Questions on finding integer/rational solutions of equations.

learn more… | top users | synonyms

0
votes
0answers
49 views

How do I solve this diophantine equation?

I would like to know how to solve this equation: $(3a + 2b + c + 4)(36a^3 + 33a^2(2b + c + 4) + a(40b^2 + b(40c + 163) + 10c^2 + 80c + 160) + 8b^3 + 2b^2(6c + 25) + b(6c^2 + 49c + 100) + c^3 + 12c^2 ...
6
votes
0answers
106 views

Twin Prime Powers

What are all the possible triplets of numbers $a$, $b$, $c$ such that $a+2=b$, $a+4=c$, and all $3$ are prime powers (where one must be a power of $3$)? I'm aware of the cases for when they are ...
0
votes
1answer
58 views

What integers can be represented by the quadratic form $4x^2 - 3y^2 - z^2$?

Actually, I need to find if $4x^2 - 3y^2 - z^2 = 12$ is solvable. But I somehow feel that applying theory of integer representation by quadratic forms in three variables would yield quicker results... ...
2
votes
0answers
57 views

Find all integers n which satisfies $1^n+9^n+10^n=5^n+6^n+11^n$

Find all $n\in\mathbb Z$ which satisfies $1^n+9^n+10^n=5^n+6^n+11^n$ for $n=2\ or\ n=4$ it is equal but are there other numbers?
4
votes
0answers
61 views

Why has $3^x+4^y=5^z$ has only one solution (2,2,2) in positive integers? [duplicate]

First, do we have to exclude the cases, where $(x,y,z)$ are not all even or odd and then show the only possibility ? or is there a geometric solution maybe ?
0
votes
1answer
28 views

find solution for the variable y

I have some problem with understaing how wolfram calculate the solution for the variable $y$ in equation $2x^2+y^2+xy+2x=-1$
2
votes
3answers
134 views

Positive integral solutions of $3^x+4^y=5^z$

Are there more integral solutions for $3^x+4^y=5^z$, than $x=y=z=2$ ? If not, how do I show that? I could show that for $3^x+4^x=5^x$, but I'm stuck at the general case? Any ideas, maybe graphs, ...
8
votes
3answers
260 views

Solving the diophantine equation $y^{2}=x^{3}-2$

It is known that the diophantine equation $y^{2}=x^{3}-2$ has only one positive integer solution $(x,y)=(3,5)$. The proof of it can be seen from the book "About Indeterminate Equation (in Chinese, by ...
2
votes
0answers
38 views

Using Graphs Changes the Solutions for Diophantine Equation? Imperfection of Graph?

Solve the Diophantine equation $$x^2+4y^2=z^2$$ The problem here is that I derived solutions using two different methods, and the both solutions do satisfy the given equation yet they are ...
6
votes
3answers
118 views

Find all real solutions for $16^{x^2 + y} + 16^{x + y^2} = 1$

Find all $x, y \in \mathbb{R}$ such that: $$16^{x^2 + y} + 16^{x + y^2} = 1$$ The first obvious approach was to take the log base $16$ of both sides: $$\log_{16}(16^{x^2 + y} + 16^{x + y^2}) = 0$$ ...
1
vote
2answers
61 views

Finding integer solutions to this equation

$p^{\; \left\lfloor \sqrt{p} \right\rfloor}\; -\; q^{\; \left\lfloor \sqrt{q} \right\rfloor}\; =\; 999$ How do you find positive integer solutions to this equation?
0
votes
0answers
66 views

System of congruences that do not satisfy CRT assumptions (via algorithm)

Let $x_i,a_i\!\in\!\mathbb{Z}$. The following procedure solves a system of congruences $$x \equiv x_i\pmod{a_i}\;\;\text{ for }i\!=\!1,\ldots,n$$ when $a_i$ are pairwise coprime. Assume that ...
3
votes
2answers
82 views

On the equation $(1-x)^2/x + (1-y)^2/y + (1-z)^2/z + 4 = 0$

The problem is to solve the equation, $$\frac{(1-x)^2}{x} + \frac{(1-y)^2}{y} + \frac{(1-z)^2}{z} + 4 = 0\tag{1}$$ in the rationals. Treating this as an equation in $z$, easy solutions would involve ...
8
votes
3answers
283 views

Integral solution of the equation $x^2+y^2+z^2 = 2xyz$

Calculation of all Integral solution of the equation $x^2+y^2+z^2 = 2xyz$ $\bf{My\; Try}::$ Let we will calculate for $x,y,z>0$. Then Using $\bf{A.M\geq G.M}$ $x^2+y^2\geq 2xy$ sililarly ...
0
votes
1answer
170 views

Do I have this right? Are these conclusions valid in this isomorphic view of $\Bbb{R}$?

Let $F = (\Bbb{R}, \oplus_d, \cdot)$ be the field with usual $\cdot$, and $\oplus_d$ is defined as $a \oplus b = (\sqrt[d]{a} + \sqrt[d]{b})^d$. This field is isomorphic to usual $\Bbb{R}$ structure ...
0
votes
0answers
61 views

What can we say about solutions in fields isomorphic to $\Bbb{R}$?

