Questions on finding integer/rational solutions of polynomial equations.

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1answer
20 views

Parametrization of $x^2+ay^2=z^k$, where $\gcd(x,y,z)=1$

$x,y,z$ be 3 coprime integers, $a \in \mathbb{Z}>0$ and $k$ an odd integer. How do I find all the non-trivial solutions of the diophantine equation? $$x^2+ay^2=z^k$$ Does the method which consists ...
0
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0answers
26 views

Simpler proof of the lemma: if $\gcd(a,b)=1$ then all odd factor of $a^2 + 3b^2$ has the same form?

Lemma: If $\gcd(a,b)=1$, then every odd factor of $a^2 + 3b^2$ has this same form. I was reviewing the proofs for this lemma online. Every proof is long and cumbersome. Is there a simpler method to ...
4
votes
1answer
53 views

Solution to Diophantine equation $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2} $

I have to prove the following, but I don't know how to start. The only solutions in positive integers of the equation $$ \frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2} \qquad \gcd(x,y,z)=1 $$ ...
-2
votes
2answers
92 views

Solve $x,y\in \mathbb{Z}$ [on hold]

Solve for $x,y\in \mathbb{Z}$ $$x^{6}=y^{2}+53$$ I tried but I couldn't complete
1
vote
1answer
45 views

All integer solutions to diophantine equation: $x^2+p y^2=z^2$?

I would like to find all integer solutions to the Diophantine equation $$ x^2+p y^2=z^2 $$ where $p\ge2$ is a given prime number. Also prove that my (probably parametric similar to Pythagorean triples ...
5
votes
8answers
173 views

Proving $x=2,y=4$ is the only solution to $x^y=y^x$ [duplicate]

Prove $x=2,y=4$ is the only solution to $x^y=y^x$ with the additional proviso that $x\ne y$ and $x,y$ are positive integers (if $(x,y)$ is a solution, so is $(-x,-y)$). Ideally I am looking for a ...
3
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0answers
72 views

Solutions to $\prod_{i=1}^{k}p_i=\sum_{i=1}^{k}p_i^2$ with $p_i$ distincts primes

Does $\prod_{i=1}^{k}p_i=\sum_{i=1}^{k}p_i^2$ with $p_i$ distincts primes and $k\geq2$ have a solution ? Here is what I already know : There is no solutions if $k\equiv0\bmod2$ or if ...
-2
votes
0answers
25 views

Diophantine equation of four variables [on hold]

Please is there one who can solve this form of Diophantine equation of four variable aw+bxy+cxz+kx=d where a,b,c,d,k are constant and w,x,y,z are variables and give reasons where there is positive ...
2
votes
5answers
116 views

How to prove $\frac{2^a+3}{2^a-9}$ is not a natural number

How can I prove that $$\frac{2^a+3}{2^a-9}$$ for $a \in \mathbb N$ is never a natural number?
2
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1answer
54 views

solutions for Diophantine equation for ${k_1} + 2{k_2} + 3{k_3} + … + n{k_n} = n$

Consider the Diophantine equation of the form ${k_1} + 2{k_2} + 3{k_3} + .... + n{k_n} = n$, where ${k_1},{k_2},...{k_n} \in Z^+$ . For a given $n$, how can I obtain the solutions of a given equation? ...
2
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0answers
76 views

How to prove every term of this sequence is not a natural number

Sorry for the repost and for my "bad" English. I made a lot of errors in the previous one, so here's my actual question: Let's take a look at this sequence: (1) $[a_1,a_2,a_3,a_4,...,a_x]$ where ...
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2answers
48 views

Solving two diophantine equations.

Find at least one 5-tuple of positive integers which satisfy the following two equations $$a^2-d^2=3(b^2-c^2)$$ $$e^2-b^2=3(d^2-c^2)$$ such that no three of the 5 positive integers $a, b, c, d, e$ ...
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2answers
53 views

All natural numbers $m, n$ such that $m = \sqrt{\frac{1}3A^2 - 3n^2}$ [closed]

I have $m = \sqrt{\frac{1}3A^2 - 3n^2}$. A is a known integer. How do I find all solutions for what m and n are if both m and n are naturals (round positive numbers)
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0answers
36 views

Exponential Diophantine equation to solve Project Euler problem

I am currently trying to solve problem 321 on project euler I know that each $n$ must exist such that $$8n^2+16n+1$$ is a perfect square. This is derived from the equation for the swapping of ...
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3answers
63 views

Multivariable Equation: $4ab=5(a+b)$

Find all Natural roots of the following multi-variable equation : $4ab=5(a+b)$ I have tried many handy candidate solutions and it seems there is no SOLUTION! Indeed we should show that it has no root ...
2
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1answer
36 views

Generalization of Erdos-Selfridge

Consider the equation $P(x)=y^d$ where $d \geq 2$ is an integer and $P$ can be written $P(x)=c(x-r_1)(x-r_2)\ldots (x-r_t)$ where $c$ and all the $r_i$ are integers not all equal (some of them can be ...
3
votes
3answers
57 views

