Questions on finding integer/rational solutions of equations.

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11
votes
4answers
241 views

Find all natural numbers $x,y$ such that $3^x=2y^2+1$.

Find all natural numbers $x,y$ such that $$3^x=2y^2+1$$ solutions are $(1,1)$, $(2,2)$, $(5,11)$. I found that parity of both is same and If $x$ Is odd it is of the form $4k+1$.
6
votes
1answer
80 views

When $n!=m(m+1)(m+2)$: A Diophantine Equation

I believe that I saw this problem not long ago in a book: Solve the Diophantine Equation $k!=n(n+1)(n+2)$, where $k,n$ are positive integers. However, I am no longer able to find this question, and ...
-13
votes
3answers
324 views

Is there any flaw in the following proposed elementary " proof'' of FLT? [closed]

Earlier on today, I received an interesting mail from a certain exceptionally talented undergraduate student, claiming an elementary proof of FLT. He has also submitted his paper to a formal journal ...
3
votes
2answers
167 views

Pell equations upper bound

Consider the Pell equation $x^2-p_n y^2=1$, where $p_n$ is the $n$th prime. Is $n^{2 \sqrt{n}}$ a reasonable upper bound for the smallest integer solution for $y$? Above is a plot of $\log x$ ...
1
vote
1answer
54 views

On the genus of a curve and its set of rational points.

The genus $g$ of a nonsingular curve $C$ of degree $n$ is defined as $g = \frac{1}{2}(n-1)(n-2)$. Let $C(Q)$ denote the set of rational points on $C$. By Faltings, we know that $C (Q) < \infty$ for ...
0
votes
0answers
14 views

Finite solutions in $x^2+D=λk^n$

In http://www.math.tifr.res.in/~saradha/saradharev.pdf it is stated that in $$x^2+D=λk^n$$ The following result of Siegel shows that the number of solutions $(x, n)$ in each case is finite: If $f(x) ...
4
votes
1answer
78 views

Is it true that for every $k\in\mathbb N$ , there exist infinitely many $n \in \mathbb N$ such that $kn+1 , (k+1)n+1$ both are perfect squares ?

Is it true that for every $k\in\mathbb N$ , there exist infinitely many $n \in \mathbb N$ such that $kn+1 , (k+1)n+1$ both are perfect squares ? What I have tried is that I have to necessarily solve ...
1
vote
1answer
58 views

Find the diophantine equation $x^2(y^2-1)=z^2-1$ solution

How can I solve (find all the solutions) the nonlinear Diophantine equation Let $x,y,z$ be postive integers ,and $x,y,z\ge 2$,find this following equation solution $$x^2=\dfrac{z^2-1}{y^2-1}$$ I ...
1
vote
0answers
32 views

Solving a Diophantine Equation with 2 variables

This is my answer for the following question: Find all natural numbers $(a,b)$ for which $a^b-b^a=1$. When $a$ or $b$ equals $1$, $(a,b)=(2,1)$ is trivial. If $a,b>1$, I generalized the problem ...
2
votes
2answers
35 views

A diophantine related query

Supposing I give you a multivariate equation $$F\in\Bbb Z[x_1,\dots,x_n]$$ Following is undecidable: 'Is there an $(a_1,\dots,a_n)\in\Bbb N^n$ such that $F(a)=0$?' However is the following always ...
1
vote
1answer
18 views

Proof that $x^2 + D = AB^y$ has in every case of $D,A,B$ a finite amount of solutions $x,y$

Could somebody please find me a proof that $$x^2 + D = AB^y$$ has in every case of $D,A,B$ a finite amount of solutions $x,y$. I forgot how this is called and would greatly appreciate it if someone ...
2
votes
0answers
133 views

Prove that $(a-b)^n\mid (a^n-b^n) \iff n=1$ under given conditions

Suppose that $a,b,(a-b)$ are pairwise co-prime (i.e. $a\perp b\perp (a-b)\perp a$), and that $\frac{a}{2}<b<a$, where $a$ and $b$ are both positive integers greater than $2$. Let $n$ be odd. ...
4
votes
3answers
88 views

