Questions on finding integer/rational solutions of polynomial equations.

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3
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1answer
34 views

For a given positive integer $n>1$ , how to find all positive integers $s,t$ such that $n^s-(n-1)^t=1$ ?

For a given positive integer $n>1$ , how to find all positive integers $s,t$ such that $n^s-(n-1)^t=1$ ? $s=t=1$ is clearly a solution . One more thing is clear that for any such $s,t$ we must have ...
0
votes
0answers
23 views

A bivariate quadratic diophantine equation

Given $a,b,c>0$, is there a procedure to solve $(x,y)\in\Bbb Z:ax^2+by^2=c$ in $O(\log^d c)$ arithmetic operations (either randomized or deterministic) with $d>0$ being fixed? Is there a ...
1
vote
0answers
20 views

Diophantine linear Equation Gaussian Integers

We know that $ax+by=c$ with $gcd(a,b)=1$ could be solved over $\Bbb Z$. Supposing if $a,b,c\in\Bbb Z[i]$, is there an analogous framework to find $x,y\in\Bbb Z[i]$ (at least of minimum norms)?
0
votes
0answers
53 views

Approximation of irrationals by rationals!

Let $\alpha$ be an irrational number and $\beta$ be an arbitrary real number, Prove that there are infinitely many pair of integers $(x,y)$ with $x\in\mathbb{N}$ such that: ...
5
votes
0answers
82 views

Solve $x^3=y^2-y+1$ in positive integers.

I recently started doing number theory and have finished with all the basic, intermediate and some of the advanced stuff with ease. However, I encountered this question and have been stuck for about a ...
5
votes
1answer
42 views

An integer sequence with integer $k$ norms

Find the maximum value of $n$(if exists) such that there exists a sequence $a_1,a_2,\ldots,a_n$ of positive integers such that for every $2\leq k \leq n$ $$\sqrt[k]{a_1^k+a_2^k+\cdots+a_k^k}$$ is ...
3
votes
0answers
32 views

A question on the Pell equation $x^2-5dy^2 = -1$

We know that a necessary but not sufficient condition such that, $$x^2-dy^2 = -1\tag1$$ is solvable is that $d$ is not divisible by a prime of form $4m+3$. It is not sufficient because the prime ...
0
votes
1answer
20 views

Question about the chakravala method on solving Pell's equation

I am currently reading on this old way of Pell's equation: http://en.wikipedia.org/wiki/Chakravala_method Looking at the section where they consider $N = 61$, it is not clear to me if the solution ...
5
votes
2answers
246 views

Diophantine system of two equations with four variables

Find all integer solutions for the system: $$\left\{\begin{array}{rcl}xy + vw &=& 5 \\ xv - yw &=& 6\end{array}\right.$$ It's supposed to be solvable by 9-graders...
0
votes
1answer
57 views

How do you solve the Problem below?

Let $u,v,w\in \mathbb{Z}>0$ denote 3 relatively prime integers(Pairwise coprime). If $(mn)$ is irrational, can we find 2 non-zero coprime (non-square) integers $u,v$ such that: ...
1
vote
1answer
41 views

$a,b$ are integers , both greater than $1$ , such that $(a^n-1)(b^n-1)$ is a perfect square for every positive integer $n$ , then $a=b$ ?

If $a,b$ are integers , both greater than $1$ , such that $(a^n-1)(b^n-1)$ is a perfect square for every positive integer $n$ , then is it true that $a=b$ ?
3
votes
2answers
56 views

How to find all positive integers $m,n$ such that $3^m+4^n$ is a perfect square?

How to find all positive integers $m$, $n$ such that $3^m+4^n$ is a perfect square? I have found $m=n=2$ is a solution, but cannot find any other and cannot prove whether there is any other solution ...
1
vote
1answer
25 views

Numbers to Powers of Number of Factors Equality

I'm trying to prove that the following equations have no solutions to finish a problem. They're intuitively impossible but I'm looking for rigorous arguments (if they are actually possible, then prove ...
4
votes
3answers
41 views

How many pairs of positive integers $(n, m)$ are there such that $2n+3m=2015$?

I know that $m$ must be odd and $m\le671$. Also, $n\le1006$. I can't go any further, any help?
-1
votes
0answers
58 views

diophantine-equations

Why there are no solutions in positive coprime integers for the following diophantine equation $$2x^3 + y^2 = z^k$$ where, (x,y,z) are (pairwise) positive coprime integers, and k is positive integer ...
3
votes
1answer
53 views

Help me finding $a+b+c$ in the given question

If $a,b,c$ are three positive integers such that $$abc+ab+bc+ca+a+b+c=1000$$ then what is the value of $a+b+c$?
1
vote
2answers
36 views

Sums of Consecutive Cubes (Trouble Interpreting Question)

Show that it is possible to divide the set of the first twelve cubes $\left(1^3,2^3,\ldots,12^3\right)$ into two sets of size six with equal sums. Any suggestions on what techniques should be used to ...
1
vote
1answer
31 views

Show that $x^2 − Dy^2 = 1$ has infinitely many integer solutions.

