Questions on finding integer/rational solutions of equations.

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2
votes
3answers
93 views

System of diophantine equations $x^2+3y=u^2$, $y^2+3x=v^2$

Solve the following system of Diophantine equations(the unknowns are positive integers): $$ \left\{ \begin{array}{c} x^2+3y=u^2 \\ y^2+3x=v^2 \end{array} \right. $$ I worked as follows: ...
5
votes
0answers
32 views

How many generators needed for Pell-equation-related group

Let $d$ be a positive integer which is not a perfect square. We have the norm multiplicative group homomorphism, $N:{\mathbb Q}[\sqrt{d}] \to {\mathbb Q}$ defined by $N(x+y\sqrt{d})=x^2-dy^2$. It ...
1
vote
3answers
34 views

Diophantine equations using Euclidean algorithm

I solved two systems of Diophantine equations using the Euclidean algorithm and I can't figure out where I went wrong because the solutions I test aren't working but I have rechecked my work several ...
5
votes
2answers
53 views

Pell equation in ${\mathbb Q}(x)$

Is it known whether the equation $A^2-(x^2+3)B^2=1$ has a solution $A,B\in{\mathbb Q}(x)$ with $B\neq 0$ ? My thoughts : I think that there is no solution, as the fundamental solution of $A^2-(x^2+3)...
-1
votes
1answer
59 views

Given $N$ find the number of natural numbers less than $N$ that may be written in the form $\frac{(k)(k+1)}{2}$

Given $N$, find the number of natural numbers less than $N$ that may be written in the form $$\frac{k(k+1)}{2},$$ where $k\in \Bbb N$. I know that the answer to this problem is approximately $\sqrt {...
4
votes
4answers
81 views

Showing that there are infinitely many integer solutions for the hyperbolic formula $|a^2 - 26 b^2| = 1$

I want to show that the formula $$ | a^2 - 26\cdot b^2| = 1$$ has infinitely many solutions $(a, b) \in \mathbb{Z}^2$. First I tried to solve the formula for one of the two variables, to get ...
5
votes
0answers
129 views

Diophantine equation with binomial coefficient

Suppose that $p$ is a prime number and $p \le q \le p^2$ is an integer. How many solutions are there to the following equation? $$\binom{p^2}{q}-\binom{q}{p}=1$$ This question was proposed ...
0
votes
1answer
44 views

Solve the following diophantic equations

I can't seem to find the solution to two problems in my textbook. They ask us to solve the diophantic equations: 1) $xy²-2y²-x-6=0$ $4x²-4xy+y²-9=0$ I tried several things but these two just ...
3
votes
2answers
110 views

Solutions to $\lfloor x\rfloor\lfloor y\rfloor=x+y$

Find all solutions to $$\lfloor x\rfloor\lfloor y\rfloor=x+y$$ and show that the non-Integral solutions lie on two unique lines. Also determine the equations of these 2 lines. I divided the problem ...
4
votes
2answers
119 views

Nature and number of solutions to $xy=x+y$

Find all solutions to $$xy=x+y$$ Initially the given condition was $x,y\in \Bbb{Z}$. $$$$In this case, I just guessed that the solutions were $(0,0)$ and $(2,2)$. As far as I can see, these are the ...
2
votes
4answers
158 views

Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.

Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?
4
votes
1answer
182 views

Three Colour Analogue of Boolean Pythagorean Triples Problem

Having read about the Boolean Pythagorean Triples Problem (see here and this question), it occurred to me that a related problem would require the integers to be coloured in three rather than two ...
3
votes
1answer
38 views

Positive integer solutions to $1\cdot m!\cdot(m^2)!\cdots (m^p)!=2(n^p)!$

Let $m$ and $n$ be natural numbers and $p$ a positive integer such that $$1\cdot m!\cdot(m^2)!\cdot\ldots\cdot(m^p)!=2(n^p)!$$ One solution is $(m,n,p)=(2,1,1)$. Are there any others?
3
votes
0answers
273 views

Sum of the cubes of a Pythagorean triple equal a cube.

