# Tagged Questions

Questions on finding integer/rational solutions of equations.

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### On representing the general solution for the diophantine equation $a_1x_1+\dotsb+a_nx_n=c$

On representing the general solution with the special solutions for the diophantine equation $$a_1x_1+a_2x_2+\dotsb+a_nx_n=c$$ here $a_1 ,a_2, \dotsb,a_n,c\in\Bbb Z,(a_1 ,a_2, \dotsb,a_n)=1$. Can ...
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### Divisors of Pell Equation Solutions

Let $d > 0$ be square-free. Let $\epsilon = x_0 + y_0 \sqrt{d}$ be the minimal solution to the Pell's equation $x ^ 2 - d y ^ 2 = 1$. Let $x + y \sqrt{d} = \epsilon ^ l, l \geq 1$ be a solution. ...
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### Show that $0 = 2a^3-5ab^2+25b^3$ has no other integer solutions than $a = b = 0$.

I am trying to solve the following problem: I have the equation $0 = 2a^3-5ab^2+25b^3$, where $a,b \in \mathbb Z$. Obviously, $a = b = 0$ is a solution of this equation. But how can I show that there ...
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### Solving equation involving the ceiling function

How can I solve the equation $$\lceil \log_{b}{1024} \rceil = n$$ where $n \in \mathbb{N}$ in terms of $b$? I have seen equations of a similar form (Solving an equation with floor function before), ...
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### Integers can be expressed as $a^3+b^3+c^3-3abc$

$$S=\{a^3+b^3+c^3-3abc|a,b,c\in\Bbb Z\}$$ Can we decide $S$? that is, we want to find all integers of the form $a^3+b^3+c^3-3abc$. obviously, if $m,n\in S$, then $mn\in S$, so we only need to ...
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### prove Diophantine equation has no solution $\prod_{i=1}^{2014}(x+i)=\prod_{i=1}^{4028}(y+i)$

show that this equation $$(x+1)(x+2)(x+3)\cdots(x+2014)=(y+1)(y+2)(y+3)\cdots(y+4028)$$ have no positive integer solution. This problem is china TST (2014),I remember a famous result? maybe is a ...
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### An algorithm for solving linear diophantine equations?

I am entering an interesting team based math contest called the purple comet, and quite a lot of questions on this contest involve Diophantine equations. For this contest, you are given a computer, ...
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### Finding all possible values

we have to find all possible prime values $(p,q,r)$ such that $pq = r + 1$ $2(p^2+q^2) = r^2 + 1$ I do not know how to start looking for an answer.
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### Can we find positive integers $a$ and $k \geq 2$ with $2^n - 1 = a^k$?

I would appreciate if somebody could help me with the following problem: For a given positive integer $n$, can we find positive integers $a$ and $k$ ($k\geq 2$) such that $2^n-1=a^k$? The ...
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### Prove that $2^x \cdot 3^y - 5^z \cdot 7^w = 1$ has no solutions

Prove that $2^x \cdot 3^y - 5^z \cdot 7^w = 1$ has no solutions in $\mathbb{Z}^+$, if $y\ge 3$.
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### What's the set p,if $37x^2-113y^2=p$ is solvable,with p a prime

if $37x^2-113y^2=p$ is solvable.with p a odd prime. What's the set of all $p$? Does it have a formula?
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### Number of Solutions to a Diophantine Equation

I am asked the following: Show that the number of integer solutions to $y^p=x^2+2$ for any odd prime $p$ is at most $p-1$. I checked that for $y^p=x^2+2$, the same method for $y^3=x^2+2$ works ...
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### Real valued function associated with the Diophantine equation $a^2(2^a-a^3)+1=7^b$

The parent question that maybe still remains to be answered at this moment is:Solve the Diophantine equation $a^2(2^a-a^3)+1=7^b$ . As far as the parent question is concerned, when generalizing to ...
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### Solution of a simple linear diophantine equation

I'm having a slight problem with a simple equation of the sort $a_1+a_2+a_3...=n$. Where $n,a_1, a_2, a_3... \in N$. I do know how to find the number of solutions to these equations when they are of ...
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### Cubic diophantine equation

How can I solve the equation $x^3+x-1=y^2$ in positive integers? I know this equation defines an elliptic curve but this seems to be a non-elementary way to solve the question. Is there a more ...
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### Equation with an infinite number of solutions

I have the following equation: $x^3+y^3=6xy$. I have two questions: 1. Does it have an infinite number of rational solutions? 2. Which are the solutions over the integers?($x=3$ and $y=3$ is one) ...
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### $\left(x+{2\over x^2+x}\right)\left(y+{2\over y^2+y}\right)$ product is equal to positive integers, general solution

Given $\left(x+{2\over x^2+x}\right)\left(y+{2\over y^2+y}\right)$ this product is equal to positive integers. $x,y$ are both positive. Conditions for general solution is required. List a few ...
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### Four integers that satisfy $a+b+c+d\; =\; -3$ and $a^{3}+b^{3}+c^{3}+d^{3}\; =\; 3$

Find a set of 4 integers that satisfy $$a+b+c+d\; =\; -3$$ and $$a^{3}+b^{3}+c^{3}+d^{3}\; =\; 3$$ I am really not sure how to proceed. I tried letting $d = -c$ to see if that would yield a possible ...
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### What is quadratic equations in Algebra?

