4
votes
1answer
350 views

On the mixed sum of three k-th powers

Let the set $S_k=\{\pm x^k \pm y^k \pm z^k \ \vert \ x,y,z \in \mathbb{Z} \}$. Note that the signs are independently positive or negative. First of all $S_2 = \mathbb{Z}$ because (see the answers ...
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, $$-a+nb+c = -d+ne+f\tag1$$ $$a+b+c = d+e+f\tag2$$ $$a^2+b^2+c^2 = d^2+e^2+f^2\tag3$$ $$a^6+b^6+c^6 = d^6+e^6+f^6\tag4$$ ...