Let $\phi: \Bbb{R} \to (\Bbb{R}, \oplus, \cdot)$ be a field isomorphism such that $\phi(\Bbb{Z}) \subset \Bbb{Z}. \ $ By FLT, $x^n + y^n - z^n = 0$ has no positive integer solutions for $n \gt 2$. ...
2
votes
0answers
55 views

fourth powers as sums of squares

Is it possible to have a fourth power that is the sum of two squares in four different ways, e.g., $w^4 = a^2 + b^2 = c^2 + d^2 = e^2 + f^2 = g^2 + h^2$ with the added restriction that $e = a+c$ and ...
3
votes
0answers
64 views

How to solve $x^4+y^4=n$?

How to solve Diophantus equation $$x^4+y^4=n $$ where $x,y$ and $n$ are positive integers. We know that Theorem: A natural numbern $n$ can be represented as a sum of two squares if and only if ...
0
votes
1answer
54 views

solution of $y^2 - x = 15$ and $x^2 -xy = 2009$

Find all the integer solutions to the equations: \begin{eqnarray} y^2 - x &=& 15 \\ x^2 -xy &=& 2009 \end{eqnarray} Not sure how to solve this :/, tried the usual algebra way ...
1
vote
1answer
54 views

General solution of equation with coefficients the symmetric polynomials

If $a,b,c$ are fixed integers, how do you find the general solution of $$X(abc)+Y(ab+bc+ca)+Z(a+b+c)=0$$ in integers $X,Y,Z$?
3
votes
0answers
145 views

Second longest prime diagonal in the Ulam spiral?

Given the Ulam spiral with center $C = 41$ and the numbers in a clockwise direction, we have, $$\begin{array}{cccccc} \color{red}{61}&62&63&64&\to\\ ...
1
vote
2answers
94 views

rational solutions of Pell's equation

1) $D$ is a positive integer, find all rational solutions of Pell's equation $$x^2-Dy^2=1$$ 2) What about $D\in\Bbb Q$ ?
3
votes
1answer
153 views

Find all positive integers m, n, p such that $(m+n)(mn+1)=2^p$

Find all positive integers m, n, p such that $$(m+n)(mn+1)=2^p$$ Please give me some hints Thanks
1
vote
0answers
58 views

How to solve the diophantine equation:$ xa^3+yb^3=c^3$

Let $a,b,c,x,y \in \mathbb{Z}> 1$. Any hint on how to solve of the diophantine equation $ xa^3+yb^3=c^3$?
2
votes
0answers
72 views

Find all Integers ($ n$) such that $n\neq 6xy\pm x\pm y$

I am interested in proving that there exist an infinite number of positive integers ($n$) which are not of the form $$ n=6xy\pm x\pm y $$ for $x,y\in\Bbb Z^+$. [Note: The $\pm$ signs above are ...
37
votes
5answers
696 views

Generalizing the sum of consecutive cubes $\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$ to other odd powers

We have, $$\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$$ $$2\sum_{k=1}^n k^5 = -\Big(\sum_{k=1}^n k\Big)^2+3\Big(\sum_{k=1}^n k^2\Big)^2$$ $$2\sum_{k=1}^n k^7 = \Big(\sum_{k=1}^n ...
0
votes
1answer
122 views

How to prove the Diophantine equation : $x^2 - 71y^2 = -1$ has no solution with x,y in the set of integers

How to prove the Diophantine equation : $x^2 - 71y^2 = -1$ has no solution with $x,y$ in the set of integers? I am stuck on this question for a long time and i cant really understand how to prove ...
1
vote
1answer
27 views

About Diophantine Equation

This is a problem about Diophantine equation. The problem is the following. If $ax+by=c$ is solvable and $b\ne0$, then prove that it has a solution $x_0$, $y_0$ with $0 \le x_0 <|b|$ First I ...
3
votes
2answers
74 views

Solve diophantine equation $x^2 - 2y^2 = x - 2y$

Thanks to internet, I found and understand how to solve diophantine $x^2 - Dy^2 = 1$. Now I would like to solve the following diophantine equation : $$x^2 - 2y^2 = x - 2y$$ but I don't know how to do ...
1
vote
1answer
84 views

Positive solutions for “squarefree” diophantine equation

I would like to find solutions in positive integers for diophantine equations having no variable squared. (And having some other limitations, but I will not consider them now.) Take, for example, ...
1
vote
1answer
79 views

Find two positive rational solutions to a Diophantine problem

I need to answer this question using the Diophantine method. The question is: Find two numbers so that the square of either number, plus twice the other number, is also a square. Give two sets of ...
3
votes
2answers
106 views

How can this equation be solved?

