Questions on finding integer/rational solutions of equations.

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0
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1answer
22 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
53 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
36 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
3
votes
0answers
53 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. Solve the following Diophantine equation:$$(p+q)^q-p^q-q^q+1=n^{p-q}$$ What I have tried: Obviously $p>q$. First let ...
5
votes
0answers
109 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 ...
4
votes
1answer
43 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 ...
11
votes
4answers
58 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 ...
1
vote
2answers
21 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 ...
3
votes
1answer
24 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
19 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 $c$ ...
0
votes
0answers
29 views

On Markov triples and the square root of $2\times 2$ matrices

Let for two Markov triples $a^2+b^2+c^2=3abc,$ and $\alpha^2+\beta^2+c^2=3\alpha\beta c$, where we (can) take $a\leq b\leq c$ and $\alpha\leq \beta\leq c$. Then one has the ratio between RHS's and ...
-2
votes
2answers
59 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
112 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
58 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 ...
0
votes
0answers
30 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 ...
0
votes
0answers
25 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 ...
1
vote
4answers
173 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
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0answers
34 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
89 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: ...
-1
votes
1answer
35 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
63 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 ...
0
votes
1answer
54 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
51 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 ...
7
votes
0answers
202 views

Seemingly easy Diophantine equation $a^3+a+1=3^b$

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
46 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
81 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 ...
1
vote
1answer
45 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
72 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 ...
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 ...
2
votes
1answer
76 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
42 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 $$ ...
2
votes
5answers
82 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\ ...
6
votes
2answers
66 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 ...
-2
votes
1answer
77 views

Prove an equation has no integer solutions… [closed]

I know that ${x^3} - 8{y^3} = 12$ has no integer solutions but how can I prove it? If I had to sit down with someone and convince them (at least, fairly) rigorously that it has no integer solutions. ...
-4
votes
1answer
35 views

No solution in naturals

Prove that there is no pair $(x,y)$ of positive integers such that $$axy-b=x(x-c)+y(y-d)$$ where $a,b,c,d$ are positive integers such that $a>b>(\frac{1}{2} \cdot max\{c,d\})^2$.
0
votes
0answers
39 views

Is there a short proof for Bézout's lemma when $\gcd(a,b)=1$?

Bézout's lemma states that there exist integers $x$ and $y$ such that $$ax+by=\gcd(a,b)$$ Is there some short proof for this when $a$ and $b$ are coprime? As opposed to something like this, like, ...
1
vote
1answer
72 views

Show that $x^3 + y^3 + z^3 + t^3 = 1999$ has infinitely many integer solutions.

Show that $x^3 + y^3 + z^3 + t^3 = 1999$ has infinitely many integer solutions. I have not been able to find a single solution to this equation. With some trial I think there does not exist a ...
3
votes
3answers
63 views

Find $(m,n)$ where $m$ and $n$ are positive integers.

Find all positive integers $m$ and $n$, such that: $$\frac 1m + \frac 1n - \frac 1{mn}=\frac 25$$ Actually, I have already solved this problem using inequality. The solutions I have found are: ...
1
vote
0answers
12 views

Bound on smallest $n$ for consistency of a system of equations?

Given small $\epsilon>0$ how small should $n\in\Bbb N$ be such that if $a,b,c,d,q,r,u,v,x,y,m,m'\in\Bbb N$ with $gcd(a,b)=gcd(a,x)=gcd(b,y)=1$ the following relations can hold with constraints ...
4
votes
3answers
164 views

How to find integer solutions to $M^2=5N^2+2N+1$?

My number theory is terrible so I don't know what "class" of problem this secretly is. I'm looking for all positive integer solutions to the equation: $M^2=5N^2+2N+1$ That is, I want positive ...
3
votes
0answers
17 views

Diophantine equation $10^x=yzwt-3$

I have resolved, brute force, the following problem someone asked me: Solve the Diophantine equation $$10^x=yzwt-3\space \text{where}\space \space y,z,w,t \space \text {are distinct primes}$$ $$ ...
1
vote
1answer
41 views

Trying to understand Diophantine equations.

I'm revising for an upcoming exam and do not understand how you solve these questions at all. It all seems like a bit of trial and error to get any type of solution. The questions I am looking at are: ...
4
votes
1answer
60 views

Diophantine equation involving factorials

$$x!+y=y^3$$ $$y=\sqrt[3]{x!+\sqrt[3]{x!+\sqrt[3]{x!+\cdots}}}$$ The only integer solutions to these identities that I have found are: $$3!+2=2^3$$ $$4!+3=3^3$$ $$5!+5=5^3$$ $$6!+9=9^3$$ I ...
2
votes
1answer
62 views

Diophantine $4x^3-y^2=3$

I am interested in how to tackle this Diophantine equation: $$4x^3-y^2=3$$ The solutions I have found so far are $(1,1)$ and $(7,37)$. Are there any more? I have looked up various material on cubic ...
3
votes
2answers
52 views

Diophantine $xy+yz+zx=4(x+y+z)$

How do you solve the Diophantine equation $xy+yz+zx=4(x+y+z)$ for positive integers $x,y,z$? My approach was to consider $d=\gcd(x,y,z)$. I could just about show that the equation has no ...