Nonnegative Integer solutions of $x+y-xy=0$

I would like to see other methods, besides the one I use here to find all the nonnegative integer solutions of an equation like $$x+y-xy=0$$. This is the one I used: First we note that for $x=1$ ...
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3answers
34 views

Find more integral points on a hyperbola

Let $\mathcal H$ be a hyperbola (in the affine plane) whose defining equation has integers coefficients. Assume that one knows 2 points of $\mathcal H$ with integral coordinates. Is there a way to ...
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0answers
40 views

On Catalan's complete solution to the equation $T^2=U^2+V^2+W^2$

Catalan proved the following: If $t,u,v,w$ are coprime integers such that \begin{equation*} t^2 = u^2 + v^2 + w^2, \end{equation*} then there exist integers $\alpha,\beta,\gamma,\delta$ such that ...
1
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1answer
33 views

Diophantine sets

I am reading the following part: Diophantine sets A subset of a power $\mathbb{Z}^n$ of the set $\mathbb{Z}$ of integers is diophantine if it can be written as $$\{\overline{x} \in \mathbb{Z}^n : ...
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2answers
50 views

A tricky diophantine equation with factorials

I am being unable to solve this diophantine equation. Does anyone have any suggestions. Let $n$ and $m$ both be non-negative integers. Find all solutions to $$n(nm - 2)! = (n!)^m$$ How would one ...
0
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1answer
24 views

How do I put up a table of conditions so that $\frac{ab}{cd} \in \mathbb{Z}$ with $abcd \neq 0$

Let $a,b,c,d \in \mathbb{Z}\neq 0$, I am trying to put up an organized method to find the exhaustive list of conditions such that:$$\frac{ab}{bc} \in \mathbb{Z}$$ like a table. How can I do that?
2
votes
3answers
92 views

Solution of a quadratic diophantine equation

I try to solve the Diophantine quadratic equation: $$X^2+Y^2+Z^2=3W^2.$$ Obviously, there is a non-trivial solution: $(1,1,1,1)$. So I tried to apply Jagy's method: Solutions to $ax^2 + by^2 = cz^2$ . ...
0
votes
1answer
47 views

Diophantine equation - Special form (quadratic)

I am dealing with a series of quadratic diophantine equations that all have the same form: $$A^2x^2 - C^2y^2 + Dx - Ey + F = 0$$ ($A,C,D,E >0$ | $A$ and $C$ have a common factor (or $C=1$) | ...
0
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1answer
60 views

prove if $xyk \neq 0$, then: $x^3=3(k+xy)(k-xy-y^3)$ has no integral solutions.

Let $\gcd(x,y)=1, k \in \mathbb{Z}$ and $x \equiv 0 \pmod 3$. Show that if $xyk \neq 0$, then: $$x^3=3(k+xy)(k-xy-y^3)$$ has no integral solutions. Any hints? I keep getting lost in my reasoning.
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4answers
50 views

How do I show that we can't write $N=114^n-1$ as sum of $3$ squares for all natural number $n>2$?

I run some computations in wolfram alpha, I see that we can't write :$$N=114^n-1$$ as sum of $3$ squares, then Hop someone who can show me how I do prove that we can't write $N=114^n-1$ as sum of $3$ ...
4
votes
0answers
70 views

Exponential diophantine equation: $2p^2-6p+7=3^n$.

I'm trying to prove that the only integer positive solutions are $(n=1,\ p=1)$ and $(n=3,\ p=5)$. Is there a simple way to do that?
4
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0answers
203 views

Integer solutions of a cubic equation

With $\mathrm {gcd}(x,y)=1$ I have the following equation: $$x^3-xy^2+1=N$$ I want to find the integer solutions, given an N, of the variables $x$ and $y$. I have tried factoring the equation into ...
0
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0answers
23 views

The Relationship Between Euler Bricks and Pythagorean Quadruples

I've recently been studying the open problem of finding a Perfect Cuboid (a cube with integral sides and integral face and space diagonals). After doing some research, I came up with a conjecture that ...
0
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1answer
76 views

Prove quadratic diophantine has no solutions?

I am trying to prove a quadratic diophantine equation has no integer solutions. Any input would be great, I am interested in the general method for this type of equation so any explanation / link to ...
4
votes
3answers
113 views

How can $p^{q+1}+q^{p+1}$ be a perfect square?

How can one find all primes $(p,q)$ such that $p^{q+1}+q^{p+1}$ is a perfect square I considered it $\mod 2$ and found a trival solution . Im curious about an eventual answer Diophantine equations ...
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2answers
50 views

Can I solve for all integer solutions of this diophantine equation?

I do not know much about this subject, but this problem is bothering me. $$ x + 33y = 2399 $$ How can I find the possible integer values of x and y? I know there are two solutions, which I ...
0
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0answers
18 views

What is the best time complexity for this case?

