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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6
votes
2answers
247 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 $j\geq5$; ...
-1
votes
0answers
23 views

Further counting integer solutions for a system of (in)equalities [closed]

After solving a previously raised problem, I am considered with the more general problem of enumerating the number of solutions of: x+y+z+w=a x+y>b y+z>b z+w>b x,y,z,w are non-negative integers.
0
votes
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&>...
0
votes
2answers
82 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 ...
2
votes
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-...
1
vote
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(...
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.
2
votes
1answer
225 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 ...
1
vote
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 ...
17
votes
2answers
918 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 ...
0
votes
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 ...
3
votes
1answer
148 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 ...
3
votes
2answers
85 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
vote
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^{\...
1
vote
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 ...
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
243 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 ...
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 ...
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 ...
2
votes
0answers
143 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 ...
6
votes
3answers
256 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
vote
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
546 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 $q$...
3
votes
0answers
145 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 $$\dfrac{a_1}{x_1}+\dfrac{a_2}{x_2}+\dfrac{a_3}{x_3} = 1$$...
0
votes
0answers
148 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 $$n!+1=m^2$$...
-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 $\chi(v)=0$ ...
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 ...
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} + \...
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 ...
5
votes
1answer
523 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: ...
12
votes
1answer
761 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 ...
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 ...
4
votes
1answer
260 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
270 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
777 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
245 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 ...
19
votes
2answers
731 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 ...
0
votes
2answers
222 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 ...
5
votes
1answer
220 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 $$ x^2+y^...
10
votes
2answers
500 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
355 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
346 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, $$-a+nb+c = -d+ne+f\tag1$$ $$a+b+c = d+e+f\tag2$$ $$a^2+b^2+c^2 = d^2+e^2+f^2\tag3$$ $$a^6+b^6+c^6 = d^6+e^6+f^6\tag4$$ ...
0
votes
2answers
172 views

How to solve the following system of diophantine equations? [closed]

We have the equations $$a_1x+b_1y+c_1z=d_1,$$ $$a_2x^2+b_2y^2+c_2z^2=d_2,$$ $$a_3x^3+b_3y^3+c_3z^3=d_3,$$ where $a_i,b_i,c_i \in\Bbb N$ at $i \in \{1,2,3\}$ are known. Is there an efficient ($O(\...
11
votes
1answer
396 views

Why does genus control the number of points

Often number theorists can bound the number of solutions to a diophantine equation based on the size of the points and the size of the coefficients. But this, as I understand it, can be a bit of a red ...
3
votes
1answer
158 views

infinite solution of a diophantine quadratic equations

Let $a,b,c,d$ be integers such that $GCD(a,b,c,d)=1$. Assume that the diophantine equation $ax^2+bxy+cxz+dyz-x=0$ has a non-zero solution.Can we assert that it admits infinitely many solutions? ...
2
votes
2answers
238 views

Parametrizing the solutions to a diophantine equation of degree four [closed]

Good evening, Consider $x^4+y^4+z^4=2t^4$ where x,y,z,t integer. Is it known how to find all parametrisation of this equation ? If you have any parametrisation or reference of this equation, please ...