Questions on finding integer/rational solutions of equations.

learn more… | top users | synonyms

1
vote
0answers
30 views

How many different triangles have side lengths $x,y,z$ that satisfy $3x^3-yz^2 = z^3+4x^2-y$?

How many different triangles have side lengths $x,y,z$ that satisfy $3x^3-yz^2 = z^3+4x^2-y$? I was wondering about this and was wondering in general are there ways to solve such a question for $f(x,...
0
votes
1answer
71 views

System of equation $x+y+z=2007; xyz=14000$

I have to solve the system of equations $$\begin{cases} x+y+z=2008,\\ xyz=14000, \end{cases}$$ where $x,y,z$ are positive integers such that $1\le x \le y \le z \le 2000.$ My work so far: Let $...
9
votes
3answers
257 views

Find all integral solutions for the Diophantine Equations $x^4 - x^2y^2 + y^4 = z^2$ and $x^4 + x^2y^2 + y^4 = z^2$.

Find all integral solutions for the Diophantine Equations $$x^4 - x^2y^2 + y^4 = z^2$$ and $$x^4 + x^2y^2 + y^4 = z^2$$ I basically think that to solve these equations we need to use the fact that ...
2
votes
1answer
65 views

Diophantine equation $n^2+n+1=m^3$

Is there an elementary method for solving Diophantine equation $n^2+n+1=m^3$ for integers $m$ and $n$? There is a similar one, which I could solve:$$p^2-p+1=q^3,$$where $p$ and $q$ are prime numbers. ...
5
votes
1answer
118 views

Link between the negative pell equation $x^2-dy^2=-1$ and a certain continued fraction

Consider the generalized continued fraction $$F(x)=(x-1)-\cfrac{(x+1)}{x+\cfrac{(-1)(5)} {3x+\cfrac{(1)(7)}{5x+\cfrac{(3)(9)}{7x+\cfrac{(5)(11)}{9x+\ddots}}}}}$$ I experimentally discovered that at ...
7
votes
2answers
114 views

Show that $x^2 + y^2$ and $x^2 - y^2$ cannot both be perfect squares at the same time where $x, y \in \mathbb{Z}^+$.

Show that $x^2 + y^2$ and $x^2 - y^2$ cannot both be perfect squares at the same time where $x, y \in \mathbb{Z}^+$. I think that $x^2 + 2xy + y^2$ and $x^2 + y^2$ are not consecutive squares since ...
4
votes
1answer
63 views

Diophantine equations $x^n-y^n=2016$

Solve equation $$x^n-y^n=2016,$$ where $x,y,n \in \mathbb N$ My work so far: If $n=1$, then $y=k, x=k+2016, k\in \mathbb N$ If $n=2$, then $2016=2^5\cdot 3^2 \cdot 7$ $x-y=1; x+y=2016$ $x-y=...
7
votes
1answer
170 views

Diophantine equation: choosing the right modulus to prove an equation cannot be satisfied

I was looking at this problem, which asks to show that there are no $m,n \in \mathbb Z$ such that $$3n^2+3n+7 = m^3.$$ The result follows immediately from considering the equation modulo $9$ and ...
12
votes
1answer
142 views

$2^n + 3^n = x^p$ has no solutions over the natural numbers

A few weeks ago, I was asked to prove that $2^n + 3^n = x^2$ has no solutions over the positive integers. My proof was: $2^n + 3^n \equiv (-1)^n \equiv \pm 1 \mod{3}\\\text{However, quadratic residue ...
1
vote
1answer
70 views

Preserving modulus residue under division

Modulus residue is preserved or honored (sorry, I don't know the correct term. Is it homomorphism?) under addition and multiplication. For example: 2 + 4 = 6 2 * 4 = 8 Then, making those values ...
7
votes
4answers
135 views

There does not exist any integer $m$ such that $3n^2+3n+7=m^3$

I have this really hard problem that I am working on and I just don't seem to get it. The question is: let $n$ be a positive integer; prove that there does not exist any integer $m$ such that $3n^2+3n+...
4
votes
1answer
81 views

