Questions on finding integer/rational solutions of equations.

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193 views

TI83+. Math-> Solver. Why does it give different solutions?

Not sure if anyone still uses this one.... Recall the keystrokes: Math 0:Solver Up arrow Enter equation Press ENTER You will now be at a prompt displaying the equation, and a line saying x=### ...
4
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2answers
156 views

Solve $x^2 + 8 = 3^n$, where $x\in\mathbb N$.

First of all, $n$ is even (find that out by trying to see what kind of remainders you can possibly get when dividing everything by 4). So $n=2m$, $m\in N$. $ x^2 + 8 = 3^{2m} \Rightarrow x^2 - 1 = ...
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1answer
132 views

Heronian isosceles triangles

This is a problem from Project Euler, problem 94. The problem asks about isosceles triangles with integer sides (differing by 1 unit, e.g, 5-5-6) and integer area, which are known to be Heronian ...
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0answers
136 views

Find all the solutions of $x^2+7=2^n$. [duplicate]

Checking for some small natural numbers $n$, I found out that $2^n-7$ is a perfect square for $n=3,4,5,7,15$. How can we find all of the numbers $n$ for which $2^n-7$ is a perfect square? What I ...
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1answer
69 views

A question on integer values of an expression in positive rational variables

i) How do we find all $x,y ∈ \mathbb Q^+$ such that $x+y+ \dfrac1{x} +\dfrac1{y}$ is an integer ? ii)Let $p$ be an odd prime and $q=\dfrac{p-1}2$ , then how do we find all $x,y ∈ \mathbb Q^+$ such ...
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3answers
377 views

Is it possible that $(x+2)(x+1)x=3(y+2)(y+1)y$ for positive integers $x$ and $y$?

Let $x$ and $y$ be positive integers. Is it possible that $(x+2)(x+1)x=3(y+2)(y+1)y$? I ran a program for $1\le{x,y}\le1\text{ }000\text{ }000$ and found no solution, so I believe there are none.
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1answer
122 views

What kind of methods there are to solve a Diophantine equation from IMO longlist?

Namely, in IMO longlist 1987 were given the equation $3z^2=2x^3+385x^2+256x-58195$ and asked to find its integer points. How can I find those? I tried to substitute $z=12k,x=6t$ to get ...
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1answer
62 views

Reducing radical congruence to polynomial congruence

I am trying to find a way to describe all integer values of $x$ for which the following holds true: $\sqrt[2]{(1/2) * x * (x - 1) + (1/4)} + (1/2)\in \mathbb{Z}$ Noting that this can be equivalently ...
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1answer
199 views

For what values of $c$ does $8x + 5y = c$ have exactly one strictly positive integer solution?

We know that any Diophantine equation of the form $ax + by = c$ has either no solutions, or infinite solutions of the form: $$x = x_0 + n\frac{b}{(a, b)}$$ $$y = y_0 - n\frac{a}{(a, b)}$$ Where $n$ ...
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1answer
115 views

What's the approach to find out if this equation has integer solutions?

The equation is $u^3(s^4 + (r-1)^4) - s^4(t - 1)^3 = 0$ Has no integer solutions for $u,s \neq 0$. How do mathematicians today approach this problem? Sorry for broadness, just looking for a ...
11
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3answers
645 views

Prove the lecturer is a liar…

I was given this puzzle: At the end of the seminar, the lecturer waited outside to greet the attendees. The first three seen leaving were all women. The lecturer noted " assuming the attendees are ...
2
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1answer
94 views

Three variable equation using only natural numbers

I am a high school student that has been working on amateur mathematics for about a year now. Lately, I have been encountering a particular type of equation that I have not yet solved. If I could ...
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1answer
118 views

Equation with seven unknowns

Let $a,b,c,d,e,f,g$ be positive integers greater than or equal to $2$. What values of these numbers satisfy the equations $$a+b+c+d+e+f+g =18 \tag 1$$ $$a(b+c+d+e+f+g+3) + b(c+d+e+f+g+3) + ...
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1answer
97 views

How many numbers N satisfy N consecutive positive integers add to 2013?

How would you find how many numbers N there exist such that N consecutive positive integers add to 2013? (Assume that N=1 is a valid case whose solution is just 2013 itself). To clarify, this, when ...
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1answer
71 views

Need a proof to show all the units are satisfied $\mathbf{Z}\sqrt{2}$ is the all the integer solution in Pell equation [duplicate]

We know the integer solutions of Pell's equation $$a^2-2b^2=\pm1$$ correspond to the units of $\textbf{Z}[\sqrt{2}]$. How can we prove this?
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2answers
91 views

The diophantine equation $a^n - 1 = (a-1)^m$.

Let $a,n,m$ be odd integers larger than one. The diophantine equation $a^n - 1 = (a-1)^m$ fascinates me. I know that Catalan's conjecture has been proven and that Pillai's conjecture has not been ...
5
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1answer
85 views

integer solutions of the equation $a^3-5b^2$=2?

