Questions on finding integer/rational solutions of polynomial equations.

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3
votes
3answers
109 views

A conjecture on products/composition of Pell forms

Based on a few brute-force calculations, I've formulated the following. Conjecture. Let $x,y,u,v,p,q,a,b,c \ge 2$ be integers such that $$ (x^2+ay^2)(u^2+bv^2) = p^2+cq^2, $$ and write \begin{align} ...
0
votes
0answers
72 views

Integer solution to Diophantine equations? [on hold]

I was challenged with the question of showing non-existence of integral solutions to the following Diophantine equations: $a^3+3 b^3-7 a b c+3 c^3=0$, and $a^4-6 a^2 b^2+9 b^4-10 a^2 c^2-30 b^2 ...
0
votes
1answer
44 views

Pell's equation for n=2

If know that $x=3$, $y=2$ is a solution of $$x^2-2y^2=1,$$ then apparently all other solutions can be calculated as $$x_k+y_k\sqrt{2}=(x+y\sqrt{2})^k,$$ which I have trouble understanding. I've been ...
1
vote
1answer
24 views

Prove for relatively prime numbers.

Prove that for relatively prime positive integers $a$ and $b$, the equation $ax+by=c$ must have non-negative integer solution if $c>ab-a-b$.
0
votes
1answer
28 views

if $k>1$, Does $a+b =k(ax+by)$ have finitely many solutions?

Let $a,b,k,x,y$ be non-zero integers, solve $a+b=k(ax+by)$. It's a rather simple problem, but I just want to make sure that I have got all the possible solutions.
1
vote
1answer
41 views

Solving for a variable in an integer divisibility problem

Say I have a problem of the form Where , , and are known integers, is some unknown variable, and is an integer output. Is there an approach I could take to determine if there is some integer ...
0
votes
2answers
65 views

Solve the equation $a+b+c=abc$ for $a,b,c\in\mathbb{Z}$

Solve for $a,b,c$ (where $a$, $b$, and $c$ are integers) the equation $$a+b+c=abc.$$ I would prefer a solution using trigonometry and I think that it might use the formula $\tan A + \tan B + \tan ...
2
votes
1answer
57 views

Solving algebraic equations for x

So I was able to find the least common denominator which is $12$ but I'm struggling to solving the equation: $$\frac{4(x - 2)}{6} - \frac{2(x + 4)}{4} = -\frac{2}{3}.$$
1
vote
0answers
50 views

Any general “formula” solutions for higher order polynomial equation?

We know that fifth (or higher) degree polynomial equation has no general solution formula using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of ...
3
votes
3answers
96 views

Solving $y^3=x^3+8x^2-6x+8$

Solve for the equation $y^3=x^3+8x^2-6x+8$ for positive integers x and y. My attempt- $$y^3=x^3+8x^2-6x+8$$ $$\implies y^3-x^3=8x^2-6x+8$$ $$\implies ...
0
votes
0answers
26 views

Characterize the set that solves the Diophantine inequality

Given that $a, b, c$ are integers and $2\max(|b|, |3a + b|) \le \min(|c|, |3a+2b+c|)$ What characterizes the solution set in $\mathbb{Z}^3$? Obviously this is equivalent to the system of linear ...
5
votes
0answers
54 views

To solve for $x,y,n$ in non-negative integers , $\dfrac{x!+y!}{n!}=p^n$ , $p$ a given prime

Let $p$ be a given prime , then how do we find non-negative integers $(x,y,n)$ $\space$ , such that $\dfrac{x!+y!}{n!}=p^n$ ?
0
votes
2answers
75 views

Parametric solution of the Diophantine equation $\frac{1}{p}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z} ,x,y,z∈Z^+.$

I have prove that, for any given positive integer $p,$ parametric solution of the Diophantine equation $$\frac{1}{p}=\frac{1}{x}+\frac{1}{y}$$ can be written in the form $x=ac(a+b),y=bc(a+b),$ where ...
0
votes
0answers
42 views

Integral point on certain cubic surfaces and rational parametric solutions

The motivation of the following question comes from the Problem D4 in the book "unsolved problems in number theory" by Richard Guy. No integral points on the surface $S:x^3+y^3+z^3=3$ is known other ...
0
votes
1answer
63 views

Diophantine equation: $2(x^3+xy+y^3)=3(x+y)$

Here is a nice equation: $2(x^3+xy+y^3)=3(x+y)$ over $ \mathbb{Z}$ x $\mathbb{Z}$. Any nice way to approach this?
1
vote
6answers
75 views

The diophantine equation $a^2+ab-b^2=0$

I first tried with brute force with $-1000 \leq a,b \leq 1000$ but found no solution. But then a simple argument showed me that there was no solution. Not only in the integers, but even for the ...
0
votes
1answer
60 views

Solve $a^2+b^n=c^2$

Let $a,b,c$ be co-prime integers >1, for all $n>2$, I need help finding the integral solutions of the above diophantine equation. I saw the result but I am curious about to how to get there. ...
5
votes
1answer
178 views

