Questions on finding integer/rational solutions of polynomial equations.

learn more… | top users | synonyms

1
vote
1answer
42 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
48 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
119 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
63 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
35 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
64 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
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)
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
42 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
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
votes
1answer
165 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
94 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
103 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
50 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 ...
2
votes
7answers
292 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
37 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
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 $$x^2+2y^2=2p?$$ ...
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
62 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
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 ...
0
votes
2answers
75 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 ...
1
vote
1answer
42 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, ...
3
votes
2answers
43 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 ...
2
votes
3answers
190 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?
0
votes
1answer
62 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
71 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
82 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 ...
1
vote
0answers
36 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
votes
1answer
79 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
votes
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
7
votes
5answers
287 views

Find all integers $x$, $y$, and $z$ such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$

Characterize all positive integers $x$, $y$, and $z$ such that: $$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{z}$$ For example, $\dfrac{1}{x+1} + \dfrac{1}{x(x+1)} = \dfrac{1}{x}$.
0
votes
3answers
34 views

Solve this equation.

$$ \dfrac{x^2-1}{x^2+9-6x-1}=\dfrac{x+2}{x-4}-\dfrac5{(x-2)^2} $$ Can you tell me what should I factorize the denominator? I thought to put $$x^2+9-6x-1=x^2-6x+8$$But I suppose they gave it in in ...
0
votes
0answers
29 views

Intgral squares for $h(ab+bc+ac-a-b-c)-abc+1=0$

I am looking for a solution to $$h^2(a^2b^2+b^2c^2+a^2c^2-a^2-b^2-c^2)-a^2b^2c^2+1=0$$ such that $\{a,b,c,h\}$ is pairwise coprime set and $a,b,c,h \ge 2$. I have run some lines of C-code for that ...
7
votes
2answers
171 views

How to solve $(2x^2-1)^2=2y^2 - 1$ in positive integers?

I encountered this question (posed by Fermat) in a letter from Fermat to Carcavi and was wondering what would be the best elementary way to solve it. Solve in positive integers$$(2x^2-1)^2=2y^2 - ...
6
votes
2answers
291 views

$x^2+y^2+z^2=5(xy+yz+zx)$ — Is this all solutions?

Problem: Find all integers that satisfy $x^2+y^2+z^2=5(xy+yz+zx)$. Does the following parametrization give all solutions?: $x=m^2+mn-5n^2$; $y=-5m^2+9mn-3n^2$; $z=-3m^2-3mn+n^2$, where $m,n$ are ...
0
votes
0answers
33 views

Number of solutions to diophantine equation involving products

How do we solve the following Diophantine equation? Find the number of positive integer solutions to $x_1x_2x_3x_4 = 3^{11}\cdot5^{9}\cdot7^{7}\cdot11^{5}$ where $x_1 \le x_2\le x_3\le x_4$. ...
8
votes
2answers
560 views

Determine if a number is the sum of two triangular numbers.

Is it possible to figure out if a number $z$ is the addition of two triangular number without recursion or finding the values to $x$ and $y$? $$\frac{x(x+1)}{2} + \frac{y(y+1)}{2} = z$$ An example ...
0
votes
0answers
32 views

Diophantine equations and the area of a triangle

prove that the area of the triangle whose vertices are (0,0), (b,a), and (x,y) is |by-ax|/2. Now this is suppose to be a number theory problem. Although I can prove this using other means I am not ...
1
vote
3answers
52 views

Solve $2^x+3^y=z^2$ in nonnegative integers.

So, we are trying to find all the solutions to $2^x + 3^y = z^2$ in nonnegative integers. Here are my insights: First of all, $z^2$ can be either $0$ or $1$ modulo $3$. If $z^2 = 3k$, then LHS ...
7
votes
0answers
96 views

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I'm working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we ...
3
votes
1answer
65 views

Diophantine equation, $x^2 = y^3$.

I have the following diophantine equation to solve: $x^2 = y^3$ I got that if we introduce $ z=a^6 $ then all numbers $ x=a^3$ and $y =a^2$ satisfy the equation. However, I am not sure whether this ...
3
votes
1answer
33 views

Non-negative integer solutions given restrictions on $x_i$ (check work)

Use Inclusion-Exclusion to determine the number of integer solutions to the equation $$x_1+x_2+x_3+x_4=14$$ Where $0{\leq}x_1{\leq}8; 0{\leq}x_2{\leq}5; x_3, x_4{\geq}0$. My thought process: I ...
2
votes
1answer
37 views

System of linear diophantine inequalites

are there any papers that deal with System of linear diophantine inequalites? I have a hard time finding any. The wikipedia entry calls it diophantine approximation, but i am not sure if this is the ...
0
votes
0answers
28 views

Solutions to this Diophantine equation.

I need to solve $x^2-x=y^3$ with $x,y \in \mathbb{Z}$. My final answer gave the solution set to be $x \in \left\{ {0,1}\right\} $. Could anyone verify this?
0
votes
3answers
68 views

Is it possible to solve for $a, b \in \mathbb{N}$?

I need to solve the following equation so that both $a$ and $b$ are natural numbers. $$ab - 2a = 2b$$ I must also prove that the solutions found are the only ones possible. Is it possible to do ...
0
votes
1answer
98 views

$x^3-9=y^2$ find integral solutions

Find all integral solutions $x^3-9=y^2$ I tried many times but still no idea how to solve it. I will be grateful for any help.
2
votes
1answer
37 views

How to prove a quadratic Diophantine equation has no solution?

Take the equation $3x^2-5y^2+7z^2 = 0$. If we take this $mod \: 4$ we get: $3x^2+3y^2+3z^2 \equiv 0 \: mod \: 4$ All of the squares modulo $4$ are either $0$ or $1$. $3x^2+3y^2+3z^2$ will never be ...
0
votes
1answer
147 views

Finding the largest 3-digit number $\; \overline{abc}\;$ s.t $\; \overline{abc}=100a+10b+c \equiv a+b^2+c^3$

This question comes from a maths contest (infer no calculators or other electronic calculating aids) for 14-16 year olds (infer no use of complicated theorems, but those accessible to high-school ...
3
votes
1answer
84 views

Number Theory : Solving $x^2$ $+$ $y^2$ $=$ $2^{10}$ - $1$

I was working my way through some basic number theory problems and was all thumbs while solving this problem : List all the pairs of integer solutions $(x, y)$ of the Diophantine equation : $x^2$ ...