Questions on finding integer/rational solutions of polynomial equations.

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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 > ...
5
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3answers
50 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 $z=a+b\omega$, ...
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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
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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\, ...
2
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3answers
76 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 $\color{blue}{\text{both}}$ $k = 2,4$ ...
1
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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
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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 ...
0
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0answers
26 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 ...
0
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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 ...
4
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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 ...
5
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0answers
87 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
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0answers
46 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?
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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
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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
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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. ...
1
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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 ...
1
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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 ...
0
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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
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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 ...
4
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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) ...
4
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1answer
84 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 ...
6
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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 ...
0
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0answers
28 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
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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
0answers
53 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 $k$, ...
8
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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 ...
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 ...
2
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1answer
138 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 + ...
0
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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 ...
4
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5answers
178 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 $k$?
1
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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
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0answers
28 views

Does this “distribution of factors” cover all possibilities?

I have the Diophantine equation $$3a^2(4a^2+1)=b(b+1). \tag{$\star$}$$ Each side can evidently be “separated” into two [integer] factors as $$3a^2 \cdot (4a^2+1) = b \cdot (b+1).$$ Now I believe I ...
3
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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 ...
3
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0answers
55 views

Diophantine $7^a+2=3^b$

I want to find the solutions $(a,b)\in\mathbb{Z}^+\times\mathbb{Z}^+$ of $7^a+2=3^b$. One such solution is $(a,b)=(1,2)$. Looking modulo $4$, we have $(-1)^a+2\equiv(-1)^b$, so $a$ and $b$ are of ...
3
votes
1answer
118 views

At which p-adic fields does the equation have no rational solution?

I have to check if the equation $3x^2+5y^2-7z^2=0$ has a non-trivial solution in $\mathbb{Q}$. If it has, I have to find at least one. If it doesn't have, I have to find at which p-adic fields it has ...
2
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1answer
33 views

Diophantine Equation 1

I want to solve for positive integral values of $x$ and $y$: $$1216562x=87654321y+a$$ Here $a$ is a positive integer. For example if $a=40642509$ then one solution is : $x=37716$ and $y=523$ How do I ...
3
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1answer
67 views

Why doesn't the equation have a solution in $\mathbb{Q}_2$?

I have to find for which primes $p$, the equation $x^2+y^2=3z^2$ has a rational point in $\mathbb{Q}_p$. According to my notes: Obviously, $\forall p \in \mathbb{P}, p \nmid 2 \cdot 3$, there is a ...
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0answers
42 views

Distinct solutions to linear Diophantine equations

I am trying to solve a linear Diophantine equation of the form: $$x_1 + x_2 + x_3 + x_4 + x_5 = 15$$ where $x_1, x_2, x_3, x_4,$ and $x_5$ are all unique, and $1 \leq x_i \leq 5$. Is there an ...
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0answers
37 views

Undecidebility in Number Theory [duplicate]

Recently one of my teachers says that it is not impossible that we find a problem in number theory that is undecidable in usual system of set theory. This was so wonderful for me. When I say this ...
4
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1answer
83 views

What are the integer solutions of the system $a^2+b^2=c^2$, $a^3+b^3+c^3=d^3$?

How to solve these equations to find the integer numbers (a, b, c, and d)? $$a^2+b^2=c^2\tag{1}$$ $$a^3+b^3+c^3=d^3\tag{2}$$ I know one of solutions which is $a=3, b=4, c=5, ...
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0answers
19 views

Why do we conclude that $(a,b)=1$, having found that $(a',b'=1)$?

Suppose that we have the equation $ax^2+by^2+cz^2=0, a,b,c \in \mathbb{Q}$. Without loss of generality, we suppose that $gcd(a,b,c)=1$. Also, we can consider that $a,b,c$ are square-free. We can ...
1
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2answers
175 views

Does the equation has a non-trivial solution?

Could you give me some hints how I can solve the following exercise? Check if the equation $3x^2+7y^2-5z^2=0$ has a non-trivial solution in $\mathbb{Q}$ . If it has a solution, find at least one. If ...
2
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1answer
20 views

Infinite geometric series whose coefficients correspond to the number of solutions of a Diophantine equation.

This is problem 2 from Polya's Problems and Theorems in Analysis I. The question is as follows, Let $A_n$ denote the number of solutions to the Diophantine equation $x+5y+10z+25u+50v=n$. What is the ...
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0answers
57 views

When $Ax^2+By^2=z^2$ has a solution in integers?

Consider the Diophantine equation $Ax^2+By^2=z^2$, with positive integer parameters $A$ and $B$ (not necessarily square-free or co-prime). When does this equation have a non-trivial solution? Can one ...
2
votes
2answers
61 views

Existence of integer solution to 63x+70y+15z=2010

I have an equation $63x+70y+15z=2010$. The question asks me to conclude whether it has an integral solution or not? Any help on how to proceed?
0
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1answer
33 views

Diophantine solution set for $\frac{n(n-1)}2 = b(b-1)$

By Diophantine solution set I mean solutions where n and b are integers. I have one solution I found by trial and error but ...
0
votes
0answers
16 views

Interval in future

I have two intervals (times). For example, t1= 0:17, in this interval, we are now and interval t2=0:12, and the time, when was the time "time2". time2WasBefore = 0:04( which means that t2 was from now ...
9
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0answers
95 views

Prove that $ x^xy^y=z^z $ has infinite integral solutions [duplicate]

Show that there exist an infinite number of solutions for $$ x^xy^y=z^z $$ where $x,y,z \gt 1$ & $x,y,z\in \mathbb Z$ I don't know how to even start,infact I am not able to find a particular ...
0
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0answers
23 views

Solve this Simple quadratic equation $cU^2-2(a+ b)U+2(a-b)V- cV^2=0$

I need help solving this symmetrical quadratic equation where $\gcd(a,b,c)=1$$cU^2-2(a+ b)U+2(a-b)V- cV^2=0$ Is there an easier method than the quadratic formula? Any hint?
6
votes
1answer
83 views

Solve $3x^2-y^2=2$ for Integers

If $x$ and $y$ are integers, then solve (using elementary methods) $$3x^2-y^2=2$$ I tried the following If $y$ is even, then $4|y^2$ and hence $2|y^2+2$ (and $4$ doesn't divide it), but ...