Solving the Diophantine Equation $x^2 - y! = 2001$ and $x^2 - y! = 2016$

I had recently faced a problem: Solve the Diophantine Equation $x^2 - y! = 2001$. Solving it was quite easy. You show how $\forall y \ge 6$, $9|y!$ and since $3$ divides the RHS, it must divide ...
1
vote
3answers
59 views

Integer solutions to $xyz = w^2(x+y+z)$

I'm looking for a way to enumerate all positive integer solutions of the equation $xyz = w^2(x+y+z)$ where $w \le W$ and $1 \le x \le y \le z$. Could anyone provide a hint at how to approach this?
5
votes
3answers
96 views

Proving that the only integer solution of $2x^2+3y^2=z^2$ is $(0,0,0)$

I'd like to prove that the only integer solutions of $$2x^2+3y^2=z^2$$ is $(0,0,0)$. By working in $\mathbb{Z}_2$ and $\mathbb{Z_3}$, I have gone as far as proving that in $\mathbb{Z}$, any integer ...
3
votes
1answer
55 views

$x^5 - y^2 = 4$ has no solution mod $m$

A common technique for proving that a diophantine equation does not have a solution is to prove that it does not have a solution mod $m$ for a suitable modulus $m$. This technique works with $m=11$ ...
1
vote
0answers
31 views

Squares in a second order linear recurrence of positive integers

Let the integer sequence $n_k$, ($k\ge 0$) be defined as $$ n_0=1$$ $$n_1=64$$ $$ n_k=38 n_{k-1}-n_{k-2}-90$$ How can one find the squares in such a sequence? Besides $ n_0=1^2, n_1=8^2$, we also ...
1
vote
1answer
51 views

Prove that there does not exist integer solutions for the diophantine equation $x^5 - y^2 = 4$.

Prove that there does not exist an integer solution for the diophantine equation $x^5 - y^2 = 4$. It's obvious that $x$ and $y$ are of the same parity. We can also claim that if $x$ is odd, then it ...
0
votes
0answers
36 views

Complete formalization of solutions to $a^2+b^2=c^2+k$ for fixed $k>0$

Is there a known complete formalization of solutions to $a^2 + b^2 = c^2 + k$ for a fixed constant $k>0$ similar to the one for primitive Pythagorean triples (i.e. $(a,b,c) = (m^2-n^2,2mn,m^2+n^2)$ ...
-4
votes
2answers
163 views

Why is minimum solution example to $x^n + y^n = z^n$ comes in the form of three successive integers? [closed]

Can we prove or disprove this conjecture by elementary mathematics: If this is a true statement: $$x^n + y^n = z^n $$where $x, y, z, n$ are positive integers, then there must be a minimum integer ...
1
vote
1answer
80 views

Computational complexity of solving linear diophantine equations?

Is there any good complexity upper bound for checking satisfiability of a matrix system $Ax=b$ where $A\in \Bbb Z^{m\times n}$? I found some estimate on computing the Smith Normal Form $N$ such that $...
1
vote
2answers
59 views

Solving the equation $x^3+y^2=4x^2y$ over integers.

$$x^3+y^2=4x^2y$$ This is a quadratic in $y$, the discriminant of which must be $>0$ $$\implies 16x^4-4x^3>0$$ $$\implies x \in (-\infty,0) \cup (1,\infty)$$ (So we have nothing new up to this ...
2
votes
0answers
39 views

Diophantine System Solution

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
3answers
34 views

Solving Diophantine Equation $xB=(2^N)-1$

If given a value for $x$, does anyone have a way to solve the diophantine equation below? $xB=(2^N)-1$ where $x,B,N\in\mathbb Z$ Where presumably a smaller $N$ is better, but any way to find a ...
0
votes
0answers
30 views

Linear Diophantine equations of several variables

I know how to solve Diophantine equations of the form $ax+by=c$ but how can I solve linear Diophantine equations having more variables. Like what are the integer solutions of $43x+23y-435z+1324w=1$? I ...
1
vote
2answers
47 views

Help answering Pell Equation questions

I understand the Pell equation is $$x^{2}-dy^{2}=1$$ However I don't understand how to use this to get $(x,y)$ for these questions. 1) Find a nontrivial solution of $x^{2} − 3y^{2} = 1.$ 2) Find ...
3
votes
1answer
70 views

How do I count the solutions of $m^2 + m n + n^2 = T$?

