Questions on finding integer/rational solutions of equations.

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0
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1answer
47 views

Diophantine equation $(n-x-y)^2 = 4xy + 1$

Given the Diophantine equation $(n-x-y)^2 = 4xy + 1$ where n is known and x, y are unknown: (1) Is there a single solution for any given n, or are there multiple solutions? (2) How does one determine ...
2
votes
1answer
76 views

Show that $x^3 + y^3 + z^3 = 4$ has no solutions?

As the title says, I've been asked to show that the equation $x^3+y^3+z^3 = 4$ has no solutions for any integers $x$, $y$, and $z$. I'm a bit stuck on where to start, so I'd appreciate any help or ...
2
votes
3answers
106 views

Find all non-negative integer solutions of $x+y+z=20$

I'm looking for all non-negative integer solutions to $x+y+z=20$ and I reason this way: I essentially take $\overbrace{1 + 1 + ... + 1}^{\text{20}}$ and segment this into three groupings. In essence ...
2
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2answers
47 views

Factorize : $(x+y+z)^p-[(-x+y+z)^p+(x-y+z)^p+(x+y-z)^p]$ where $p$ is an odd prime.

I am trying to factorize the expression: $$(x+y+z)^p-[(-x+y+z)^p+(x-y+z)^p+(x+y-z)^p]$$ where $p$ is an odd prime and $x,y,z$ are any non-zero integers. I know that it is divisible by $pxyz$. How do ...
0
votes
1answer
34 views

Is there a general formula to compute the number of integer solutions of an equation?

recently, I asked a question concerning the number of solutions of a diophantine equation that used the rounding function. This question, however, dealt with a linear function, and I was wondering if ...
7
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2answers
64 views

Writing an integer as a sum of two square in many ways, with consecutive arguments

Let $n\in{\mathbb N}$. I call $n=x_1^2+y_1^2=x_2^2+y_2^2=\ldots x_r^2+y_r^2$ (where $(x_1,y_1),(x_2,y_2),\ldots,(x_r,y_r)$ are distinct uples in ${\mathbb N}^2$) a multi-decomposition of $n$, of ...
2
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0answers
43 views

Show that $x^4+py^4+p^2z^4=p^3w^4$ has no solutions, where $p$ is any prime.

I am trying to show that the equation: \begin{equation}x^4+py^4+p^2z^4=p^3w^4\end{equation} has no solutions. Assuming there is a nonzero solution $(x_0,y_0,z_0,w_0)$, with $w_0$ minimal, then it ...
2
votes
1answer
29 views

why this Diophantine equation $2ks=(5t+3)(16t+9)$ has always a solution for every $k$?

I would like to solve the following Diophantine equation and show that it has always a solution; i.e. for every positive integer $k$, there exists an integer $t$ such that the fraction is an integer: ...
1
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1answer
76 views

Solutions of diophantine eq: $x^4-2x^3y+2xy^3+y^4=2s^2$

I'm examining solutions of this diophantine equation: $$x^4-2x^3y+2xy^3+y^4=2s^2$$ It looks like all the solutions are of the form $(x,y,s) = (t,\pm t, \pm t^2)$ where $t$ is any integer. But how do ...
2
votes
2answers
88 views

Parametrization of $a^2+b^2+c^2=2d^2$

Is a complete parametrization of primitive solutions to the equation $$a^2+b^2+c^2=2d^2\qquad a,b,c,d \in \mathbb{Z}$$ known? A reference would be great. Solutions to $a^2+b^2=c^2$ give solutions ...
1
vote
1answer
37 views

How to solve the following Diophantine equation $(s+1)(3s+2)(2s+1)^2=r^2?$

How to solve the following equation $(s+1)(3s+2)(2s+1)^2=r^2,$ where $s,\space r \in \mathbb{Z},\space s>0,\space r>0$? If it is not possible to solve it, probably something can be said about ...
17
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4answers
720 views

Why can't $p^p-(p-1)^{p-1}=n^2$ be a square?