Multiples that are one less than Squares

I was inspired to ask this problem after trying to find all $(x,y,u,v)$ for which $xy+1,xu+1,xv+1,yu+1,yv+1,uv+1$ are all sqaure. After some basic calculation I was easily able to find $x=n, y=n+2, ...
2
votes
1answer
95 views

Euler`s Theorem and Diophantine Equations

Euler`s Theorem says that for all coprime intergers $a,b$ $a^ {φ(b)} \equiv 1 \pmod b$. This implies that for any $z,x$ which satisfies $\gcd(x,y)=\gcd(z,y)=1$ $x^{φ(y)}-z^{φ(y)} \equiv 0\pmod y$ ...
0
votes
1answer
47 views

Rational points on $4x^5 + y^2 = z^2$

Does the title curve have any nonzero rational points ? I have to admit that i didn't find any significant insight to this problem.
1
vote
4answers
43 views

Does every linear integer polynomial give a square at some integer?

My question is, if you have some function $$f(x)=nx+c$$ which accepts only integer inputs of $x$, where $n>0$ and $c$ are fixed integer constants, can you always find an $x$ such that $$f(x)=k^2$$ ...
2
votes
3answers
159 views

Find all solutions to the diophantine equation $(x+2)(y+2)(z+2)=(x+y+z+2)^2$

Solve in postive integer the equation $$(x+2)(y+2)(z+2)=(x+y+z+2)^2$$ It is rather easy to find several parametric solutions, (such $(a,b,c)=(2,1,1),(2,2,2)$).but it seems harder to find a complete ...
3
votes
2answers
130 views

The Archimedes Cattle Problem and how to find $x^2-dp^2y^2=1$?

This was inspired by the Archimedes Cattle Problem. A crucial step is to solve the Pell equation, $$u^2-(609)(7766)v^2=1\tag1$$ and whose fundamental solution is, ...
1
vote
1answer
28 views

Difference between function and equation

What is the precise difference between function and equation ? In which case will it be wrong if used( common mistakes )? Also will the Venn diagram overlap if I were to draw one ? Any help and ...
1
vote
1answer
39 views

Show relations are Diophantine

Show that the following relations are diophantine. (a) $x_3$ is the remainder when $x_1$ is divided by $x_{2}+1$. (b) $x_3$ is the integer part when $x_1$ is divided by $x_{2}+1$ I'm not sure how ...
2
votes
0answers
79 views

An interesting equation in natural numbers

Let $n$ be a fixed natural number. How to solve the following equation in natural numbers: $$ \frac{1}{x_1} + \frac{2}{x_2} + \cdots + \frac{n}{x_n} = 1 $$ (I can find many soltions but I am looking ...
14
votes
3answers
219 views

Finding the common integer solutions to $a + b = c \cdot d$ and $a \cdot b = c + d$

I find nice that $$ 1+5=2 \cdot 3 \qquad 1 \cdot 5=2 + 3 .$$ Do you know if there are other integer solutions to $$ a+b=c \cdot d \quad \text{ and } \quad a \cdot b=c+d$$ besides the trivial ...
0
votes
2answers
73 views

Solutions to Diophantine Equations

I am looking for integer solutions to the equation $$x^2 = 5y^2 + 14y + 1$$ I know that Pell's Equation is of the form $x^2 - ny^2=1$ and that there exist algorithms to solve this equation. I was ...
2
votes
2answers
91 views

Can we find $x_{1}, x_{2}, …, x_{n}$?

Consider this. $$x_{1}+x_{2}+x_{3}+....+x_{n}=a_{1}$$ $$x_{1}^2+x_{2}^2+x_{3}^2+....+x_{n}^2=a_{2}$$ $$x_{1}^4+x_{2}^4+x_{3}^4+....+x_{n}^4=a_{3}$$ $$x_{1}^8+x_{2}^8+x_{3}^8+....+x_{n}^8=a_{4}$$ ...
0
votes
1answer
36 views

How to solve quadratic Diophantine equation with 3 variables

Given the equation: $3x^2 - x - 3y^2 + y = 3n^2 - n$ I'd imagine solving this involves techniques for solving Diophantines? Or am I wrong? Could someone point me in the right direction?
1
vote
0answers
51 views

