Questions on finding integer/rational solutions of polynomial equations.

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-2
votes
0answers
15 views

Diophantine equation of four variables

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 ...
1
vote
5answers
105 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
votes
1answer
34 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
votes
0answers
64 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 ...
1
vote
2answers
45 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 $a, b, c, d, e$ ...
-3
votes
2answers
50 views

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

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)
0
votes
0answers
33 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 ...
1
vote
3answers
60 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
votes
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 $x=1$ ...
0
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3answers
33 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 ...
1
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0answers
37 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
vote
1answer
31 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 : ...
0
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2answers
48 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
votes
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
91 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
46 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
votes
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.
3
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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
69 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
votes
0answers
159 views
+50

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
votes
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
111 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 ...
0
votes
2answers
49 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
votes
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
19 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 ...
1
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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 ...
1
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2answers
65 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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votes
1answer
48 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
30 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
votes
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
106 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 ...
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0answers
26 views

Diophantine Equation with gcd. [duplicate]

Find all positive integers $a,b$ such that $\gcd(3^a+1,3^b+1)$ is a multiple of $ab$. I've given this problem many attempts but I can't seem to make any progress, there doesn't appear to be any way ...
-2
votes
1answer
46 views

Solve for b and d

Solve for b and d in the following equation. A triangle with sides $(a, a, b)$ has the same area and the same perimeter as a triangle with sides $(c, c, d)$ where $a, b, c$ and $d$ are positive ...
3
votes
2answers
126 views

Showing there is no triplet of positive integers $(a,b,c)$ satisfying $a^7+b^7=7^c$ [duplicate]

Show that $$a^7+b^7=7^c$$ has no positive integer solutions $(a,b,c)$. I've posted a general and way too long approach as an answer. How may one prove the claim more briefly and specifically?
0
votes
1answer
71 views

Are all even numbers the difference of prime powers

Does there exist an even positive integer greater than $100$ (to eliminate trivial cases) that cannot be expressed in the form: $p^2-q$ $p-q^2$ $p^2-q^2$ $p^3-q^3$ where $p$ and $q$ are primes.
0
votes
1answer
64 views

Can the solvability of a single Diophantine equation be undecidable (in any sense of the word)?

Apologies in advance for asking the following "philosophical" question, which falls dramatically short of any reasonable standards of mathematical rigour: Is it possible that there should exist a ...
2
votes
1answer
158 views

Solve $x^n+z^n=(x+1)^n$ for $n\ge 3$ without FLT

Is there a way to prove that for $x,z,n \in \mathbb{Z}$, $x > 0$, $z > 0$, $n > 2$, the equation $$ x ^ n + z ^ n = (x + 1) ^ n $$ has no solution, without using Fermat's Last Theorem? ...
4
votes
2answers
75 views

Substitutions that transform Fermat Equations to Elliptic Curves

I was reading Chapter 1 of Elliptic Curves - Number Theory and Cryptography by Lawrence C Washington. He was considering Fermat equations $$a^4+b^4=c^4\text{ and }a^3+b^3=c^3.$$ For the 1st equation, ...