Let $p$ be a prime number. Show that $p^p-(p-1)^{p-1}$ can't be a square. In other words, there is no $n\in\mathbb{N}^{+}$ such that $$p^p-(p-1)^{p-1}=n^2.$$
2
votes
1answer
62 views

Diophantine equation: $y^2=1+12x+16x^2$

The diophantine equation $$y^2=1+12x+16x^2$$ only has solutions $x=0, y=\pm1$ according to wolfram alpha. How would I go about proving these are the only solutions? Similarly the equation ...
2
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1answer
49 views

Using Hensel's Lifting Lemma to Solve $x^2 + x + 34 \equiv 0 \pmod{81}$

As in the title, I'm trying to solve $$x^2 + x + 34 \equiv 0 \pmod{81}.$$ Let $f(x) = x^2 + x + 34$ throughout. I'm using Hensel's lemma, but it's a bit dense and I'm not sure my interpretation is ...
2
votes
1answer
195 views

General method for determining if $Ax^2 + Bx + C$ is square

Is there a general method for solving Diophantine equations in the form $Ax^2 + Bx + C = k^2$, preferably turning them into Pell's equations, when possible? For example, $2x^2 + x + 1 = k^2$ or $5x^2 ...
0
votes
2answers
57 views

Algorithm for finding all roots of linear Diophantine equation with finite solution space

I have the following Diophantine equation: $$17a_1+16a_2+\dots+2a_{16}+a_{17}+c=0$$ with $c$ being a constant integer value, where I have two concrete cases: $c=-200$ and $c=-40$. I am looking for ...
2
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3answers
61 views

Find all the solutions of diophantine eq: $x^3-2xy^2+y^3-s^2=0$

Given $x,y,s$ are natural numbers: $$x^3-2xy^2+y^3-s^2=0$$ I found the solutions using wolfram alpha $$(x,y,s) = (1,2,1), (6,10,4), (4,8,8)$$ But how do I prove these are the only solutions? Any ...
3
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0answers
57 views

Parametrizing the solutions to a diophantine equation of degree four [closed]

Good evening, Consider $x^4+y^4+z^4=2t^4$ where $x,y,z,t$ integer. Is it known how to find all parametrizations of this equation? If you have any parametrization, or reference of this equation, ...
3
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1answer
62 views

A Diophantine Equation

Finding the number of $(a, b, c)$, where $a, b, c$ are positive integers, that $$ \frac{a^2+b^2-c^2}{ab}+\frac{c^2+b^2-a^2}{cb}+\frac{a^2+c^2-b^2}{ac}=2+\frac{15}{abc} $$ I factored it ...
3
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3answers
76 views

Solve $2n^3 + 3n^2 + n = d$

how to solve the following Diophantine equation? $$ 2n^3+3n^2+n=d $$ Here, $n$ and $d$ are non-negative integers. It would also be great if it were possible to formulate an algorithm for $$ an^3 +bn^2 ...
0
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1answer
44 views

Two sets of three integers, whose sum of squares are equal, and sum of fourth powers are equal

Consider the following two constraints on a set of integers: $$a^2+b^2+c^2=d^2+e^2+f^2$$ $$a^4+b^4+c^4=d^4+e^4+f^4$$ There are simple solutions like $a^2=b^2=c^2=d^2=e^2=f^2$, or pairs are the same ...
1
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2answers
43 views

Find the sum of all $x$ such that $y = 2 - \frac{9}{x+1}$ has an integer solution $(x, y)$

Find the sum of all $x$ such that $y = 2 - \frac{9}{x + 1}$ has an integer solution $(x, y)$ I made $x$ the subject of the formula, i.e., wrote $$x = \left(\frac{9}{2 - y}\right) - 1 .$$ I don't ...
3
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0answers
83 views

How can I find values for which a given expression gives a perfect square?