Let $D$ be a non-square positive integer. Suppose there are positive integers $a$ and $b$ such that $a^2 − Db^2 = 1$. Show that the Diophantine equation $x^2 − Dy^2 = 1$ has infinitely many integer ...
2
votes
3answers
73 views

Find rational points on $x^2 + y^2 = 3$ and on $x^2 + y^2 = 17$

$(a)$ Find all rational points on the circle $x^2 + y^2 = 3$, if there are any. If there is none, prove so. $(b)$ Find all rational points on the circle $x^2 + y^2 = 17$, if there are any. If there ...
0
votes
0answers
23 views

Exponential and regular Diophantines?

I am looking for a reference on connections between exponential and "regular" (polynomial) Diophantine equations. For example, I was wondering about the Catalan-Mihailescu problem and I thought of the ...
4
votes
1answer
40 views

How to solve $a\sqrt{x}\pm b\sqrt{y}=c\sqrt{z}$

Let $a,b,c,x,y,z \in \mathbb{Z}>1$ How do I prove if $x,y,z$ are square-free integers and: $$a\sqrt{x}\pm b\sqrt{y}=c\sqrt{z}$$ Then $\gcd(x,y,z)>1$? I know for some of you it may be ...
0
votes
1answer
97 views

Can you explain this identity's secret with this Equation $n-th$ powers.

For $k = 0,1,2,3,4,5,6,7,8$, we have the equality, $$(-5)^k + (-119)^k + (-101)^k + (-215)^k + (-197)^k + 43^k + 157^k + 31^k + 217^k + 169^k\\ =\\ (-47)^k + (-161)^k + (-35)^k + (-221)^k ...
4
votes
1answer
203 views

No of right angled triangles [closed]

How many right angled triangles are possible with the perpendicular side equal to 36 units. I took the side $x$ and $y$ and using Pythagoras theorem you have $(x+y)(x-y) = 1296$ and $1296$ has $25$ ...
2
votes
0answers
45 views

Exponential diophantine: $2^x-7^y=z^2$

Find all integers $x,y,z$ such that $2^x-7^y=z^2$. For example: $2^3-7^1=1^2$ $2^5-7^1=5^2$ $2^7-7^1=11^2$ (But note that $\sqrt{2^9-7^1}\not\in \mathbb{Z}$.) The problem with this particular ...
1
vote
1answer
47 views

Exponential diophantine: $(a^r+1)(b^s+1)=c^t+1$?

I've been trying to solve this for a while to no avail. Problem: Find all integers $a,b,c,r,s,t$ such that $(a^r+1)(b^s+1)=c^t+1$. (In fact, the problem I was trying to solve had $a^r+1,b^s+1\in ...
1
vote
2answers
58 views

Integral solutions of $x^\alpha+y^\alpha=z^\alpha$

The problem is Is it true that the equation $x^\alpha+y^\alpha=z^\alpha$ has no solution in integers (except $0$) where $\alpha\in \mathbb{R}\setminus\mathbb{Z}$ ? I am for sometime with this ...
0
votes
0answers
25 views

Diophantine equation of type $ax^2+bx+cy^2=n$

Is there a recipe for, or are there practical examples of, solving Diophantine equations of type $ax^2+bx+cy^2=n$. How would I prove that a particular equation has no ( Integer ) solutions for $x, y$? ...
2
votes
0answers
46 views

Large initial solutions to $x^3+y^3 = Nz^3$?

Let $x,y,z$ be non-zero integers. Is it true that the initial or smallest solution (in terms of absolute value) to, $$x^3+y^3 = Nz^3\tag1$$ for $N=94$ is, $$15642626656646177^3 + ...
1
vote
1answer
32 views

How many positive solutions are there (Positive 3 tuples)?

I want to find how many positive solutions for the Diophantine equation $4x + 2y + 5z = 100$ I found a particular solution $(x,y,z) = (50,-50,0)$ then I found a general solution (basis) $s(-2,-1,2) + ...
3
votes
0answers
45 views

Probability of another 3 integers with same sum and product as the first 3 integers

Let us suppose $3$ integers are selected at random from a large range, say $$-1000\leq x\leq y\leq z\leq 1000$$ Now, we define the sum and product: $$\begin{align*}s&=x+y+z ...
9
votes
3answers
111 views

If $a$ and $b$ are positive integers such that $a^n+n\mid b^n + n$ for all positive integers $n$, prove that $a=b$.

I ran into this problem in a math camp, but I can't seem to solve it via elementary techniques. If $a$ and $b$ are positive integers such that $a^n+n\mid b^n + n$ for all positive integers $n$, ...
-2
votes
2answers
53 views

Volume and surface area of a drilled out cube (BM01 2010/11 Contest Question 2)

Let $s$ be an integer greater than $6$. A solid cube of side $s$ has a square hole of side $x < 6$ drilled directly through from one face to the opposite face (so the drill removes a cuboid). The ...
1
vote
2answers
31 views

Integer solutions of $x^2-5y^2=1342$ with $0\leq x,y<400$

$x^2-5y^2=1342$, where $x,y \in \mathbb N \ and \ x,y<400$., how many pairs of $(x,y)$ possible here. what would be my approach here?
1
vote
2answers
60 views

algorithm for positive integer solutions of equation $a^3+b^3=22c^3$

This is a look-a-like to Fermat's last theorem for $n=3$, but it has solutions! I believe that its solution requires knowledge of the techniques of algebraic or analytic number theory which I don't ...
3
votes
2answers
131 views

How can we solve $y^2=x^3+23$ without trial and error?

$$y^2=x^3+23$$ Are there any easy ways to solve this problem with number theory, abstract algebra, etc.? (trial and error for mods by the way)
0
votes
1answer
28 views

I need to have an result of 36 to 47 from from an input of 0 to 127 - all using the same equation.

