Questions on finding integer/rational solutions of polynomial equations.

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1
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1answer
34 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 ...
6
votes
2answers
132 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 $gcd(x,y)=1$.
1
vote
3answers
37 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
52 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 ...
4
votes
4answers
252 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 $(2)$ ...
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 } ...
3
votes
3answers
121 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} ...
2
votes
1answer
93 views

Can the equation $ax^2+by^2=cz^2$ be solved in integers (excluding trivial solutions)?

Suppose $a,b,c\in\mathbb{N}$ and are each squarefree. Is there a general solution for this equation? I found that for this equation to be soluble in integers there are three necessary and sufficient ...
2
votes
2answers
3k views

Count the number of positive solutions for a linear diophantine equation

Given a linear Diophantine equation, how can I count the number of positive solutions? More specifically, I am interested in the number of positive solutions for the following linear Diophantine ...
13
votes
1answer
264 views

Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?

Are there solutions in integers $a,b>1$ to the following simultaneous congruences? $$ a^4\equiv 1 \pmod{b^2} \quad \mathrm{and} \quad b^4\equiv1 \pmod{a^2} $$ A brute-force search didn't turn up ...
3
votes
1answer
74 views

Find all $x,y\in\mathbb{Z}$ s.t $2x^3-7y^3=3$

Find all $$x,y\in\mathbb{Z}$$ such that $$2x^3-7y^3=3$$ Solution: We consider first $$2x^3-7y^3\equiv3 \pmod 2$$ $$5y^3\equiv 1 \pmod 2$$ $$y^3\equiv 1 \pmod2$$ which has solution $y\equiv 1 ...
1
vote
3answers
133 views

Solving $a^2+3b^2=c^2$

I'm looking for how to solve the equation $a^2+3b^2=c^2$ where $a,b,c$ are integers and $b$ is even, I'm looking for the algorithm used to solve this kind of equations, not just the solution. Regards ...
0
votes
0answers
72 views

Integer solution to Diophantine equations? [closed]

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
46 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 ...
0
votes
1answer
30 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.
11
votes
3answers
11k views

How to find solutions of linear Diophantine ax + by = c?

I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ? I tried to play around with it: ...
3
votes
2answers
262 views

Whenever Pell's equation proof is solvable, it has infinitely many solutions

Prove that whenever the equation $x^2 - dy^2 = c$ is solvable, then it has infinitely many solutions. I consider that, if $u$ and $v$ satisfy $x^2 -dy^2 = c$ and then $r$ and $s$ satisfy $x^2 ...
1
vote
1answer
25 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 $c>ab-a-b$.
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 ...
3
votes
3answers
100 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 ...
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 ...
2
votes
1answer
59 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
51 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 ...
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
140 views

If $p$, $q$ are naturals, solve $p^3-q^5=(p+q)^2$.

In If $p,q$ are prime, solve $p^3-q^5=(p+q)^2$., the author asks to solve the equation $x^3-y^5=(x+y)^2$ for primes $p$ and $q$. A proof is given that $p=7, q=3$ is the only solution. In this ...
0
votes
4answers
57 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 ...
0
votes
2answers
76 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 ...
5
votes
0answers
56 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 $\dfrac{x!+y!}{n!}=p^n$ ?
2
votes
1answer
111 views

Diophantine equation: x^2+2=y^3

just so this doesn't get deleted, I want to make it clear that I already know how to solve this using the UFD $\mathbb{Z}[\sqrt{-2}]$, and am in search for the infinite descent proof that Fermat ...
9
votes
1answer
184 views

Solve the Diophantine equation $a^2(2^a-a^3)+1=7^b$.

The problem is to find all positive integers $a$ and $b$ such that $a^2(2^a-a^3)+1=7^b$. I found a=10, and my intuition tells me there are no more solutions. I've also shown that $a=42k+10$ for some ...
0
votes
1answer
495 views

Using recurrences to solve $3a^2=2b^2+1$

Is it possible to solve the equation $3a^2=2b^2+1$ for positive, integral $a$ and $b$ using recurrences?I am sure it is, as Arthur Engel in his Problem Solving Strategies has stated that as a method, ...
5
votes
3answers
291 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 $n$. ...
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 ...
33
votes
4answers
820 views

Conjecture: Only one Fibonacci number is the sum of two cubes

As the title says, I need help proving or disproving that there is only one Fibonacci number that's the sum of two (positive) cubes, $2$. I did a small brute force test with Fibonacci numbers below ...
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?
5
votes
1answer
179 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 ...
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
61 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. ...
6
votes
2answers
610 views

Are there any $n$ for which $ n^4+n^3+n^2+n+1$ is a perfect square?

Are there any positive $n$ for which $ n^4+n^3+n^2+n+1$ is a perfect square? I tried to simplify \begin{align*} n^4+n^3+n^2+n+1 &= n^2(n^2+1)+n(n^2+1)+1\\ &= (n^2+n)(n^2+1)+1 \\ &= ...
1
vote
0answers
111 views

How can we prove that this equation cannot be solved?

How can we prove that this equation cannot be solved? $ 25k^3+30k^2+23k+3=x^2$ where x,k are integer numbers
13
votes
5answers
758 views

Integral solutions of $x^2+y^2+1=z^2$

I am interested in integral solutions of $$x^2+y^2+1=z^2.$$ Is there a complete theory comparable to the one for $x^2+y^2=z^2?$
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
2answers
260 views

A diophantine equation $x^3+y^3-xy^2=1$

What kind of methods there are to find integer solutions of $x^3+y^3-xy^2=1$? I tried some inequalities and congruences without success. I also found on Wikipedia that this might be a Thue equation ...
4
votes
1answer
329 views

Factorial equaling a polynomial

Are there any positive integer solutions $(n,x)$ to the equation $(x)(x+1)=n!$ except $(2,1)$ and $(3,2)$? If not (as I suspect is the case), how do you prove that? In general, is there a way to ...
4
votes
0answers
98 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 ...
38
votes
5answers
787 views

Generalizing the sum of consecutive cubes $\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$ to other odd powers

We have, $$\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$$ $$2\sum_{k=1}^n k^5 = -\Big(\sum_{k=1}^n k\Big)^2+3\Big(\sum_{k=1}^n k^2\Big)^2$$ $$2\sum_{k=1}^n k^7 = \Big(\sum_{k=1}^n ...
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