There have been several posts on this topic on math.se, such as this one with the same title. However all the posts I found contained coefficients to $x^2$, that were perfect squares. I am looking for ...
1
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1answer
82 views

Find all solutions to $2^n+n=m!$. [closed]

Find all natural $m,n$ such that: $$2^n+n=m!$$ I think the first $m$ factors to be discussed.
3
votes
1answer
60 views

Finding all solutions to $32x + 5y= 1$.

I am trying to find all $x,y \in \mathbb{Z}$ such that $32x + 5y = 1$. Here's how I see the situation. Since $(32,5) = 1$, we know there exists infinitely many integer solutions. Since $x_0 = -2$ and ...
1
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0answers
22 views

What is Mordell's conjecture in simple terms?

I am reading Ian Stewart's The Great Mathematical Problems, and come across the chapter regarding the Mordell's conjecture. After reading I couldn't draw a clear picture about the conjecture and also ...
1
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1answer
65 views

Positive integer solutions to $5a^2 + 8a + 4 = 4b^2$

I'm convinced $5a^2 + 8a + 4 = 4b^2$ can be somehow turned into Pell's equation. My first steps: Rewrite as $5(a + 4/5)^2 + 4/5 = 4b^2$. Rewrite $B = 2b, A = 5a + 4$ to get $A^2 + 4 = 5B^2$. I'm ...
6
votes
3answers
178 views

Solve symmetric equations with 4 variables

Is there a method to find or count the number of unique integer solutions $(n, H, L, W)$ to symmetric equations such as, $$x = 4n^2 + 2 + 4n(H + L + W) + 2HL + 2HW + 2LW$$ given $x$? All variables ...
3
votes
4answers
93 views

Finding all positive integer solutions for $x+y=xyz-1$

How do I manually solve $x+y=xyz-1$ assuming that $x, y$ and $z$ are positive integers? I was able to guess all possible solutions, but I do not know how to show that these are the only ones: $x=1, ...
3
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2answers
54 views

Find solutions of $2^m\cdot p^2+1=q^5$

$2^m\cdot p^2+1=q^5$ $p$ and $q$ are prime numbers find $p$ and $q$ I think it will be useful to transfer $1$ to the other side of the equation $2^m\cdot p^2=(q-1)(q^4+q^3+q^2+q+1)$ and we know ...
3
votes
4answers
47 views

Minimum number of fractions to be summed up to $\frac45$

What is the minimum number of fractions having numerator 1 and a natural number as denominator to be summed up to $\frac 45$? I have tested with 2 fractions: $\frac1a + \frac1b = \frac45$ and get into ...
0
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1answer
48 views

Does it possible to show that the Diophantine equation $X^2-Y^2=N$ has no solution except trivial?

Does it possible to show that the Diophantine equation $X^2-Y^2=N$ (N - odd)has no non-trivial solution?
3
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1answer
70 views

$\sum_i \frac{1}{a_i}=1$ implies $\prod_i a_i$ is a square

Let $a_1<\cdots<a_k$ be positive integers such that $\sum_i \frac{1}{a_i}=1$. Prove that if $a_k/2$ is prime then $\prod_i a_i$ is a square.
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2answers
41 views

show this diophantine equation has at least is $3n+3\lfloor \frac{n+1}{3}\rfloor+1$ postive integer solution

For any postive integer $n\ge 4$, let $s(n)$ denote the number of ordered pairs $(x,y,z)$ of positive integers for which $$\color{red}{xy+yz+xz=n(x+y+z)}$$ show that $$s(n)\ge 3n+3\lfloor ...
1
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1answer
45 views

Determining the highest value of c in a linear diophantine equation for which there exists three positive solutions

Given $$5x+7y=c$$What is the largest value of c for which there exists exactly 3 solutions (x,y)? I've tried researching how to find the exact number of positive integer solutions for linear ...
3
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4answers
128 views

Equation - what first?