Using a formula, I need to have a result between 36 and 47 - depending on the input: the input will be an integer between 0 and 127 as follows... 0, 12, 24, 36, etc MUST = 36 1, 13, 25, 37, etc ...
2
votes
1answer
41 views

Find all integer solutions of equality

Find all integer solutions of equation $$x^3+(x+1)^3+...+(x+7)^3=y^3$$ I've solved it by opening brackets and consideration of signs but I think there is simpler way of solving it .
0
votes
1answer
29 views

solving the equation $x^{n}-dy^{n}=1 $ in integers

how could we solve the equation $x^{n}-dy^{n}=1 $ by knowing the continued fraction expansion of $ d^{1/n} $ ?? in case $ n=2 $ is pell's equation if I divide all by $ y^{n} $ then $ ...
6
votes
1answer
144 views

Amount of solutions to the Diophantine equation of Frobenius

The Diophantine equation of Frobenius is any equation of the form: $$\sum_{i=1}^k a_i x_i = n$$ where the $a_i$'s are given and so are $k$ and $n$. I'm looking for an algorithm to compute the number ...
1
vote
2answers
86 views

On number of solutions of $\frac1x+\frac1y=\frac1n$

Let $S(n)$ denote the number of ordered pairs $(x,y)$ satisfying $\frac{1}{x}+\frac{1}{y}=\frac{1}{n}$, where $n>1$ and $x,y,n∈N$ 1) Find the value of $S(6)$. 2) Show that if $n$ is prime then ...
2
votes
3answers
99 views

Integer solution to $19x^3-84y^2=1984$

Show that there exist no integer values $x,y$ such that $19x^3-84y^2=1984$. Please help me in understanding no solution problems. I tried to check the modulo $7$ of both sides but couldn't reject ...
1
vote
1answer
48 views

How can I intuitively understand the algorithm for finding the integer solutions to $ax+by=c$?

Recently I've started to take interest in linear diophantine equations (they play a key role in a math puzzle I stumbled upon). I don't have a strong math background, and at first I had no clue how ...
0
votes
1answer
16 views

How to solve this Diophantine equation (involving natural logarithms)?

The equation is $r = \ln{a} + b \ln{c}$ where $r \in \mathbb{R}$ is fixed and $a,b,c \in \mathbb{N}$. In other words, for arbitrary real r, how can one say whether a solution (in form above) exists ...
1
vote
7answers
223 views

Find all integral solutions to $a+b+c=abc$.

Find all integral solutions of the equation $a+b+c=abc$. Is $\{a,b,c\}=\{1,2,3\}$ the only solution? I've tried by taking $a,b,c=1,2,3$.
2
votes
0answers
34 views

About Runge's method

I have been reading about some Diophantine equations (like Runge's theorem and Cassel's theorem) and in the text says that these theorems are solved using Runge's method, but it doesn't say what ...
1
vote
1answer
31 views

Diophantine equations involving prime numbers

If $p$ is a prime number, such that there is $a\in \mathbb{Z}$ $$a^2\equiv -2 \ (\text{mod }p).$$ how do I show that one of the equations has an integer solution $$x^2+2y^2=p$$ and $$x^2+2y^2=2p?$$ ...
2
votes
2answers
35 views

Simple Modular equation

Let $s,t,n$ be 3 non-zero positive integers. We set $s+1=nt$. If $n$ is odd,find $n$ such that: $$s \equiv 1 \pmod 3$$ $$t\equiv 1 \pmod 3$$ I know the answer is very likely simple. I just suck at ...
0
votes
1answer
42 views

Diophantine equation in $a, b, c,$and $d$

I'm looking for positive integers $a, b, c,$ and $d$ such that $$ (ad - bc)(ac + bd) \: | \: abcd$$ One partial solution that I found is $$(a, \, b, \, c, \, d) = (2x + 1,\, 2x, \, 2x + 2, \, 2x + ...
0
votes
2answers
30 views

$yx^2=z$ For any interger $z$, find a whole number solution.

Given any integer $z$, what are all the integer solutions possible that create a square prism of length $x$ with a height of length $y$? For example, if $z=25$, some possibles solutions are a ...
0
votes
2answers
74 views

Unwind the equation

Let $x, y, z, t$ be positive integers. Given that $$68(xyzt+xy+zt+xt+1)=157(yzt+y+t)$$ Find the value of the product $xyzt$. I couldn't even start with the problem. I just know that the expression n ...