Questions on finding integer/rational solutions of equations.

learn more… | top users | synonyms

3
votes
0answers
210 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
68 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
195 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
33 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
52 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
13 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
51 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
36 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
65 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
50 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
348 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
302 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
61 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
117 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
73 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
30 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.
8
votes
1answer
122 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.
2
votes
3answers
60 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
34 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
29 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
178 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
38 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
94 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
41 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
2answers
70 views

Number of positive integer solutions to the equation $(a+b+c)(x+y+z+w) = 15$ [closed]

What is the total number of positive integer solutions to the equation? $$(a+b+c)(x+y+z+w) = 15$$ I could not find a way to solve this algebraically. The way which all other answers are telling i ...
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
57 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
244 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
16 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 ...
4
votes
1answer
52 views

Power Diophantine equation: $(a+1)^n=a^{n+2}+(2a+1)^{n-1}$

How to solve following power Diophantine equation in positive integers with $n>1$:$$(a+1)^n=a^{n+2}+(2a+1)^{n-1}$$
1
vote
1answer
82 views

Sum of two consecutive squares equal square

$N^2 + (N+1)^2 = K^2$, find all solutions for $N<200$ I have done some factoring and also realized that $ K=[n\sqrt{2}]+1$ in eventual solutions, where $[x]$ denotes the greatest integer less than....
1
vote
1answer
53 views

solve pairs of two variable simultaneous linear modular equations

I’m looking for a method to solve pairs of simultaneous linear modular equations, such as 323x + 37y = 0 Mod 243; -397x + 683y = 0 Mod 32 I’ve simplified this to 80x+37y = 243g; 19x+11y = ...
2
votes
2answers
81 views

Odd binomial sum equality has only trivial solution?

Suppose $$\sum_{k\ {\rm odd}}^n {n \choose k} 2^{(k-1)/2} = \sum_{k\ {\rm odd}}^m {m \choose k} 2^{(k-1)/2} 3^{(m-k)/2}.$$ Does $m=n=1$? Clearly $m \leq n$, and for every $n$ there is at most one $m$....
2
votes
1answer
59 views

the number of integer solutions to $y^p = x^2 +4$

Let $p>2$ be prime, investigate the number of integer solutions to $$y^p = x^2 +4$$. The first part of the question was find solutions to the equation $y^3 = x^2 +4$, I could do this and I see the ...
0
votes
0answers
19 views

Question about $F(x,y)=m$

Let $F(x,y)$ be a homogeneous polynomial of degree $\ge3$ with mutually prime coefficients, then we consider the problem $$F(x,y)=m\tag1$$ such that $m$ is an integer, we set $f(x):=F(x,1)$ then why ...
2
votes
1answer
77 views

Determine all integral solutions of $a^2+b^2+c^2=a^2b^2$

I started like this : $a^2+c^2=b^2(a^2-1)\\c^2 +1=(a^2-1)(b^2-1)$ but it's leads to nowhere. can you help please ?
1
vote
3answers
46 views

Solving $rX_1^2+sY_1^2+tZ_1^2=rX_2^2+sY_2^2+tZ_2^2$ completely in integers

Given pairwise relatively prime integers $r,s,t$, I’m looking for a complete solution (i.e., integer parameterization or similar) for the Diophantine equation $$ rX_1^2+sY_1^2+tZ_1^2=rX_2^2+sY_2^2+...
2
votes
5answers
85 views

Find all positive integers $n$ such that $n^2+n+43$ becomes a perfect square

Find all positive integers $n$ such that $n^2+n+43$ becomes a perfect square. Since $n^2+n+43$ is odd,if it's a perfect square it can be written as: $8k+1$,then: $$n^2+n+43=8k+1\Rightarrow\ n^2+n+42=...
6
votes
2answers
67 views

number of integer solutions to $2x_1 + x_2 + x_3 = n$

I'm working on a problem for which I need to efficiently compute the number of integer solutions to equations of the form $x_1 + \cdots + x_k = n$ with some subset of $\{x_1, \dots, x_n\}$ equivalent. ...
0
votes
1answer
34 views

A diophantine equation of degree 3

Find the integer solutions of $y^2+6=x^3$. I guess it does not have integer solutions but I cannot prove it. By $\pmod 8$, I can know that $y$ is odd and $x\equiv7 \pmod 8$. Then what else can I do?
1
vote
3answers
133 views

Find all integral solutions of the equation $x^n+y^n+z^n=2016$

Find all integral solutions of equation $$x^n+y^n+z^n=2016,$$ where $x,y,z,n -$ integers and $n\ge 2$ My work so far: 1) $n=2$ $$x^2+y^2+z^2=2016$$ I used wolframalpha n=2 and I received the ...