Questions on finding integer/rational solutions of equations.

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1answer
54 views

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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0answers
69 views

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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3answers
116 views

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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1answer
474 views

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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2answers
195 views

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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4answers
492 views

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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0answers
91 views

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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3answers
75 views

Solving for $3^x - 1 = 2^y$

Besides $x=2, y=3$, are there any other solutions? I know that if there is another solution: $y$ is odd since $2^y \equiv -1 \pmod 3$ $x$ is even since $3^x - 1 \equiv 0 \pmod 8$ $3 | y$ since $-1 ...
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1answer
86 views

Gap between smooth integers tends to infinity (Stoermer-type result)?

Consider the following claim : (*) Let $P$ be a finite set of primes, let $S$ be the set of natural numbers all of whose divisors are in $P$, and let $s_n$ denote the $n$-th element of $S$. Then $s_{...
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1answer
183 views

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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3answers
93 views

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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1answer
104 views

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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1answer
85 views

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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0answers
126 views

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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1answer
47 views

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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1answer
78 views

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 ...
3
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1answer
479 views

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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4answers
407 views

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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1answer
120 views

$\left(x+{2\over x^2+x}\right)\left(y+{2\over y^2+y}\right)$ 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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3answers
100 views

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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1answer
110 views

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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1answer
117 views

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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2answers
178 views

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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4answers
105 views

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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1answer
110 views

$x^x + y^x=x^y + y^y$ 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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4answers
48 views

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 ...
4
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1answer
205 views

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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1answer
86 views

Problem with Diophantine equation

Let $a,b \in \mathbb N$ be coprime. Prove that for all $n\in \mathbb N$ such that $n>ab$ there are $r,s\in \mathbb N$ such that $n=ra+sb$. I'm really stuck on this problem. I know that since $(a,b)...
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4answers
116 views

Diophantine equation: $(x-y)^2=x+y$

I have to solve the following equation: $(x-y)^2=x+y$, where $x$ and $y$ are non-negative integers. This equation has an infinite number of solutions, but how to prove that there exists a positive ...
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3answers
385 views

Solving $x^p + y^p = p^z$ in positive integers $x,y,z$ and a prime $p$

The question is from Zeitz's ''The Art and Craft of Problem Solving:" Find all positive integer solutions $x,y,z,p$, with $p$ a prime, of the equation $x^p + y^p = p^z$. One thing I noticed is ...
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0answers
113 views

15th power Diophantine equation

I'd appreciate some help (a hint) for the following. If $x,y>1$ are so that $2x^2-1=y^{15}$ then $x$ is a multiple of $5$. Don't know if this helps but the equation can be rewritten as $2x^2=(y^5+...
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1answer
85 views

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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1answer
212 views

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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1answer
93 views

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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2answers
73 views

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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1answer
81 views

Diophantine Equation with 15th power

So I'm working on the Diophantine equation $2x^2-1=y^{15}$ (1) with $x,y>1$ In particular I want to show that x must be a multiple of 5. I have found that it suffices to show that for $y=1 \pmod{...
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3answers
150 views

Integer solutions of $ x^3+y^3+z^3=(x+y+z)^3 $

Consider the equation $$ x^3+y^3+z^3=(x+y+z)^3 $$ for triples of integers $(x, y, z) $. I noticed that this has infinitely many solutions: $ x, y $ arbitrary and $ z=-y $. Are there more solutions?
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1answer
149 views

How to solve equations like $x^2+y^2=2004^{2005}$?

I've found this kind of equation but I think I haven't enough mathematical tools to solve it. What would you do? $$x^2+y^2=2004^{2005}$$ Another kind: $$x^2+y^2=2005^{2004}$$
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0answers
70 views

How to solve diophantine equation $\frac{x^p-y^p}{x-y}=n$

$$\frac{x^p-y^p}{x-y}=n$$ whit $p$ a prime greater than or equal to $3$,for what value to $n$, it's solvable and how to solve,and whether $\frac{x^p-y^p}{x-y}=q_1$ $\frac{x^p-y^p}{x-y}=q_2$ is ...
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5answers
247 views

If $(m,n)\in\mathbb Z_+^2$ satisfies $3m^2+m = 4n^2+n$ then $(m-n)$ is a perfect square.

I came across this question on another forum. The question is: $$ \text{If $m,n\in \mathbb{Z}_+$ such that $3m^2+m=4n^2+n$, then $(m-n)$ is a perfect square.}$$ I have managed to partially prove ...
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1answer
54 views

Given the sum of a geometric progression and the number of terms, can we recover the progression?

Consider a set of numbers which are in geometric progression: $n, nd, nd^2, \ldots ,nd^{M-1}$ Their sum is $S=\frac{n(d^M-1)}{d-1}$. Now if we know the values of $S$ and $M$, can we find values of $...
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1answer
64 views

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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3answers
76 views
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238 views

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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0answers
108 views

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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1answer
28 views

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 .
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4answers
401 views

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 ...
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2answers
89 views

nonlinear diophantine equation $x^2+y^2=z^2$

how to solve a diophantine equation $x^2+y^2=z^2$ for integers $x,y,z$ i strongly believe there is a geometric solution ,since this is a pythagoras theorem form or a circle with radius $z$ $x^2+y^2=...
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3answers
112 views

Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$.

Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$. E.g. $p(7)=2\cdot3\cdot5$ (and $n=7$ is a solution). Let $$p(n)=2\...