Questions on finding integer/rational solutions of equations.

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11 views

If $a\not\equiv 0\mod{p}$ then there are $p-1$ solutions (ordered pairs) to $x^2-y^2\equiv a\mod{p}$

Let $p$ be an odd prime, and let $a\in\mathbb{Z}_p$ such that $a\not\equiv 0$. I need to show that there are $p-1$ ordered pairs $(x,y)$ such that $x^2-y^2\equiv a \mod{p}$. As I see it, the ...
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0answers
14 views

Linear Transformations between solutions to different hyperboloids

Is there a way to develop a linear transformation which will always send solutions of one hyperboloid to another? (for example the hyperboloids: $$a^2+b^2-c^2=4$$ and $$d^2+e^2-f^2=9$$ )I know that ...
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4answers
42 views

Under certain conditions $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a'}+\frac{1}{b'}+\frac{1}{c'}\Rightarrow \{a,b,c\}=\{a',b',c'\}$

Let $a,b,c,a',b',c'\in \mathbb{Z}_{\geq 1}$ be such that $$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}<1,\quad \frac{1}{a'}+\frac{1}{b'}+\frac{1}{c'}<1. $$ Suppose $$ ...
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3answers
52 views

$x^2+y^2=2z^2$, positive integer solutions

Determine all positive integer solutions of the equation $x^2+y^2=2z^2$. First I assume $x \geq y$, and I have $x^2-z^2=z^2-y^2$. Then I have $(x-z)(x+z)=(z-y)(z+y)$, but from here, I don't know how ...
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0answers
26 views

intersection of a line and a curve [on hold]

I have this question let $3x+4y=100$ find $x,y\in$ $\mathcal {} \mathbb{Z}$ such that $y^2+x^2=c $ is the smallest. using calculus I can find that $x=12, y=16$. But this is the only solution it ...
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2answers
29 views

Completeness proofs for the solutions of Diophantine Equations

In general, what are the strategies for showing the completeness of a solution set for Diophantine Equations? For example, take the $\textit{Pell-type}$ equation $x^2 - dy^2 = a$. Say, you have a set ...
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1answer
17 views

Does this Diophantine inequality have any solutions for $p, q \in \mathbb{N}$?

Does this Diophantine inequality have any solutions for $p, q \in \mathbb{N}$? $$p^2 q^2 \geq 3 p^2 q + 3p^2 + 3pq^2 + 3pq + 3p + 3q^2 + 3q + 3$$ I tried to use Wolfram Alpha, and it says that ...
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0answers
13 views

Finding the fundamental Pell solution from a system of Pell-like equations

Assume $d$ is a non-square integer, and $r,s,t,w$ are integers, and $n$ and $m $ are integers with $n,m \neq 0,\pm 1$, satisfying the system of Pell-like equations \begin{align} r^2-ds^2 &= m, \\ ...
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1answer
65 views

The number of integral solutions $(x,y)$ of $x^3+3x^2y+3xy^2+2y^3=50653$

This was a wonderful question given to me by professor in my last class test. He asked for the solution with the least number of steps. Find the number of integral solutions $(x,y)$ of the ...
2
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3answers
111 views

Integer solutions to $x^2-xy+y^2=1$

What are the integer solutions to $x^2-xy+y^2=1$? (I found the solution below while working on another problem, so I thought I'll add it to the knowledge base here.)
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2answers
113 views

Find all positive inegers solution for $x^2-xy-y^2=1$

Find all positive inegers solution for the following diophantine equation $$x^2-xy-y^2=1$$ My work so far 1)$$x^2-xy-y^2-1=0$$ $$D=y^2+4(y^2+1)=5y^2+4=k^2, k \in \mathbb Z$$ 2)$$ ...
2
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1answer
35 views

Solving $(ap)^2-d(bq)^2=1$ for distinct primes $p,q$

I'm pondering the following claim regarding special cases of the Pell equation. Conjecture: For every pair of distinct primes $p$ and $q$, there exist integers $a$ and $b$, and a non-square integer ...
1
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1answer
18 views

Hilberts tenth problem over $\mathbb R$ with coefficients in $\mathbb Q$

Consider the following decision problem: Given: An equation $f(x_1, \dots, x_n) = 0$ where $f(x_1, \dots, x_n)$ is a polynomial with variables $x_1, \dots, x_n$ and coefficients in $\mathbb Z$. To ...
2
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1answer
72 views

A seemingly-trivial divisibility conjecture

While working on another problem, I stumbled on the following divisibility claim. Conjecture: No integers $a,b,c,d$ satisfy all of the following conditions: $a^2+b^2-c^2-d^2 = 2(ad-bc)-1$; ...
3
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1answer
112 views

How many pairs $ (a,b)$ of integers such that , $a^2b^2=4a^5+b^3 $

I would appreciate if somebody could help me with the following problem: $Q$: How many pairs $ (a,b)$ of integers such that $$a^2b^2=4a^5+b^3 $$
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0answers
27 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 ...
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3answers
144 views
+50

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 ...
2
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1answer
63 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
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1answer
86 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
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2answers
102 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 ...
4
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1answer
57 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 $2016=2^5\cdot 3^2 \cdot 7$ $x-y=1; x+y=2016$ ...
7
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1answer
160 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 ...
13
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1answer
127 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 ...
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1answer
59 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
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4answers
127 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 ...
4
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1answer
53 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 ...
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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
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3answers
86 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
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1answer
46 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
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0answers
26 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 ...
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1answer
46 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 ...
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0answers
26 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. $(a,b,c) = (m^2-n^2,2mn,m^2+n^2)$ ...
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2answers
147 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 ...
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1answer
77 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 ...
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1answer
44 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 $>0$ $$\implies 16x^4-4x^3>0$$ $$\implies x \text { belongs to } (-\infty,0) \cup (1,\infty)$$ (So we have ...
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0answers
36 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 ...
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3answers
33 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
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0answers
23 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 ...
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2answers
42 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 ...
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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
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1answer
111 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 ...
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0answers
15 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
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1answer
61 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
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3answers
39 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
89 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
409 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 = ...
3
votes
1answer
74 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
86 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 ...
0
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1answer
38 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
59 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 ...