Finding every triplet $(n,a,b)$ such that $n!=2^a-2^b$

Question : Let $n,a,b$ be positive integers. Are there infinitely many triplets $(n,a,b)$ which satisfy the following equality?$$n!=2^a-2^b$$ If Yes, then how can we prove that? If No, then how ...
0
votes
4answers
56 views

Finding the number of two digit numbers

I was solving questions from a book and it had a question : Find all two digit numbers such that the sum of digits constituting the number is not less than 7; the sum of squares of digits is not ...
5
votes
3answers
287 views

Generating Functions and Linear Diophantine Inequalities

The following exercise is from Analytic Combinatorics by Philippe Flajolet and Robert Sedgewick, page 46. A $k$-composition of $n$ is an ordered $k$-tuple of non-negative integers whose sum is $n$. ...
9
votes
1answer
178 views

Minimum of $|az_x-bz_y|$

I am trying to minimize the following function: \begin{align} &f(z_x,z_y)=|az_x-bz_y| \\ &\text{ s.t. } z_x,z_y \in \mathbb{Z},1 \le z_x \le N_x \text{ and } 1 \le z_y \le N_y \text{ and } ...
0
votes
4answers
66 views

Solve $c^2-b^2-a^2=2N$

Is there anyone that can help solving this equation: $c^2-b^2-a^2=2N$ where $a,b,c,N$ are natural numbers. Edit: We need to express $a,b,c$ for a certain $N$. Regards
0
votes
4answers
53 views

An example of how to solve equation over $\mathbb{Z}$

I found an example of how to solve equation over $\mathbb{Z}$ Example Solve the equation over $\mathbb{Z}$ : $$xy + 1 = 3x + y. $$ $$ xy = 1 + 3x + y \Longleftrightarrow (x-1) (y-3) = ...
4
votes
0answers
97 views

Diophantine: $x^3+5=y^5$

Find all integers $x$ and $y$ such that $x^3+5=y^5$. I found this after solving the equation $3^a+5=2^b$. For this equation, since $(a,b)=(3,5)$ is a solution, it is possible to write it as ...
2
votes
1answer
28 views

Intersection of two recurrences.

I have two sequences obtained by recurrences: $$f(0) = 1, f(1) = 9, f(n+2) = 10f(n+1) - f(n)$$ $$g(0) = 1, g(1) = 7, g(n+2) = 6g(n+1) - g(n)$$ How can I prove that apart from $f(0) = g(0) = 1$, these ...
2
votes
2answers
103 views

coin problem with two coins, inductive proof

Adjustment This proof is flawed. I want to ask something about the coin problem with two coins. Let $a,b$ be to numbers in $\mathbb{N} \setminus \{0\}$ (elsewhere I include zero) which have no prime ...
-7
votes
0answers
53 views

Solving a system of polynomial equations in three variables (x^2-yz=18, y^2-zx=8, z^2-xy=-7)

Solving a system of polynomial equations in three variables (x^2-yz=18, y^2-zx=8, z^2-xy=-7 I've tried rearranging each equation to isolate for one variable ex: z^2-xy=-7 --> z= x^2-18/y after, I ...
0
votes
1answer
66 views

Solve $i^3j-j^3i=x^3y-y^3x$

Do anyone have an idea about how to solve this kind of equation: $i^3j-j^3i=x^3y-y^3x$ where $i,j,x,y$ are distinct natural numbers and $i>j$ and $x>y$ Regards
0
votes
0answers
25 views

What are some [mostly trivial] Pell transformations?

Euler looked at some transformations which turned one Pell[-type] equation into another. Example 1: $$u^2-av^2=-1 \quad\iff\quad (2u^2+1)^2-a(2uv)^2=1.$$ Example 2: $$u^2-av^2=-2 \quad\iff\quad ...
2
votes
1answer
46 views

solve the diophantine equation: $x^3-3xy^2=z^3$

Let $ x,y,z$ be 3 integers greater than 1,if $x$ and $y$ are relatively prime, solve the diophantine equation: $x^3-3xy^2=z^3$.
-1
votes
1answer
22 views

How can we generate a $2$-digit number $XY$ on base $B$, such that $BX+Y=Y^X$?

For example, $25$ on base $10$ is equal to $5^2$. This should be pretty easy to solve using fairly simple arithmetic. But I'm finding it hard to generate any other solutions besides the one ...
1
vote
1answer
28 views

“Descent” on binary quadratic forms?

Let's say I have the Diophantine equation $$ x^2+3n^2 = y^2+3z^2. \tag{$\star$} $$ where $n$ is a known integer, and we're trying to determine solutions in integers $x,y,z \ge 1$. Rewrite ($\star$) ...
0
votes
0answers
26 views

Superelliptic curves

I'm trying to find information on superelliptic curves and how to solve them over the integers. The equation is $$y^k = f(x)$$ where $k=3$ and $f$ has degree $d=3$. Does anyone know any ...
1
vote
0answers
25 views

Powerful numbers in Pell solutions (or, more generally, any Lucas sequence)

There are several definitive results regarding perfect powers in the Pell numbers — e.g., the only perfect power is $P_7=169=13^2$. On the other hand, when it comes to powerful numbers, I've only ...
5
votes
1answer
46 views

Conditions on solutions of a diophantine equation.

