Questions on finding integer/rational solutions of polynomial equations.

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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 $a=b$ ?
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2answers
58 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 ...
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1answer
39 views

Diophantine Equations with Factor Exponents

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 ...
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3answers
47 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?
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0answers
67 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 ...
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1answer
44 views

is it possible to find $x$ where $y$ is equal to a whole number in a non iterative fashion

Given the equation $$\frac{635x+326}{637+x} = y$$ where $$x>0$$ Is it possible to find all positive values of $x$ (there is only one) where $x$ is positive and $y$ is a whole number. While I ...
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1answer
54 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 $a+b+c$?
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2answers
55 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 ...
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1answer
32 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 ...
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3answers
92 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 ...
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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
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1answer
41 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 ...
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1answer
102 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
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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 $25$ ...
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0answers
46 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 ...
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1answer
63 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 ...
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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 ...
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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 $x, y$? ...
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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 + ...
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1answer
37 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) + ...
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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 ...
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3answers
115 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 $n$, ...
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2answers
58 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 ...
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2answers
34 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?
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2answers
61 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 ...
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2answers
135 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)
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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
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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 .
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1answer
31 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
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1answer
152 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 ...
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2answers
89 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 ...
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3answers
100 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 ...
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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 ...
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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 ...
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7answers
236 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 $a,b,c=1,2,3$.
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0answers
35 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 ...
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1answer
32 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 $$x^2+2y^2=2p?$$ ...
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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 ...
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1answer
46 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 + ...
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2answers
31 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 ...
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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 ...
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1answer
39 views

The Method of Ascent in Diophantine Equations

Can someone help me to prove there are infinitely many solutions to the Diophantine equation: $$x^2 − 3y^2 = 1$$ using the method of ascent. The Method of Ascent: We can do this by showing how, ...
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2answers
42 views

Proof by induction that $P_n(a) \neq 0$ for $n>3$.

Let $a,b,c$ be 3 non-zero coprime integers and $P_n(a)=a^n+\sum_{k=1}^{n}{{n\choose{k}}a^{n-k}(c^k-b^k)}$ Show that if $P_3(a) \neq 0$ then for all $n \geq 3, P_n(a)\neq 0$ Using mathematical ...
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3answers
185 views

Polynomial division challenge

Let $x,y,n \in \mathbb{Z} \geq 3$, Find $A,B$ such that $$x^{n-1}+x^{n-2}y+x^{n-3}y^2+\cdots+x^2y^{n-3}+xy^{n-2}+y^{n-1}= A(x^2+xy+y^2)+B$$ What is the best method to approach this?
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1answer
54 views

Pell-Type Diophantine Equation Solving using the method of ascent [duplicate]

Can someone help me to prove there are infinitely many solutions to the Diophantine equation: $$x^2 − 3y^2 = 1$$ using the method of ascent. We can do this by showing how, given one solution $(u, v)$, ...
4
votes
2answers
65 views

solving cubic diophantine equation

Can someone show me how to find all solutions in positive integers to the diophantine equation: $$x^3 + y^3 = 35$$ I know how to do it algebraically, but I want to know how you solve it in number ...
2
votes
1answer
79 views

Method of ascent to prove that $x^2 − 3y^2 = 1$ has infinitely many solutions [duplicate]

Use the method of ascent to prove there are infinitely many solutions to the Diophantine equation: $$x^2 − 3y^2 = 1$$ We can do this by showing how, given one solution $(u, v)$, we can compute another ...
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0answers
35 views

How can I obtain a solution for the equation $a^2 + b^2 = c^2 + 1$? [duplicate]

For the equation $a^2 + b^2 = c^2$, the solution is: $a = m^2 - n^2, b= 2mn, c = m^2 + n^2$ $m,n\in\mathbb{Z}$ and $m > n$, free to choose How is a similar solution obtained for the equation ...
2
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1answer
75 views

Ways to solve in integers $\frac{2x^2+5y^2}{xy-14}=11 $

Consider the diophantine equation $$\frac{2x^2+5y^2}{xy-14}=11.$$ I have successfully found all its integer solutions, but in view of different equations, I was wondering if there are other ...
0
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0answers
23 views

Solving Diophantine equations of the form $am^x +b n^y = ab z^2$

How can Diophantine equations of the following form be solved? $$am^x +b n^y = ab z^2$$ Can you suggest articles dealing with this type of problem