Yesterday someone asked a question in SE about indeterminate quadratic equations(of the form $x^2−ny^2=1$ which got me really interested in them and I thought I would try to learn something related to ...
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### rational points on particular elliptic curve

I do have a few books that discuss elliptic curves, however... What are the rational points on $$y^2 = 4 x^3 - 4 x = 4 x(x-1)(x+1)?$$ I think it ought to be $(-1,0), (0,0), (1,0).$ Maybe it's ...
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### There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

According to Wikipedia, There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. I cannot find the proof in the citation provided. I am ...
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### Solving $6ab+a+b-36pq-19p-13q=7$ where $a,b,p,q \in \mathbb{N}$, $a,b,p,q \neq 0$

Is there an efficient way to find solutions to the equation: $6ab+a+b-36pq-19p-13q=7$ where $a,b,p,q \in \mathbb{N}$ and $a,b,p,q \neq 0$ If the equation has no solutions, how could you prove that, ...
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### $x^x + y^x=x^y + y^y$ positive integer solutions?

required is positive solutions for $x^x + y^x=x^y + y^y$? And negative integer solutions as well if possible?
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### how do i prove $ab|n$ if $\gcd(a, b) = 1$ and $a|n$ and $b|n$?

Suppose that, for integers $a, b,$ and $n,$ $$\gcd(a, b) = 1\text{ and }a|n\text{ and }b|n.$$ How do I prove that $ab|n$ using linear Diophantine equations? Can I extend the above result to the ...
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### solving $x^3-2y^3=1$ using cubic number field

I am trying to solve the diophantine equation $x^3-2y^3=1$ using $\mathbb{Q}(\sqrt[3]{2}).$ I've read this link: Solve $x^3 +1 = 2y^3$ The following is what i have tried: Finding all integer ...
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### For which primes $p$ does $px^2-2y^2=1$ have a solution?

Let $p$ be an odd prime. If $px^2-2y^2=1$ is solvable, we can get Jacobi symbol $(\frac{-2}{p})=1$, so $p=8k+1,8k+3$. But when $k=12$, $p=97$, the Pell equation $97x^2-2y^2=1$ is unsolvable. I think ...
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### Solve the Diophantine equation $a^2(2^a-a^3)+1=7^b$.

The problem is to find all positive integers $a$ and $b$ such that $a^2(2^a-a^3)+1=7^b$. I found a=10, and my intuition tells me there are no more solutions. I've also shown that $a=42k+10$ for some ...
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### Co-primality of coefficients of coprime integers

Given that $a,b$ are co-prime, we have infinitely many solutions for $x,y$ to the equation $$ax+by=c.$$ Furthermore, solutions have the form: $x=ca^{-1}+tb,y=cb^{-1}-ta$. Given that $c$ can ...
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### Can 11 be represented by $a^2 - 3b^2$ where a and b are integers?

I know the answer is no, just wan't to know how. From a similar question on the site I got that $a^2 - 3b^2$ should always equal a square modulo 3 which 11 is not. But I don't understand how to get to ...
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### Equation in rational numbers

I can't seem to find the way to solve the following equation so help would be much appreciated.. $x^2+y^2=x^3+y^3$ over $\mathbb{Q}$
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### For a Pythagorean triple $a, b, c \in N$, if a is even and a, b, c have no common factors, prove that $c-a$ and $c+a$ are perfect squares.

What property(s) should I examine to approach this problem?
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### A conjecture about Pythagorean triples

I noticed for the integer solutions of $a^2 + b^2 = c^2$, there don't seem to be cases where both a and b are odd numbers. In fact, I saw this property pop up on a nice question, which required you to ...
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### Diophantine equation using Pell's equation

I asked this question some days ago: Is there a way to find for which A the system $X^2+Y^2=Z^2+T^2+1$ $XZ−YT=A$ has only one solution in positive integers? Looking for the solution of the problem,...
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### Number of solutions for (a-x)(b-y)-1=0

How to find number of integer solutions for x and y for given values of a and b . is it related to number of divisors , i read it from a post bit didn't get it . Anybody can explain with example .
### Solve $x^2+2=y^3$ using infinite descent?
just so this doesn't get deleted, I want to make it clear that I already know how to solve this using the UFD $\mathbb{Z}[\sqrt{-2}]$, and am in search for the infinite descent proof that Fermat ...