I've come across this problem in my studies. I was wondering if there is a better algorithm for it: Given a fixed positive integer $T$, count the solutions of $$n^2 + n m + m^2 = T$$ where $m$ and $n$ ...
6
votes
1answer
124 views

Solve $x^2 = 2^n + 3^n + 6^n$ over positive integers.

Solve $x^2 = 2^n + 3^n + 6^n$ over positive integers. I have found the solution $(x, n) = (7, 2)$. I have tried all $n$'s till $6$ and no other seem to be there. Taking $\pmod{10}$, I have been ...
2
votes
2answers
100 views

How to enumerate 2D integer coordinates ordered by Euclidean distance?

The square of Euclidean distance between $(x, y)\in\mathbb{Z}^2$ and origin is $d = x^2+y^2$. How to enumerate the coordinates $(x, y)$ in ascending order of $d$? For example, the first 14 sets of ...
2
votes
1answer
69 views

Numbers expressible as sum of 2 squares in 2 distinct ways

I was trying this question here which goes like: Find numbers which are squares and can be expressed as $x^2y^2-x^2-y^2+2$ for non-consecutive positive integers only. Let the number be $a$ \...
0
votes
3answers
41 views

Solutions to the diophantine equation $6x^2 - 6x - y^2 + y=0$?

Are there any positive integer solutions to the diophantine equation in the title other than $(1,1)$? This equation looks easy enough so it could be that there is some simple argument that shows ...
4
votes
4answers
90 views

Find all $x,y$ so that $\dfrac{x+y+2}{xy-1}$ is an integer.

I am trying to find the integers $x,y$ so that $\dfrac{x+y+2}{xy-1}$ is an integer. What I have done: I suppose there exists $t$ such that $$t=\dfrac{x+y+2}{xy-1}$$ where $xy\neq 1$ then consider ...
3
votes
3answers
413 views

Find all integer solutions to $\frac{1}{x} + \frac{1}{y} = \frac{2}{3}$

Find all integer solutions $(x, y)$ of the equation $$\frac{1}{x} + \frac{1}{y} = \frac{2}{3}$$ What have done is that: $$\frac{1}{x}= \frac{2y-3}{3y}$$ so, $$x=\frac{3y}{2y-3}$$ If $2y-3 = +1 \...
3
votes
1answer
80 views

Finding solutions to $x^2+y^2+z^2=w^3$

Suppose $w$, $x$, $y$, and $z$ are all positive integers less than $100$. Find all such solutions to the equation $x^2+y^2+z^2=w^3$. This problem was in a competition I participated in this past ...
4
votes
2answers
101 views

Show that $x^2 + y^2 + z^2 = x^3 + y^3 + z^3$ has infinitely many integer solutions.

Show that $x^2 + y^2 + z^2 = x^3 + y^3 + z^3$ has infinitely many integer solutions. I am not able to find an idea on how to proceed with the above questions. I have found only the obvious solution $...
0
votes
1answer
39 views

Coprime - Irreducibility - Natural numbers

In reference to this question, is anyone could deduce that if $x^2+2=y^3$ and $x,y \in \mathbb{N}$, then $x=5$ and $y=3$. I already prove that the only natural number $x$ for which $x+\sqrt{-2}$ is a ...
2
votes
1answer
63 views

Non-linear Diophantine equation on integer quadruples

Find all integer quadruples $\{a,b,c,d\}$ such that $$ad = b + c$$ $$bc = a^2 - d$$ Working $\bmod 8$ (very messy) gives $d = 3 - 8k \quad \forall k \in \mathbb{N}$. Numerical searching has so ...
-1
votes
2answers
68 views

Solve equation over $\mathbb{N}\setminus\{0\}$ [closed]

