Questions on finding integer/rational solutions of equations.

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2
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0answers
135 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
94 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
102 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
50 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
163 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
137 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
31 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
82 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
223 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
78 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
55 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
40 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
116 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
77 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
164 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
72 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
88 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
44 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
84 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
148 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
124 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
95 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
49 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
52 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
95 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
227 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
55 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
81 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
106 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
96 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
101 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 + ...
0
votes
0answers
17 views

Is DEHP a kind of Multivariate hard problem?

Please correct me if I am wrong. To my understanding , given a '$m$' multivariate set of equations in '$n$' variables in a integer field '$F$' is hard to solve, even in case of $MQ(multiquadratic)$ ...
1
vote
0answers
97 views

Strategies for solving rational Diophantine equations

Are there any strategies for solving Diophantine equations where the solutions can be any rational number, not just an integer, besides substituting $x=p/q$ and $y=r/s$, with $p,q,r,s$ integers with ...
9
votes
3answers
325 views

Three pythagorean triples

Are there any solutions for $a, b, c$ such that: $$a, b, c \in \Bbb N_1$$ $$\sqrt{a^2+(b+c)^2} \in \Bbb N_1$$ $$\sqrt{b^2+(a+c)^2} \in \Bbb N_1$$ $$\sqrt{c^2+(a+b)^2} \in \Bbb N_1$$
4
votes
4answers
81 views

What are all the concordant forms $n$ such that $a^2+b^2 = c^2,\,a^2+nb^2=d^2$ for $n<1000$?

Part I. The list of congruent numbers $n<10^4$ such that the system, $$a^2-nb^2 = c^2$$ $$a^2+nb^2 = d^2$$ has a solution in the positive integers is known (A003273) $$n = 5, 6, 7, 13, 14, 15, ...
1
vote
3answers
116 views

$abx^2+bcy^2+acz^2=(xyz)^2+2abc$ has no integral solutions if $a,b,c,x,y,z >1$?

let $a,b,c,x,y,z$ be all pairwise coprime integers . Show that: $$abx^2+bcy^2+acz^2=(xyz)^2+2abc$$ has no integral solutions if $a,b,c,x,y,z >1$. I tried to confirm the results in wolfram but I am ...
0
votes
2answers
84 views

The diophantine equation $m(n-2016)=n^{2016}$

How many natural numbers, $n$, are there such that $$\frac{n^{2016}}{n-2016}$$ is a natural number? HINT.-There are lots of solutions HINT.-$\frac{n}{n-2016}=m \iff \frac{2016}{n-2016}=m-1$ and if, ...
4
votes
1answer
122 views

Finding integer solutions to $y^2=x^3+7x+9$ using WolframAlpha

I am an unconditional admirer of WolframAlpha and for this reason I want to let the people of this error (or is it really the fault of mine?). If I'm not mistaken, I would be very happy to contribute, ...
2
votes
3answers
100 views

How can one show the inequality

Let $a,b,n$ be natural numbers (in $\mathbb{N}^*$) such that $a>b$ and $n^2+1=ab$ How can one show that $a-b\geq\sqrt{4n-3}$, and for what values of $n$ equality holds? I tried this: We suppose ...
4
votes
1answer
43 views

What is the Diophantine Prime-Representing Polynomial with the Least Variables?

Recently I was reading Jones et al.'s famous paper "Diophantine Representation of the Set of Prime Numbers." They present a Prime-Representing Polynomial in 26 variables, and outline the construction ...
7
votes
1answer
123 views

Find all primes $p$ such that $ p^3-4p+9 $ is a perfect square.

Find all primes $p$ such that $ p^3-4p+9 $ is a perfect square. I tried a few different values for $p$, namely $2,3,5,7,$ and $11$. The prime $p =2,7,11$ all worked but $p =13$ didn't so it ...
0
votes
2answers
89 views

two questions involving $x^3+y^3+z^3-3xyz$ factorization

(1) Given that $x^3+y^3+z^3=3xyz+1$, determine the minimum of $x^2+y^2+z^2$. I know that Lagrange multiplier can solve this but I believe there is a way out using the factorisation: ...