Questions on finding integer/rational solutions of polynomial equations.

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2
votes
3answers
73 views

Sinha's Theorem for Equal Sums of Like Powers $x_1^7+x_2^7+x_3^7+\dots$

Sinha’s theorem can be stated as, excluding the trivial case $c = 0$, if, $$(a+3c)^k + (b+3c)^k + (a+b-2c)^k = (c+d)^k + (c+e)^k + (-2c+d+e)^k\tag{1} $$ for $\color{blue}{\text{both}}$ $k = 2,4$ ...
2
votes
1answer
133 views

What are the necessary and sufficient conditions for a cubic equation to have integers roots

Let's start with Fermat equation with the lowest power, $x^3 + y^3 = z^3$. Now let's set $y = x + a, z = x + b$ with $b > a$ and $a,b$ integers. then the equation becomes $$x^3 + (3a-3b)x^2 + ...
4
votes
2answers
360 views

Finding solutions to $(4x^2+1)(4y^2+1) = (4z^2+1)$

Consider the following equation with integral, nonzero $x,y,z$ $$(4x^2+1)(4y^2+1) = (4z^2+1)$$ What are some general strategies to find solutions to this Diophantine? If it helps, this can also be ...
1
vote
3answers
20 views

Let $ a$ be a positive integer. Show that $\text{gcd}(a,a-1) = 1$. use the result of par t $ a)$ to solve the Diophantine equation $ a+b=ab$

Not sure if I did part a right, not sure how to complete part $b)$ $a)$ Let $a$ be a positive integer. Show that $\text{gcd}(a,a-1) = 1$. Proof by contradiction suppose $\text{gcd}(n, n-1) = p > ...
4
votes
2answers
297 views

Positive integer solutions of $x^2+21y^2=z^4 $

Can one find all positive integer solutions of $$x^2+21y^2=z^4 ?$$ I am not sure if this is possible. I just saw this problem and this problem came to my mind.
0
votes
0answers
24 views

Finding set of integer pairs for which two integer polynomials intersect

I am wondering if there is a theorem in number theory that addresses the following issue: Suppose we have two polynomials, f and g, with integer coefficients. Is there a general way to find elements ...
5
votes
3answers
48 views

Eisenstein integers and applications to Diophantine equations

Solve the equation $7\times 13\times 19=a^2-ab+b^2$ for integers $a>b>0$. How many are there such solutions $(a,b)$? I know that $a^2-ab+b^2$ is the norm of the Eisentein integer $z=a+b\omega$, ...
0
votes
2answers
15 views

Two equivalent equation

Recently, one of my friends have told me that the following two equations are equivalent on the basis of the number of solutions. I checked the number of solutions to the two equations and found ...
1
vote
0answers
33 views

Theorem for Equal Sums of Like Powers $x_1^8+x_2^8+x_3^8+\dots$

Kindly see the question at the end of post. Solutions to the system of three equations, $$\begin{aligned} a^2+b^2+c^2+d^2\, &= e^2+f^2+g^2+h^2\\ a^4+b^4+c^4+d^4\, &= e^4+f^4+g^4+h^4\\ abcd\, ...
20
votes
3answers
845 views

Does an elementary solution exist to $x^2+1=y^3$?

Prove that there are no positive integer solutions to $$x^2+1=y^3$$ This problem is easy if you apply Catalans conjecture and still doable talking about Gaussian integers and UFD's. However, can this ...
5
votes
0answers
85 views
+50

Number of ways to express a binary number in a certain way

So I'm working on a problem where I get to a point where I have to count the number of solutions to an equation or at least find a decent upper bound to be used in an estimate I need later. The ...
1
vote
2answers
23 views

how many positive integer solutions to the following equation?

$a^2 + b^2 + 25 = ab + 5a + 5b$ I have tried looking for a factorisation that could solve this question but couldn't find anything useful - found $(a+b+5)^2$ - don't know if this is useful The ...
0
votes
1answer
22 views

How to show there are infinite solution to a given Pell's equation?

I was asked to prove the Pell's equation $$x^2-7y^2=1$$ has infinitely many solution. Here is what I did By using Brahmagupta method we can generate infinitely many integer solutions. Is that ...
4
votes
1answer
42 views

Solve $x^4 - 2x^3 + x = y^4 + 3y^2 + y \wedge (x,y) \in \mathbb{Z}^2$

I want to solve equation $x^4 - 2x^3 + x = y^4 + 3y^2 + y$ in integers. The task comes from the LXVI Polish Mathematical Olympiad. Series with this task ended twenty days ago, so it is legal to talk ...
1
vote
3answers
120 views

how do you solve $a^2+b^2+c^2=d^3$

let $ a,b,c,d$ be 4 integers such that $\gcd(a,b,c,d)=1$. How do you find the integral solutions of the equation: $$a^2+b^2+c^2=d^3$$
0
votes
0answers
12 views

Number of solutions of a diophantine inequality

In my problem i am looking for the number of the nonnegative integer pairs $(x,y)$ which satisfy the inqueality $x+my\leq n$, where $m$ and $n$ are coprime integers. The answer in the book is given ...
0
votes
3answers
81 views

Diophantine eqn, general solution?