Has the equation $a^3-5b^2$=2 any integer solutions ? With brute force, I checked that a must be greater than 10^9.
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0answers
70 views

Find all rational solutions to $x^3 - y^2 = 2$. [duplicate]

Find all rational solutions to $x^3 - y^2 = 2$. The only integers solutions are $(3,\pm5)$: http://mathforum.org/library/drmath/view/51569.html
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1answer
202 views

Prove or disprove the following proposition

Prove or disprove the following proposition: There are no positive integers $x$ and $y$ such that $$x^2 - 3xy + 2y^2 = 10$$
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97 views

Need a proofreading why all the units are satisfied $a^2-2b^2 =\pm1$ for $\mathbf{Z}[\sqrt{2}]$

All the units are satisfied Pell's equation $a^2-2b^2=\pm1$ for $\mathbf{Z}[\sqrt{2}]$, $a,b\in\mathbf{Z}$. Here is my proof: Let $a+b\sqrt{2}$ be a unit $\in\mathbf{Z}[\sqrt{2}]$. This implies ...
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2answers
58 views

What method would I use for this proof?

Show there are no integer, $x$ and $y$, that satisfy $x^{2} + 3y^{2} = 8$. I have no idea where to start unfortunately or what kind of method to start off with.
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1answer
93 views

Find all the natural solutions to this diophantine equation

Find all the natural solutions to this diophantine equation $968m =n^2-54257$
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1answer
163 views

negative pell's equation

If $d$ is divisible by a prime $p \equiv 3 \pmod{4}$. show that the equation $x^2-dy^2=-1$ has no solution. So far I have learn only positive Pell's equation but not negative Pell's equation. We know ...
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1answer
108 views

$\frac{x^5-y^5}{x-y}=p$,give what p ,the diophantine equation is solvable

for$$\frac{x^3-y^3}{x-y}=x^2+xy+y^2=p$$$p=6k+1$give p prime, On what conditions,the diophantine equation $$\frac{x^5-y^5}{x-y}=p$$ is solvable in integers.does it have a linear expression.for ...
2
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1answer
286 views

There are only finitely many integer solutions to $ax^n+by^n=c.$

I am trying to prove the following: Fix $n \geq 3 \in \mathbb{Z}$, then for any non zero integer $a,b,c$, there are only finitely many integer solutions to $ax^n+by^n=c.$. I think the solution uses ...
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3answers
4k views

Solving linear Diophantine equations in 3 variables

I have to solve a linear Diophantine equation in more than 2 variables> I sort of have an idea of how to solve it, but I'm not clear how. One of the problems is this: $$12x + 21y +9z + 15w = 9$$ How ...
4
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2answers
332 views

General quadratic diophantine equation.

Here is my problem: I am given a general quadratic diophantine equation: $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$ where $x$ and $y$ are variables with integers $a,b,c,d,e,f$. I have to show that if the ...
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2answers
430 views

Solve equation $ 1+2^x=3^y$

Find integers $x$ and $y$ such that$$ 1+2^x=3^y.$$ It is obvious that $x = y = 1$ and $x = 3, y = 2$ are solutions. I think others are not. How to show that?
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0answers
159 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
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3answers
134 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... ...
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2answers
106 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?
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0answers
67 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 ?
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1answer
34 views

find solution for the variable y

I have some problem with understaing how wolfram calculate the solution for the variable $y$ in equation $2x^2+y^2+xy+2x=-1$
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3answers
241 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, ...
11
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2answers
499 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 read from the book "About Indeterminate Equation" (in Chinese, by ...
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0answers
115 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
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2answers
148 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: $$\log_{16}(16^{x^2 + y} + 16^{x + y^2}) = 0$$ ...
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2answers
74 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?
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0answers
77 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
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2answers
102 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 ...
10
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3answers
1k views

Integer solutions of the equation $x^2+y^2+z^2 = 2xyz$

Calculate all integer solutions $(x,y,z)\in\mathbb{Z}^3$ of the equation $x^2+y^2+z^2 = 2xyz$. My Attempt: We will calculate for $x,y,z>0$. Then, using the AM-GM Inequality, we have $$ ...
0
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1answer
177 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 ...
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0answers
129 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 ...
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0answers
112 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 ...
2
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2answers
115 views

The Diophantine equation $x^2 - 97 - 40 = 0$

I am trying to determine whether the equation below has a solution or not $$x^2-97y-40 =0.$$ If a solution exists, $x^2-40$ must be congruent to 0 modulo $97$. If I could show the congruence above ...
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1answer
65 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
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1answer
107 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 $X,Y,Z$?
5
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1answer
413 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\\ ...
4
votes
2answers
272 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
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1answer
308 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