Apart from (3, 4, 5, 6) are there any more primitive solutions to $x^3+y^3+z^3=w^3$ where $x^2+y^2=z^2$ ? I’ve noted that if gcd(x ,y ,z) = k, then k divides w, so non-primitive Pythagorean triples ...
3
votes
1answer
69 views

Finding all pairs of integers that satisfy a bilinear Diophantine equation

The problem asks to "find all pairs of integers $(x,y)$ that satisfy the equation $xy - 2x + 7y = 49$. So far, I've got \begin{align} xy - 2x + 7y &= 49 \\ x\left(y - 2\right) + 7 &= 49 \\ ...
4
votes
2answers
197 views

Minimum of $|ax-by+c|$

Find the minimum of the function $$ f(x,y)=|ax-by+c|$$ where $a,b,c \in \mathbb N$ and $x,y \in \mathbb Z$. The questions here and here are similar but they are in cases where $x, y$ are ...
0
votes
0answers
28 views

Finding smallest positive value of a function.

Given four positive integers $A,B,C$ and $D$, we have to find the minimum absolute difference between $A+qC$ and $B+wD$ where $q$ and $w$ are non-negative integers. I know it has something to do with ...
0
votes
0answers
35 views

Solution of diophantine equation with lowest c

Lets say I have a diophantine equation , aX - bY = c Now, for some (a,b,c) I may not have any integer solution at all. But lets say , I write the equation in this way , aX - bY = c + p p is an ...
-1
votes
1answer
55 views

The numbers that can be written as the sum of squares of two **natural** numbers [closed]

It's easy to solve for sum of two squares.but it becomes hard when we want numbers that can written as sum of squares of two natural number.For example given number $n$ can be written as the sum of ...
0
votes
1answer
15 views

A bilinear diophantine problem

Suppose we know $a,b,c,d,e,f,m\in\Bbb Z$ in $$(a^2c+b^2d)y+ab(vy)+(a^2e+b^2f)v=m$$ how do we find $v,y\in\Bbb Z$?
3
votes
1answer
50 views

Show that the equation $x^2+y^2+z^2=x^2y^2$ has no integer solution,except $x=y=z=0$

Show that the equation $x^2+y^2+z^2=x^2y^2$ has no integer solution,except $x=y=z=0.$ Let one of the $x,y,z$ be even number.Let $x=2p$ $x^2+y^2+z^2=x^2y^2$ This gives $y^2+z^2$ is also even,which ...
0
votes
3answers
47 views

Find the value of $a$ if $x^2+y^2=axy$ has positive integer solution.

Find the value of $a$ if $x^2+y^2=axy$ has positive integer solution. My try: Let g.c.d of $x$ and $y$ is $d$ i.e.$(x,y)=d$ and let $x=dx',y=dy'.$ Then $x'^2+y'^2=ax'y'$ I am stuck here.The answer ...
4
votes
1answer
52 views
0
votes
1answer
54 views

General Conic and its Rational Solutions

Suppose you have a rational conic $ax^2+bxy+cy^2+dx+ey+f=0$. There is a theorem that states if a conic has 1 rational solution it has infinitely many rational solutions. How can you prove this ...
2
votes
0answers
37 views

Integers of the form $m^k-n^k$ [closed]

We know that an integer number is the difference of two squares if and only if it is not congruent to 2 mod 4. As a generalization, do we have a similar statement for integers of the form $m^k-n^k$, ...
0
votes
1answer
27 views

Find all $(a,b) \in \Bbb Z^2$ such that $b \equiv 2a \pmod 5$ and $28a+10b=26$

I'm stuck with this exercise: Find all $(a,b) \in \Bbb Z^2$ such that $b \equiv 2a \pmod 5$ and $28a+10b=26$ It's from my algebra class, we are looking into diophantic and congruence equations. ...
1
vote
1answer
66 views