I have no idea how to solve this equation. $$x^2y^2+324y^2+64x^2-36xy^2-16x^2y+144xy = 0 $$ Sorry $x,y \in \mathbb{+Z}$
7
votes
0answers
101 views

Geometric intuition behind the Hasse principle

Let $f(X,Y) \in \mathbb{Q}[X,Y]$ be a quadratic polynomial. The Hasse-Minkowski theorem says that $f(X,Y) = 0$ has a solution $(x,y) \in \mathbb{Q}^2$ iff it has a solution in $\mathbb{R}^2$ and ...
0
votes
1answer
66 views

Counting Solutions of a Quadratic Diophantine Equation

How can one construct a function $f(n)$ that counts the number of solutions of the equation $$x^2+y^2-n(x+y) = 0,\quad x,y\in\mathbb{Z},$$ where $n\in\mathbb{Z}^+$? For example, we have ...
1
vote
3answers
142 views

Solutions to $x+y+z=31$ and $x+2y+3z=41$

For the equations $$x+y+z=31$$ $$x+2y+3z=41$$ is there a elegant way or method to find all the positive solutions in integers? Thus far, I have been using trial and error (which is time consuming). ...
8
votes
1answer
294 views

How find this value of $x+y+z+u+v+w$

let $x,y,z,u,v,w$ be positive integer numbers,and such $$1949(xyzuvw+xyzu+xyzw+xyvw+xuvw+zuvw+xy+xu+xw+zu+zw+vw+1)=2004(yzvw+yzu+yzw+uvw+y+u+w)$$ Find this value of $$x+y+z+u+v+w=?$$ My try: maybe ...
4
votes
1answer
83 views

Exponential Diophantine Equation $3^x5^y-2^s7^t=1$.

How to solve $3^x5^y-2^s7^t=1$ completely? Does there exists any general techniques dealing with such exponential equations? For equations like $a^x-b^y=1$, Mihăilescu's theorem (Catalan conjecture) ...
1
vote
1answer
57 views

Find the general solution and positive integral solutions of 775x -711y =1

Problem : Find the general solution and positive integral solutions of 775x -711y =1 My approach : For the equation ax -by =c , the general solution can be given as : Let h,k be a solution of ...
1
vote
1answer
54 views

Two equations & three unknowns (in $\mathbb{Z}$)

I just want to know this system-equation has answer $(x,y,z)$ in Integers Set or not? $a_1x+b_1y+c_1z=d_1$ $a_2x+b_2y+c_2z=d_2$ (in Real Number Set, we just need to check this two plate (plane) ...
0
votes
0answers
27 views

Is there a way to solve system of homogenous polynomial equations of degree 2 mod composite n?

I have quite specific system of equations, that I need to solve, each of the equations in the system is of following form: $$ \sum_{j=1}^{n}b_{ij}u_{j}^{(k)} = c_i^{(k)} $$ with both $i,k$ live in ...
2
votes
1answer
212 views

Generating all solutions for a negative Pell equation

How to get all solutions for a negative Pell equation? For example, the equation $x^2 - 2 y^2 = -1$ has two solutions - $(7, 5)$ and $(41, 29)$, and the $(7, 5)$ is the fundamental one, right? How to ...
0
votes
1answer
55 views

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$.

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and either of the following is true: $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$. We see that if $(a,b,c)$ is a solution, then so is ...
1
vote
4answers
101 views

Are there any integers $a,b$ s.t. ${ a }^{ 2 }-{ b }^{ 2 }=8$?

For $a$ and $b$ are integers greater than $1$, ${ a }^{ 2 }-{ b }^{ 2 }=8$ holds?
6
votes
1answer
190 views

Are differences between powers of 2 equal to differences between powers of 3 infinitely often?

Consider the equation $2^a-2^b=3^c-3^d$ where $a>b>0$, $c>d>0$, and $a,b,c,d$ are all integers. A computer search for solutions with $a,c\le20$ only finds 8-2=9-3, 32-8=27-3, and ...
1
vote
1answer
340 views

Solving an equation with floor function

I need some help regarding this question. Solve the following equation in natural number $x$, where $m,k$ are fixed naturals: $m\left\lfloor \sqrt{\dfrac{x}{k}}\right\rfloor = x$. I think ...
0
votes
1answer
867 views

Linear Diophantine equation - Find all integer solutions

Using the linear Diphantine equation 121x + 561y = 13200 (a) Find all integer solutions to the equation. (b) Find all positive integer solutions to the ...
9
votes
2answers
1k views

Fermat's Last Theorem near misses?

I've recently seen a video of Numberphille channel on Youtube about Fermat's Last Theorem. It talks about how there is a given "solution" for the Fermat's Last Theorem for $n>2$ in the animated ...
0
votes
0answers
67 views

How to solve an equation of the form $ax^2 - by^2 + cx - dy + e =0$?

I am trying to find out how to solve $ax^2 - by^2 + cx - dy + e = 0$ to get integer solutions, failing this the rational solutions. Thanks!
0
votes
1answer
94 views

Diophantine Equation: solving $x^2-y^2=45$ in integers

How should I solve $x^2-y^2=45$ in integers? I know $$(x+y)(x-y)=3^2\cdot 5,$$ which means $3\mid (x+y)$ or $3\mid (x-y)$, and analogously for $5$.
1
vote
1answer
131 views

How to prove Greatest Common Divisor using Bézout's Lemma

The problem is to prove the following If $\gcd(a,b) = c$, then $\gcd(a^m, b^m) = c^m$ I know that this can be solved easily by proving that $c\mid a \implies c^m \mid a^m$ and $c\mid b \implies ...