Here's the equation: $$ 4 \left( x^2+y^2-z^2 \right)=\left( 2k+1 \right) \left( x+y-z \right) $$ Is there a nontrivial solution for this in integers? If not, why not? If there is, can a general ...
1
vote
2answers
54 views

Find all solutions of the equation $n^m=x^2+py^2$ which satisfy the following properties

Prove or disprove that, There always exists a solution of the equation, $$n^m=x^2+py^2$$ with odd $x$ and $y$ and for all $m\geq k$ for some positive integral $k$. Here $p$ is an odd prime and ...
6
votes
1answer
256 views

Ramanujan-Nagell Theorem Proof Question

I'm currently working through Stewart and Tall's Algebraic Number Theory. In particular, section 4.9 of this book provides a proof of the Ramanujan-Nagell Theorem, which states that the only integer ...
1
vote
0answers
45 views

Does this system of simultaneous Pell equations have any non-trivial positive integer solutions?

Let $a,b,c$ be positive integers satisfying \begin{align} 2a^2-1 &= b^2, \\ 2a^2+1 &= 3c^2. \end{align} The trivial solution is $(a,b,c)=(1,1,1)$. Are there others?
1
vote
1answer
35 views

Using stars and bars to find how many solutions there are to an equation with 3 variables

I'm trying to make an efficient algorithm to find how many solutions there are to the equation $$Ax+By+Cz=D$$ where $A,B,C,D\in \mathbb Z$ and the range for $x,y,z\in \mathbb Z$ are given by the user. ...
0
votes
0answers
23 views

$Ax+By+Cz=D \text { has a solution iff } \gcd(\gcd(A,B),C)\mid D$

I read today that $Ax+By+Cz=D \text { has a solution iff } \gcd(\gcd(A,B),C\mid D$ but I can't find it again, I also can't find any Diophantine equations with 3 variables that doesn't have solutions ...
0
votes
1answer
33 views

An effcient method of solving a Diophantine equation with 3 variables $Ax+By+Cz=D$?

I'm trying to make an efficient algorithm to find one of the solutions and how many solutions there are to the equation $$Ax+By+Cz=D$$ where $A,B,C,D\in \mathbb Z$ and the range for $x,y,z\in \mathbb ...
1
vote
0answers
28 views

Generalization of a Diophantine Equation Problem

I've been working a lot with Pythagorean triples and sums of squares that are themselves squares, specifically interlocking sums (where one square is part of two or more sums). As part of my work I ...
4
votes
3answers
82 views

For what $a,b$ such that $ax^2+by^2 = z^2$?

This post made me think about this question. What is the criterion on positive integer $a,b$ such that, $$ax^2+by^2 = z^2$$ can be solved in positive integers $x,y,z$? (Three broad classes are: 1) ...
0
votes
1answer
24 views

What are the general solutions of the Diophantine equation $ ax+by+cxy+d=0 $

Does the diophantine equation $$ ax+by+cxy+d=0 $$ always have solutions ?
3
votes
1answer
30 views

Integer solution for $Rx^2+Sy^2=1$ .

Is there any integer solution in-terms of $R,S$ for the equation $Rx^2+Sy^2=1$ , . For example $(\frac{1}{\sqrt {2R}},\frac{1}{\sqrt {2S}})$ is a solution but not integer solution . Is there any ...
3
votes
2answers
271 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 ...
4
votes
1answer
83 views

If $a,b > 1$ and $r>2$ does $ax^2+by^2=z^r$ have any rational solutions?

I have been trying to solve the following equation for months without much success. It has been so far a very frustrating endeavor.Please help. Consider the diophantine equation: $x^2+y^2=z^r$ where ...
4
votes
2answers
132 views

Solving $x^p + y^p = p^z$ in positive integers $x,y,z$ and a prime $p$

The question is from Zeitz's ''The Art and Craft of Problem Solving:" Find all positive integer solutions $x,y,z,p$, with $p$ a prime, of the equation $x^p + y^p = p^z$. One thing I noticed is ...
1
vote
3answers
100 views

Diophantine equation on an example

I do have one task here, that could be solved my guessing the numbers. But the seminars leader said, also Diophantine equation would lead to solution. Has anyone an idea how it works? And could you ...
6
votes
2answers
80 views

Equation $a^5+15ab+b^5=1$

What are the integer solutions of $a^5+15ab+b^5=1$? The equation is symmetric in $a$ and $b$, so let's assume $a\geq b$. When $a=b$, we have $2a^5+15a^2=1$, which has no solution by the Rational Root ...
1
vote
0answers
27 views