I wonder whether there are any solutions besides considering $c=2^{5k+1}$ for this equation: $a^5+b^5=c^{2016}$, where $a,b,c\in\mathbb{N}\setminus\{0\}$
2
votes
1answer
90 views

Quadratic Diophantine Equation $x^2 + 2y^2 = 2013$ [closed]

Find integer values of $x$ and $y$ (if any) such that $x^2 + 2y^2 = 2013$.
3
votes
1answer
58 views

Parametrization of $a^2+b^2+c^2=d^2+e^2+f^2$

Is there an existing parametrization of the equation above that is similar to Brahmagupta's identity for $a^2+b^2=c^2+d^2$? I need either a reference to look it up or a hint to solve it. Thanks.
8
votes
1answer
498 views

Why there isn't any solution in positive integers for $z^3 = 3(x^3 +y^3+2xyz)$?

Consider the following Diophantine equation $$z^3 = 3(x^3 +y^3+2xyz)$$ Is there any elementary proof for the non solubility in positive integers for this Diophantine equation, where $x, y$ and $z$ ...
3
votes
1answer
56 views

Find all nonnegative integer solutions to $x^3 + 8x^2 − 6x + 8 = y^3$.

Find all nonnegative integer solutions to $x^3 + 8x^2 − 6x + 8 = y^3$. The only solution I have found is $x=0$. I have tried proving it by congruences and have had no success. I don't know how to ...
1
vote
1answer
86 views

Help Project Euler Problem 269

I am stuck on prob 269 Project Euler. I've just tried brute force method to attempt this problem the example provided by PE For example, $P_{5703}(x)$ = $5x^3 + 7x^2 + 3$. We can see that: P$_n(0)$...
2
votes
1answer
98 views

Transforming Diophantine quadratic equation to Pell's equation

I have been discussing the fastest and most efficient ways of solving QDEs in a separate question record (Alternative method to solve quadratic Diophantine equations). However, as suggested by individ,...
0
votes
0answers
25 views

Looking for info on representation of a diophantine equation as system of equations over finite field/boolean algebra

Suppose that $x$ is a positive integer. Fix some prime $p$. Then there exists some non-negative integer, $L$, and $\{x_0, x_1, . . . , x_L\} \subseteq \{0,1,...,p-1\}$ such that, $$x = \sum_{n=0}^{L}...
0
votes
1answer
66 views

Alternative method to solve quadratic Diophantine equations

For most types of quadratic Diophantine equations there exists an algorithm which makes it possible to find a solution (or solutions) over integers (good reference is here: https://www.alpertron.com....
3
votes
1answer
68 views

Solve the equation $x^3 + 117y^3 = 5$ over the integers.

Solve the equation $x^3 + 117y^3 = 5$ over the integers. I have tried solving this. It is clear that one of $x$ or $y$ must be negative. $117$ seemed a strange number. So I found out that $117 = 125 ...
2
votes
1answer
59 views

Why chord and tangent method does not give further points on Fermat's curve?

(0,1) and (1,0) are two rational points on x^3 + y^3= 1. But why doesn't chord and tangent method yield any further points on the curve?
4
votes
4answers
117 views

Does the Pell-like equation $X^2-dY^2=k$ have a simple recursion like $X^2-dY^2=1$?

If $d \ne 0$ is a non-square integer, and $(u,v)$ is an integer solution to the Pell equation $$ X^2 - dY^2 = 1, \tag{$\star$} $$ then each solution $(x_i,y_i)$ can be recursively calculated using ...
1
vote
1answer
47 views

An extension to Pell's equation

During my number theory seminary, I found this interesting problem and I didn't know how to solve it. Given Pell's equation $$x^2-3y^2=1,$$ where $x,y \in \Bbb N,$ show that there are infinitely many ...
1
vote
1answer
34 views

Solving the Diophantine equation $t^n + 2 \equiv 0 \bmod s^n - 1$

My problem is this. find the maximal integer n, so the equation: $t^n+2\equiv0 \mod (s^n-1). $ has a solution (s,t>1 have to be integers). I would like to read your solution and even just an opinion....