I wanted to list all the natural number solutions $(d_1,d_2,...,d_n)$ to the equation: $$\sum_1^n \frac1{d_i} = 1$$ I could not succeed. I noted that for $n=4$, $(2,4,8,8), (3,3,6,6), (2,3,12,12), ...
3
votes
0answers
71 views

Looking for help with this elementary method of finding integer solutions on an elliptic curve.

In the post Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$, the single positive integer solution $(x,y)=(18,7)$ is found using algebraic integers. In one of the ...
1
vote
0answers
41 views

Given a Pell “solution” in [integer] polynomials, what can be said about the components?

Let $x,y$ be integers, and $f(x,y)$, $g(x,y)$, and $h(x,y)$ be polynomials in $x$ and $y$ with integer coefficients such that $$ f(x,y)^2 - g(x,y)h(x,y)^2 = 1. \qquad(\star) $$ Furthermore, assume it ...
2
votes
1answer
58 views

The Diophantine equation $x^p - 4y^p = z^2$ with $(x, y) = 1$ and $x, y, z > p.$

If $p \geq 5$ is a prime, are there any integers $x, y, z > p$ such that $(x, y) = 1$ and $$x^{p} - 4y^{p} = z^{2}$$
0
votes
0answers
41 views

The Diophantine equation $x^2 - y^2 = 4z^n$

If $n \geq 3$ is an integer, then under what conditions on $x, y, z$ does the equation $$x^2 - y^2 = 4z^n$$ have no solution in integers? (If there is any known result, please feel free to share with ...
0
votes
2answers
74 views

Find the non-trivial solutions of the diophantine equation: $a^3+3a^2b=c^3$

If $ a$ and $b$ are co-prime integers >2, can $a^3+3a^2b$ be a cube?
4
votes
3answers
120 views

Find all positive integers satisfying $\frac{2^n+1}{n^2} =k $

Find all positive integers satisfying $$\frac{2^n+1}{n^2} =k $$ where $k$ is a integer. I can't just come up with a solution.
9
votes
2answers
127 views

Solving $n! + 3n = k^2$

Let $n$ and $k$ be integers. Need to solve $n! + 3n = k^2$, where $n!$ denotes $n$ factorial. I do not have any ideas about this equation, except I suppose the only $6$ roots are $(0,1), (0, -1), ...
0
votes
1answer
51 views

Solution to Diophantine equation with constraint.

solve the following equation over $z_x,z_y$ \begin{align} &az_x=bz_y\\ &\text{s.t. } a,b,z_x,z_y \in \mathbb{Z} \text{ and } 1 \le z_x \le N \text{ and } 1 \le z_y \le N \end{align} How ...
4
votes
1answer
50 views

Find integral solutions for $2x^2+y^2=2\times(1007)^2+1$

Find integral solutions to the equation $$2x^2+y^2=2\times(1007)^2+1$$ I tried: I rewrote the equation as $2x^2+y^2=2028099$. I found that $y_{max}=1424$ and $y$ must be odd, so I set ...
1
vote
2answers
45 views

Given a Pell solution $(u_k,v_k)$, is there a closed form “descent” to $(u_{k-1},v_{k-1})$?

Given: a solution $(u_k,v_k)$ to the Pell equation $$U^2-dV^2=1, \qquad(\star)$$ where $d$ is a non-square integer, and $k \ge 1$ is an arbitrary integer. There are well-known recurrences to ascend ...
2
votes
0answers
51 views

The Diophantine equation $x^n - y^n = z^2$

Darmon-Merel theorem (DMT) ensures that if $n \geq 4$ is an integer and $x, y, z > 0$ are integers such that $(x, y, z) = 1$ then $x^n + y^n \neq z^2.$ The question is: Does DMT apply to the ...
4
votes
0answers
78 views

Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be ...
2
votes
1answer
60 views

Hard Simultaneous Diophantine Equations

Find all positive integers $a,b,c,d,e,f$ such that : $de^2=ab^2+1$ and $df^2=ac^2+1$. I tried subtracting them, it factors quite nicely. But after that, haven't a clue. I'm not sure if it's even ...
0
votes
3answers
53 views

how can I find Integer solutions for the two variables equation without searching factors?

If $(30X+7)(30Y+1) = 50437$, then what are the integer solutions? Any way to solve it without searching $50437$ factors?
-1
votes
1answer
37 views

how can I find Integer solutions for the two variables equation?

how can I find Integer solutions for the following equation: $$900 X \, Y + 210 Y + 30 X - 3 \times 10^{100} + 7 = 0$$