Questions on finding integer/rational solutions of polynomial equations.

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0
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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
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
156 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
90 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
101 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
238 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
36 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
53 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
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 ...
1
vote
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, ...
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
187 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
55 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
66 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
80 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
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
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
273 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
28 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
165 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
288 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
32 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
550 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
28 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
51 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
93 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
64 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
27 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
33 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
93 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
32 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
138 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
83 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 : $x^2$ ...
1
vote
1answer
44 views

Solution to integers of the form $x^n\:+y^n\:=z^n\:+w^n$.

I am wondering if integers of the form $x^n\:+y^n\:=z^n\:+w^n$ have a solution if none of $x,\:y,\:z,\:or\:w$ are equal.
-2
votes
3answers
89 views

how to solve equations $x^3+y^3+z^3=x+y+z=8 $ over integers. [closed]

How to solve this equations $\left\{\begin{matrix}x^3+y^3+z^3=x+y+z&\\x+y+z=8&\end{matrix}\right.$ $x,y,z\in\mathbb{Z}$
0
votes
0answers
25 views

Minimize multivariate (multivariable) polynomial over the integers

I'd like to minimize the following polynomial in 6 variables $h_0,h_1,g_0,g_1,g_2,g_3$: $$ g_3^2\cdot h_0^3\cdot h_1^3 - g_2\cdot g_3\cdot h_0^2\cdot h_1^4 + g_1\cdot g_3\cdot h_0\cdot h_1^5 - ...
1
vote
2answers
54 views

A system of quadratic Diophantine equations with four variables

Is the following system has any positive integer solution $(x,y,u,v)$? $$\begin{cases} x^2+y^2=u^2\\ x^2-y^2=v^2 \end{cases}$$ I can prove that any pair of these integers can be relatively prime, but ...
0
votes
1answer
27 views

Impossibility of Equation

Prove that there are no solutions to $ k^2 = x^4 + 2x^3 + 2x^2 + 2x + 1 $ in $ \mathbb Z^+$. I have tried a bounding argument so far, placing $k^2$ in between $x^4$ and $(x+1)^4$, but I am unable to ...
1
vote
3answers
91 views

Why does $5x^2+6xy+2y^2=2yz+z^2$ have no integer solutions?

Why does $5x^2+6xy+2y^2=2yz+z^2$ have no primitive integer solutions? Modulo $2$ says that $x$ and $z$ are odd. Modulo $3$ says that $x=0 \bmod 3$ and $y=-z \bmod 3$. I cannot get anything modulo $5$. ...
3
votes
0answers
65 views

Solving quadratic diophantine equations in two variables

I've looked at the recommended questions, but none of them seem to match my question. Consider the equation $2015 = \frac{(x+y)(x+y-1)}{2} - y + 1$. This can trivially be simplified to $4030 = x^2 + ...