I only want to know if the following system has any integer solution or not. Actually, I do not need to know the solution(s), and only need to know the answer of question "Does the system have any ...
0
votes
0answers
27 views

Unique integer solutions for $\frac{1}{x} + \frac{1}{y} = \frac{1}{z} $, with $x,y,z \ne 0 $, given $z$ [duplicate]

What are the unique integer solutions for a given integer $z$ for $\frac{1}{x} + \frac{1}{y} = \frac{1}{z} $, with $x,y,z \ne 0 $? From what I can tell, $x|yz,\ y|xz,\ z|xy$, so $x,y,z$ must ...
2
votes
0answers
20 views

$L$-existential and $L$-diophantine

Could you explain to me the last sentence: "Whenever we want to stress dependence on the language, we will use the self-explanatory terms and $L$-existential and $L$-diophantine" ? What does ...
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2answers
40 views

Find all natural values n, that $\sqrt{P_{2}(n)}$ is also a natural number

I have a polynomial of the second degree $a\cdot n^2 + b \cdot n + c$ and I need to find out natural numbers $n$, such that $\sqrt{a\cdot n^2 + b \cdot n + c}$ is also a natural number. After ...
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2answers
66 views

Finding a Solution to a Equation that Ends up as a Weird Repeating Series

I need to find the solution to this equation that ends up in a weird repeating series. The equation in question is: $$ \ln(y)=\frac{K}{\alpha}+\frac {x^{2}}{2\alpha\sigma}+\frac{\ln(\ln(y))}{2\alpha} ...
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1answer
50 views

Example of a diophantine polynomial

A diophantine set is a subset of a power $\mathbb{Z}^k$ of the set $\mathbb{Z}$ of integers which can be written as $$\{x \in \mathbb{Z}^k : \exists y \in \mathbb{Z}^m : P(x, y)=0\}$$ where $P$ is a ...
0
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0answers
31 views

Single variable diophantine equations with many solutions.

Do significantly non-trivial (read on for clarification), many solution, single variable diophantine equations exist? Diophantine equations are equations where all variables must be integers. The ...
3
votes
3answers
83 views

Integer solutions for $x^2+y^2=208$

Which steps I can follow to find the integer solutions for the equation $x^2 + y^2 = 208$?
0
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1answer
44 views

question on quadratic equations.

Let $p, q , r$ be distinct real nos such that $ap^2 + bp + c = (\sin(\theta))p^2 +(\cos(\theta))p$ similarly we get a total of three equations if we replace $q$ and $r$ in place of p. where $a, b, c$ ...
3
votes
1answer
52 views

Solve $2b(b-1) = t(t-1)$ as Pell's equation

I know the method of continued fractions to solve the Pell's equation. I need help turning $2b(b-1) = t(t-1)$, with $b, t$ as integers, into the form $x^2 - ny^2 = 1$, if possible. This is a Project ...
2
votes
1answer
122 views

Show that the equation $y^2 = x^3 + 3$ has infinitely many rational solutions in $x$ and $y$.

Show that the equation $y^2 = x^3 + 3$ has infinitely many rational solutions in $x$ and $y$. I'm really not sure how to go about this question. I've been using trial and error and have not got ...
2
votes
1answer
30 views

System of Congruences with Special Symmetry

Show that the following system of congruences \begin{align} \begin{cases} 3 x^4 - 7 x^2 y^2 - 7 x^2 z^2 - 35 y^2 z^2 \equiv 0 \pmod{p} \\ 3 y^4 - 7 x^2 y^2 - 7 y^2 z^2 - 35 x^2 z^2 \equiv 0 \pmod{p} ...
0
votes
1answer
36 views

Simple Linear Diophantine Equation - problem with proof.

I'm reading up on diophantine equations and one of the theorems is that "if $x,y$ is any solution of $ax + by = c$, then it is of the form $x_0 +\dfrac{b}{d}t ,\, y_0 - \dfrac{a}{d}t$ where $d = ...
2
votes
1answer
43 views

How do you calculate variables as exponents in a polynomial without a calculator?

Good day The problem is as follow: Find all solutions $(x, y)$, where $x, y \in \mathbb {Z^+}$ to the equation: $$1+3^x=2^y$$ Two solutions are $(0,1)$ and $(1,2)$ but how do you go about ...
3
votes
1answer
71 views

A golden trigonometric diophantine equation

After answering this question I reflected on the identity $$\cos\frac{\pi}{5}=\phi\cos\frac{\pi}{3}$$ and thought of looking for all the quadruplets of positive integers $(a,b,c,d)$ satisfying $$\cos ...
3
votes
2answers
50 views

Three variable, second degree diophantine equation

I am trying to solve this diophantine equation: $x^2 + yx + y^2 = z^2$ In other words, I am trying to find integers $x$ and $y$ such that $x^2 + yx + y^2$ is a perfect square. So far, the only ...
7
votes
1answer
99 views

Distribution of the sum reciprocal of primes $\le 1$

$$\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\cdots \le 1 $$ This is an interesting infinite summation. This is very closely resembling my other problem with has to do with the distribution of ...
7
votes
2answers
109 views

Does the sum of the reciprocals of composites that are $ \le $ 1

The sum itself: $$ \frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+ \frac{1}{15}+ \frac{1}{39}... \le 1 $$ These are all sums of reciprocals of composites that ...