Find all positive integers that solve Mordell's equation $y^2=x^3+37$

Find all Mordell's equation: $$y^2=x^3+k$$ where $k=37$ positive integer numbers,I can't find the when $k=37$ the mordell equation solution with some result,and we can known this equation have ...
2
votes
1answer
51 views

Prove this diophantine equation $2^a-3^b=5~,a,b\in N^{+} $ has no postive integers solution

show that the diophantine equation $$2^a-3^b=5~~~~,a>5,b>3,a,b\in N^{+} $$ has no postive integers solution maybe is old problem,But I try somedays,can't solve it by now
10
votes
0answers
356 views

Power Diophantine equation involving primes: $(p+q)^q-p^q-q^q+1=n^{p-q}$

Suppose $p$ and $q$ are prime numbers, and $n>1$ is a positive integer. Find all solutions to the following Diophantine equation:$$(p+q)^q-p^q-q^q+1=n^{p-q}$$ What I have tried: Obviously $p>q$...
6
votes
0answers
430 views

The Boolean Pythagorean triples problem, a $200$-terabyte proof, and $d=163$

I came across this interesting math article, "Computer cracks 200-terabyte maths proof" where one phrase caught my attention and I quote, "... all triples could be multi-coloured in integers up to $...
5
votes
1answer
64 views

Solving a Diophantine equation in three variables as a parametric equation in one variable

Let’s say that $a$, $b$, and $c$ are integers such that $$(b^2+2)^2=(a^2+2c^2)(bc-a). \tag{$\star$}$$ By brute force search, I think I’ve discovered that $$(a,b,c)=(5d+1,3d+1,d+2), \qquad d=\dots,-2,-...
3
votes
0answers
118 views

A power Diophantine equation $(p+q)^n-p^n-q^n+1=m^{p-q}$

Suppose $p$ and $q$ are prime numbers, $n>1$ and $m>1$ are positive integers. Solve the following Diophantine equation:$$(p+q)^n-p^n-q^n+1=m^{p-q}$$I made this problem and I was trying to find $...
12
votes
3answers
74 views

Solve for integers $x, y, z$ such that $x + y = 1 - z$ and $x^3 + y^3 = 1 - z^2$.

Solve for integers $x, y, z$ such that $x + y = 1 - z$ and $x^3 + y^3 = 1 - z^2$. I think we'll have to use number theory to do it. Simply solving the equations won't do. If we divide the second ...
1
vote
2answers
23 views

Find $b \in \Bbb Z$ for which exists $a \equiv 4 \pmod 5$ such that $6a+21b=15$

I'm starting to study diophantic equations and congruence and I have found this problem that I don't know how to solve: Find $b \in \Bbb Z$ for which exists $a \equiv 4 \pmod 5$ such that $6a+21b=...
3
votes
1answer
25 views

Bilinear diophantine equations

Is there a fast way ($O((\log n)^c)$) to solve $$ax+by+xy=n$$ over integers when $a,b$ are known and $0<x,y<a,b$ holds?
0
votes
1answer
34 views

Integer solution to linear equation [duplicate]

I need to find a good configuration for my computational kernel, which forces me to find some integer solutions to the following simple equation: $a \cdot x - b \cdot y = c$, where $a$, $b$ and $c$ ...
-2
votes
2answers
61 views

Simple but hard 2 by 2 system in $x$ and $y$ [duplicate]

Is there a systematic way of solving this system, analytically? $$\begin{cases} x \ + \ y^2=11\\ x^2+y\ \ =\ 7\\ \end{cases} $$ I mean, other than brute-force.
7
votes
1answer
123 views

A very difficult Diophantine problem $n^2 \mid 3^n+2^n+1$

Prove that $n=3$ is the only positive integer greater than $1$, for which$$n^2 \mid 3^n+2^n+1$$This is a conjecture.
3
votes
3answers
64 views

Prove that the diophantine equation $x^2 + (x+1)^2 = y^2$ has infinitely many solutions in positive integers.