When is sum of squares a perfect square? [duplicate]

Recall that $$\sum_{j=1}^nj^2=\frac{n(n+1)(2n+1)}{6}.$$ When is this quantity a perfect square? It appears that the only solutions are $n=0,1,24.$ By setting $x=12n+6$, the problem reduces to finding ...
1
vote
1answer
39 views

Problems & Solutions on Fermat Theorem of Multiple of 3

I am working on an assignment in elementary number theory, in which I have to come up with original problems and then work out their solutions on Fermat theorem of multiple of 3, that is, the equation ...
2
votes
2answers
111 views

Diophantine Equation with 2017th powers: $a^{2017}+a-2=(a-1)(b^{11})$

This problem stems from a recent student-created olympiad contest. Find all integer (not simply positive) solutions to $a^{2017}+a-2=(a-1)(b^{11})$. My multiple attempts modulo many small primes ...
0
votes
1answer
74 views

How to solve these system of linear equations?

I am having a problem to solve the following set of n equations: $$a_1 - k_1*b_1 = a_2 - k_2*b_2 = a_3 - k_3*b_3 = \dots = a_n - k_n*b_n$$ Given all the values of $a_i \ and \ b_i$, the question is ...
4
votes
4answers
160 views

Solve $x^n+y^n=2015$

Determine the natural numbers $x,y,n$ matching equality $$x^n+y^n=2015.$$ I noticed for $n = 1$ the equation has solutions $(x, 2015-x), x$ integer. For $n = 2$, given that $x$ and $y$ are different ...
3
votes
1answer
71 views

$3ab + a^3 - 2b^3 - 4a + 5b - 7 = 0$

I came across this problem: Prove there arent't any $a$, $b$ integers that satisfy equation $3ab + a^3 - 2b^3 - 4a + 5b - 7 = 0$ Firstly, I've thought something like this: $$(a^3 + b^3)-3b^3 ...
5
votes
1answer
59 views

A “flowchart” for handling Diophantine equations

There's no algorithm that correctly decides if a Diophantine equation does or doesn't have a solution. Still, many equations can be successfully analyzed, and I'm wondering if anyone wrote down a ...
3
votes
1answer
37 views

Birational Equivalence of Diophantine Equations and Elliptic Curves

A while ago I saw this question Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$ which was very relevant to a undergraduate research paper I am currently working on. The answer given ...
3
votes
2answers
82 views

Solve this equation $xy-\frac{(x+y)^2}{n}=n-4$

Let $n>4$ be a given positive integer. Find all pairs of positive integers $(x,y)$ such that $$xy-\dfrac{(x+y)^2}{n}=n-4$$ What I tried is to use $$nxy-(x+y)^2=n^2-4n\Longrightarrow ...
14
votes
1answer
133 views

Prove that ${x^7-1 \over x-1}=y^5-1$ has no integer solutions

I want to show that $${x^7-1 \over x-1}=y^5-1$$ cannot have any integer solutions. The only observation I have made so far is that the left hand side is the $7$th cyclotomic polynomial $$\Phi_7(x)= ...
5
votes
1answer
120 views

Rational points on $y^5 = x^4 + x^3 + x^2 + x + 1$

Doess the above curve have only two rational points namely $(x,y)=(0,1)$ and $(-1,1)$ ?
5
votes
0answers
93 views

Describe the integral solutions to this cubic equation.

Consider the following cubic equation in $c$: $c^3 - 3c^2(a+b) + 3c(a+b) -3ab(a+b)-3=0$ Does this equation have infinitely many integer solutions $(a,b,c)$ ? EDIT: My attempt was rerwriting it as a ...
0
votes
1answer
47 views

Solve in set of natural numbers

Solve in set of natural numbers the following systems: \begin{align} &\text{(a)} && x + y = 150,\quad \gcd(x, y) = 30\\[12px] &\text{(b)} && \gcd(x, y) = 45,\quad 7x = ...
2
votes
1answer
50 views

Describe the integral solutions to $y^2 = 12x^3 - 39$

Does the above Diophantine equation have infinitely many integer solutions ? One such solution is $(x,y) = (4,27)$.
5
votes
3answers
91 views

If $x-y = 5y^2 - 4x^2$, prove that $x-y$ is perfect square

Firstly, merry christmas! I've got stuck at a problem. If x, y are nonzero natural numbers with $x>y$ such that $$x-y = 5y^2 - 4x^2,$$ prove that $x - y$ is perfect square. What I've ...
0
votes
0answers
141 views

Is the following theorem useful?