Number of solutions of $xy^2-y^2-x+y=k$ [closed]

Let $k$ be a positive integer. How many solutions does the equation $xy^2-y^2-x+y=k$ have in integers? The equation can be written as $x(y^2-1)-(y^2-y)=k$, or $(y-1)(x(y+1)-y)=k$.
4
votes
2answers
36 views

$8x +9y = 5$ where $x,y \in \mathbb{Z}$

Solve the following Diophantine equation algebaically: $$8x+9y=5$$ Give 3 possible solutions for the equation I have the following: The Diophantine equation has solutions $x,y \iff ...
2
votes
2answers
97 views

Triplets of distinct integers > 1 that return integer values.

If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer? ...
2
votes
0answers
52 views

What is stopping every Mordell equation from having a [truly] elementary proof?

The Mordell equation is the Diophantine equation $$Y^2 = X^3-k \tag{1}$$ where $k$ is a given integer. There is no known single method — elementary or otherwise — to solve equation $(1)$ for all $k$, ...
3
votes
5answers
59 views

Linear Diophantine equation $3x + 5y = 11$

Solve the Diophantine equation $3x + 5y = 11$ I know how to calculate GCD $$5 = 1\cdot 3 + 2$$ $$3 = 1\cdot 2 + 1$$ $$2 = 2\cdot 1 + 0$$ But how do I use this theorem to derive the correct ...
8
votes
1answer
67 views

Positive integer solutions to $a^{a^a}=b^b$

What are all positive integer solutions to $a^{a^a}=b^b$? $(a,b)=(1,1)$ works. If we take log on both sides, we get $a^a\log a=b\log b$, which is still hard to analyze. (It helps in equations like ...
0
votes
2answers
2k views

Linear Diophantine equation - Find all integer solutions

Using the linear Diphantine equation 121x + 561y = 13200 (a) Find all integer solutions to the equation. (b) Find all positive integer solutions to the ...
6
votes
7answers
1k views

Solutions to Linear Diophantine equation $15x+21y=261$

Question How many positive solutions are there to $15x+21y=261$? What I got so far $\gcd(15,21) = 3$ and $3|261$ So we can divide through by the gcd and get: $5x+7y=87$ And I'm not really ...
10
votes
5answers
387 views

Finding a Pythagorean triple $a^2 + b^2 = c^2$ with $a+b+c=40$

Let's say you're asked to find a Pythagorean triple $a^2 + b^2 = c^2$ such that $a + b + c = 40$. The catch is that the question is asked at a job interview, and you weren't expecting questions about ...
4
votes
5answers
177 views

When does $x^3+y^3=kz^2$?

For which integers $k$ does $$ x^3+y^3=kz^2 $$ have a solution with $z\ne0$ and $\gcd(x,y)=1$? Is there a technique for counting the number of solutions for a given $k$?
4
votes
1answer
25 views

Is there an $n$ such that $p|n^2+1$ with $2n<p<2n+\sqrt n$?

Is there an integer $n$ such that $n^2+1$ is divisible by a prime $p$ with $2n<p<2n+\sqrt n$? It's complicated to describe my interest, but these are near-missed for arc-cotangent reducible ...
0
votes
2answers
76 views

Prove there are no non-trivial solution to $3x^2 - 5y^2 + 7z^2 = 0$

I've tried using modulo $3$, and I get it down to $y^2 + z^2 = 0 \pmod 3$ ; I don't know where to go from here though. I justified my answer by stating that, because we're in $\pmod 3$ and we ...
3
votes
2answers
79 views

An annoying Pell-like equation related to a binary quadratic form problem

Let $A,B,C,D$ be integers such that $AD-BC= 1 $ and $ A+D = -1 $. Show by elementary means that the Diophantine equation $$\bigl[2Bx + (D-A) y\bigr] ^ 2 + 3y^2 = 4|B|$$ has an integer ...
1
vote
2answers
524 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, ...
1
vote
4answers
319 views

General Solution of Diophantine equation

Having the equation: $$35x+91y = 21$$ I need to find its general solution. I know gcf $(35,91) = 7$, so I can solve $35x+917 = 7$ to find $x = -5, y = 2$. Hence a solution to $35x+91y = 21$ is ...
7
votes
5answers
388 views

Find all solutions of the equation $x! + y! = z!$ [duplicate]

Not sure where to start with this one. Do we look at two cases where $x<y$ and where $x>y$ and then show that the smaller number will have the same values of the greater? What do you think?
5
votes
3answers
546 views

Solve : $x^2-92 y^2=1$

As some of you might know,this is Brahmagupta's equation . How to find solution for this ? I mean integral solution? How to solve it using programming ? I tried something like $x^2=1+92y^2$ ...
1
vote
3answers
109 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 ($\star$) ...