Prove that the diophantine equation $x^2 + (x+1)^2 = y^2$ has infinitely many solutions in positive integers. Now, that's a Pythagorean Triplet. So, we have to prove that there are infinitely many ...
0
votes
0answers
36 views

Solving an integer equation (equi-energy transition)

In chemistry, we came across an equation as follows: $$\frac{Z_1^2}{n_1^2}-\frac{Z_1^2}{n_2^2}=\frac{Z_2^2}{n_3^2}-\frac{Z_2^2}{n_4^2}$$ We were supposed to assume that this implied that $$\frac{...
0
votes
0answers
31 views

Two variables diophantine equation and divisibility

Let $n\in\mathbb{N}$ such that $n\mid35m+26$ and $n\mid 7m+3$. Find $m\in\mathbb{Z}$ I dont know how to start, i tried by writting $n=k_{1} (35m+26)=k_{2} (7m+3)$ for some $k_{1} , k_{2} \in \mathbb{...
1
vote
4answers
188 views

Proving that an equation doesn't have integer solutions

I need to prove that there are no integer solutions for a bunch of equations like the following: $$15x^2 - 7y^2 = 9$$ I was able to solve some simpler ones by picking a dividend and looking into it's ...
1
vote
0answers
40 views

Normalizing an elliptic curve to find integer solutions

I have an elliptic curve $$ c_1y^2 + a_1xy + a_3 = c_2x^3 + a_2x^2 + a_4x + a_6 $$ with integers $a_1,a_2,a_3,a_4,a_6,c_1,c_2$ and I would like to find all integer solutions of this elliptic curve. I ...
2
votes
1answer
95 views

Find all solutions to the Diophantine equation $x^2-7y^2=-3$

I want to find all integer solutions of the equation $$x^2-7y^2=-3$$ I don't really know where to start... I tried the one trick I know which is to factor in some quadratic ring: $$(x+\sqrt{-3})(x-\...
-2
votes
1answer
43 views

Meta-Pythagorean Triple

How can I find all Pythagorean triples $(a,b,c)$ such that the hypotenuse $c$ is a leg in another Pythagorean triple? For example, $(3,4,5)$ is such a Pythagorean triple because the length of the ...
0
votes
0answers
22 views

Quaternary quadratic modular problem.

Consider quadratic form $$Q(w,x,y,z)=w^2-x^2-y^2+z^2$$ and fix $r\in(0,\frac12)$ and pick a large enough $n\in\Bbb N$. How do we find a solution to $$Q(w,x,y,z)\bmod n=0$$ on condition that $$\sqrt n\...
0
votes
1answer
57 views

Find all integers $a,b,c$ that satisfy: $a^3 - 3a^2b - 3c+2b^2 = c^3 -3ab^2 + 3c^2 +1 $

(From a math competition) Question: Find all integers $a,b,c$ that satisfy: $$a^3 - 3a^2b - 3c+2b^2 = c^3 -3ab^2 + 3c^2 +1 $$ What I have tried/attempted basically I've been looking for ...
-2
votes
2answers
70 views

A quick method to solve $89y-273x=40$

how to solve this equation $$89y-273x=40$$ I saw this question somewhere and this obviously can be solved by hit and trial but is there an easier method to solve it, something more definite? I need ...
8
votes
0answers
246 views

Positive integers $a,b$ satisfying $a^3+a+1=3^b$

How to prove that $a=b=1$ is the only positive integer solution to the following Diophantine equation?$$a^3+a+1=3^b$$
0
votes
1answer
17 views

Exponential equation with square variable as an exponent?

I am trying to solve the following exponential equation where the variable is squared. Most likely it is not difficult, but I am just missing the technique: what is the way to solve an exponential ...