Theorem: Odd integer $N=6p+5$ is a prime number if and only if no one of two diophantine equations $$6x^2-1+(6x-1)y=p$$ $$6x^2-1+(6x+1)y=p$$ has solution. Odd integer $N=6p+7$ is a prime number if ...
3
votes
2answers
51 views

Prove that $\gcd(k,n) = 1$ if and only if $ \exists m,d \in \mathbb{Z}: mk+nd=1$

I need to understand why $\gcd(k,n) = 1 \Leftrightarrow \exists m,d \in \mathbb{Z}: mk+nd=1 $. Any help would be appreciated.
1
vote
0answers
74 views

Positive integer solutions to $\frac{(x+y)^{x+y}(y+z)^{y+z}(x+z)^{x+z}}{x^{2x}y^{2y}z^{2z}}=2016^m$

Do there exist positive integers $x,y,z$ and positive rational number $m$ such that: $$\frac{(x+y)^{x+y}(y+z)^{y+z}(x+z)^{x+z}}{x^{2x}y^{2y}z^{2z}}=2016^m$$
9
votes
2answers
102 views

Find a example such $\frac{(x+y)^{x+y}(y+z)^{y+z}(x+z)^{x+z}}{x^{2x}y^{2y}z^{2z}}=2016$

Assume $x,y,z$ be postive integers,and Find one example $(x,y,z)$ such $$\dfrac{(x+y)^{x+y}(y+z)^{y+z}(x+z)^{x+z}}{x^{2x}y^{2y}z^{2z}}=2016$$
2
votes
1answer
38 views

For any coprime integers $(x, y)$, $m\geq 2$, $\delta\geq 1$, is it true that $\mid x^m - y^{m+\delta} \mid \geq \delta$? [closed]

Is it true that if $x,y,m,\delta$ are integers, $\gcd(x,y)=1$, $m\ge2$, $\delta\ge1$, then $$|x^m-y^{m+\delta}|\ge\delta?$$ Any proofs or references will be most welcome.
2
votes
2answers
120 views

Describe the nonzero integer solutions to the equation $a^3 + b^3 + c^3 + d^3 + e^3 + f^3 + g^3 =0$

Can someone describe all the integer solutions to the above equation such that $abcdefg\neq 0$ ?
6
votes
2answers
91 views

Solve for Rationals $p,q,r$ Satisfying $\frac{2p}{1+2p-p^2}+\frac{2q}{1+2q-q^2}+\frac{2r}{1+2r-r^2}=1$.

Find all rational solutions $(p,q,r)$ to the Diophantine equation $$\frac{2p}{1+2p-p^2}+\frac{2q}{1+2q-q^2}+\frac{2r}{1+2r-r^2}=1\,.$$ At least, determine an infinite family of ...
2
votes
1answer
97 views

Integer solutions to $\frac{d^3}{r}+r=a^2$

What are the positive integer solutions $(a,d,r)$ to $\frac{d^3}{r}+r=a^2$? This is a revised version of my deleted question. Alternate forms are $d^3 = r(a^2-r)$ and from the quadratic formula ...
0
votes
2answers
22 views

How to solve a diophantine equation pair?

I know how to solve basic linear diophantine equations, but how would one solve this?: $$ \begin{equation} \begin{array}{rrrr} x &+& y &+& z &=& 31 \\ x &+& 2y ...
1
vote
1answer
42 views

Positive Integer solutions to $y = \frac{x z}{-x - z + x z}$

I'm trying to find positive integer solutions to the following diophantine equation: $$y = \frac{x z}{-x - z + x z}$$ The first thing I did was split the fraction as follows: $$ \frac{x z}{-x - z + ...