I have this equation: $$ (x+y)(x^x + y^y) = 2009. $$ I must designate all pairs of integers satisfying the equation. What first? I tried multiply brackets , but to no avail
2
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0answers
21 views

Constellations of three powers

How can I prove that for all $i, j \in \mathbb{N}$ there are only a finite number of solutions to $x^a + i + j = y^b + j = z^c$ with $a,b,c,x,y,z \in \mathbb{N}$ and $a,b,c \ge 2$? This is a weaker ...
0
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0answers
39 views

Linear diophantine equations with arbitrary powers as coefficients

In an algebra problems book I came across a type of equations I can't find a way to solve. For example, a linear diophantine equiation $14^{23} x - 33 y = 5$. This can be solved with a straightforward ...
1
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1answer
65 views

Find the integer values of c

Find all possible positive integer values of c that $\frac {a^2+b^2+ab} {ab-1}$=c can take in $\mathbb N$. I know that the solutions are c=7 and c=4 but I don't know how to prove this with Vieta ...
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0answers
31 views

Diophantine equations - how to go about it?

What are the methods of solving such tasks? For example: $$ \begin{cases} x^2+y-z=100\\ y^2+x-z=124 \end{cases} $$ What first?
3
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0answers
49 views

Classifying Diophantine Equations

Take a given Diophantine equation. Chances are, we can't find any solutions. But if it's an equation of a certain form, we may get lucky and may be able not only to find a solution, but be able to ...
2
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1answer
125 views

Number of solutions of a diophantine equation using the rounding function

With the equation defined as: $$\left\lfloor \frac {(x-1)(a_n-a_1)}{n-1}\right\rceil+\left\lfloor \frac {(y-1)(a_n-a_1)}{n-1}\right\rceil=N$$ How many integer values can $x$ and $y$ take as a ...
0
votes
1answer
39 views

How to get a solution of $119x-71y=19$ in natural numbers?

I'm trying to solve the following equation in natural numbers: $119x-71y=19$. I used extended Euclid for this: $119 = 71 \cdot 1 + 48$ $71 = 48 \cdot 1 + 23$ $48 = 23 \cdot 2 + 2$ $23 = 2 \cdot ...
1
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2answers
95 views

Diophantine Equation : $x+y+z=3$ and $x^3+y^3+z^3=3$

Solve, in integers, the system of equations. $x+y+z=3$ $x^3+y^3+z^3=3$ I'm not sure how to approach this question, as I have only dealt with linear diophantine equations.
0
votes
1answer
28 views

Solution of quadratic diophantine equations

Is there any algorithm so that solution to the following equation can be found? $(x+a)^2-y^2=c$ where $c$ and $a$ is a constant. It is similar to Pells eqution with a variation where $D=1$. I am new ...
0
votes
1answer
65 views

How do I solve this over the integers

I have this equation: $\frac{2(2^{p} + n^{2} - 1 - 2n)}{n(n-1)} == s, p >= 1, n >= 3, s >= 3$ For which $n$ and $p$ is $s$ an integer? I dont really have a good idea on how to solve this ...
0
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0answers
22 views

Find if an integer can be written as a sum of n positive integers with only k unique integers k<=10

Given a set of $n$ positive integers $A$ ($n \leq 10^4$), where the number of unique integers is $k$ ($k \leq 10$), I need to find if a positive integer $W$ can be written as a sum of a subset of the ...
4
votes
2answers
75 views

How to generate another solution $\in \mathbb N$ of $x^2=2y^2+7$ knowing one.

This is from a new book I started reading; it asks whether I can construct another solution for $x^2=2y^2+7$, given that $(x_0,y_0)$ is one. It gives the following diagram as a hint: However, this ...
2
votes
4answers
71 views

$\frac{x^3+y^3+z^3}{x+y+z}\in \mathbb N$ has infinitely many non-trivial solutions

Trying to solve this I find out the following problem in which it is not necessary the condition $x^3=y^3=z^3$ in some $\mathbb F_p$: Prove there are infinitely many pairwise coprime triples of ...
0
votes
0answers
18 views

Existence of positive solution of a diophantine

Is there any way to find if there are any positive solutions of a diophantine equation ($ax$ $+$ $by$ $=$ $c$)? I don't want to find $x$ and $y$. I just wan't to if such a solutions exist that $x$ and ...