Questions on finding integer/rational solutions of polynomial equations.

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2answers
57 views

How do you solve $k(a^2-b^2)=2(ax-by)$?

let $a,b,c,d,x,y,k$ be all non-zero positive integers >1. If $a^2-b^2 \neq0$,how do you find all the pairs $(x,y)$ such that $k(a^2-b^2)=2(ax-by)$. I have found so far only solutions where ...
2
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3answers
46 views

Integer solutions to $k^2m^2 -k^2 - m^2 +1 = n^2$

Can the positive integer solutions to $$ k^2m^2 -k^2 - m^2 +1 = n^2 $$ be characterized (in the sense that the solutions to $a^2+b^2 = c^2$ are characterized by $a=r^2-s^2, b=2rs, c=r^2+s^2$ with ...
1
vote
0answers
77 views

Mordell Diophantine: $x^2+11=y^3$

I've been trying to solve the diophantine $$x^2+11=y^3$$ recently but to no avail. I tried the "UFD trick", re-writing as $(x-i\sqrt{11})(x+i\sqrt{11})=y^3$, but it didn't give me all the solutions. I ...
6
votes
2answers
170 views

When $x^2+6xy+y^2$ a square number?

Find all natural numbers $x$ and $y$ such that $x^2+6xy+y^2$ is a square number. For example, $(x,y)=(2,3)$ or $(x,y)=(3,10)$. Obviously, we can consider $gcd(x,y)=1$.
5
votes
4answers
286 views

Solving Diophantine equations involving $x, y, x^2, y^2$

My father-in-law, who is 90 years old and emigrated from Russia, likes to challenge me with logic and math puzzles. He gave me this one: Find integers $x$ and $y$ that satisfy both $(1)$ and $(2)$ ...
4
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3answers
149 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
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1answer
58 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
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1answer
39 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
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1answer
31 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
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1answer
42 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
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2answers
67 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
61 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
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0answers
72 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 ...
4
votes
3answers
108 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
27 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 ...
6
votes
0answers
61 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
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2answers
81 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
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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
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1answer
66 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
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6answers
94 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 ...
1
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2answers
88 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 diophantine equation $a^2+b^n=c^2$. I saw the result but I am curious about to how to get ...
6
votes
1answer
193 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
60 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
304 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
188 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
68 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
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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) = ...
5
votes
0answers
118 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
31 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
106 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 ...
0
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1answer
72 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
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0answers
26 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
49 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
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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
2answers
95 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
28 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
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0answers
32 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
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1answer
62 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
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0answers
73 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 ...
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0answers
42 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
59 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 $$x^{p} - 4y^{p} = z^{2}$$
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0answers
42 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
77 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?
5
votes
3answers
134 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
129 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
53 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
51 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
50 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
52 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
81 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 ...