Questions on finding integer/rational solutions of equations.
4
votes
2answers
60 views
Foundation on Diophantine Analysis and Number Theory
I want to read particularly about diophantine Analysis and Elementary Number Theory from a novice level.
The books which I found on net:
A Guide to Elementary Number Theory by Underwood Dudley
...
7
votes
3answers
199 views
Proving there are no integers $a, b, c$ satisfying $12a + 18b + 27c = 227$
Given $12a + 18b + 27c = 227$, how can we prove that $a, b, c$ can never be integers? I don't have many ideas. Can someone give me some ideas?
3
votes
1answer
91 views
sum of three cubes and parametric solutions
The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$. In other words, there are finite many polynomial triples ...
0
votes
3answers
53 views
Solving the algebraic equation
I am trying to solve this: $$x-40={-400\over x}$$
The answer must be $x=20$
Please give step by step explanation.
36
votes
5answers
1k views
Solutions to $\binom{n}{5} = 2 \binom{m}{5}$
In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says:
On National Public Radio, the Weekend Edition program posed the
following probability problem: Given a certain number of ...
0
votes
5answers
166 views
Generate solutions of Quadratic Diophantine Equation
Recently I've asked a question for how to solve Quadratic Diophantine Equation and I got one interesting answer. Link to question: How to solve Quadratic Diophantine Equation
Here's the answer:
$$ ...
1
vote
2answers
58 views
How to solve $5^{2x}-3\cdot2^{2y}+5^x\cdot2^{y-1}-2^{y-1}-2\cdot5^x+1=0$ in $\mathbb{Z}$
how to solve in $\Bbb Z$:
$$5^{2x}-3\cdot2^{2y}+5^x\cdot2^{y-1}-2^{y-1}-2\cdot5^x+1=0$$
3
votes
3answers
168 views
How to solve Quadratic Diophantine Equation
Here's the problem.
Find the solutions of the following equation:
$$ k^2 - 1 = 5(m^2 - 1).$$
Here's my idea:
The original equation can be written as:
$$ k^2 = 5m^2 - 4 \Longleftrightarrow k^2 - ...
4
votes
1answer
75 views
Necessary and sufficient conditions that the difference of two quadratic equations has no solutions in $\mathbb{N}$
Suppose you have an equation of the form
$$
a(n^2 - m^2) + b(n-m) + c = 0
$$
With given integers $a$, $b$ and $c$.
Is there a necessary and sufficient condition that the equation has no solutions ...
2
votes
2answers
187 views
$x^4-4y^4=z^2$ has no solution in positive integers $x$, $y$, $z$.
How do I prove that the diophantine equation $x^4-4y^4=z^2$ has no solution in positive integers $x$, $y$, $z$.
1
vote
1answer
123 views
$x^4+y^4=2z^2$ has only solution, $x=y=z=1$ .
How do I verify that the only solution in relatively prime positive integers of the equation $x^4+y^4=2z^2$ is $x=y=z=1$?
0
votes
0answers
28 views
Non-negative integral solutions to a single equation
Assume we have a integral vector $c\in \mathbb{Z}^n$ and an integer constant $b\in \mathbb{Z}$. Is there a necessary and sufficient condition for whether or not there exists a non-negative integer ...
5
votes
2answers
115 views
Does there exist $a,b,c\in \mathbb Q$ such that $(a+b+c)^2 + 3(a+b+c)+5=2(ab+bc+ca)$
Does there exist $ a,b,c\in \mathbb Q$ such that $(a+b+c)^2 + 3(a+b+c)+5=2(ab+bc+ca)$
I think the answer is no
4
votes
1answer
206 views
$x^4 - y^4 = 2z^2$ has no solution
How do I prove that the equation $x^4 - y^4 = 2 z^2$ has no solutions using the fact that the equations $x^4 + y^4 = z^2$ and $x^4 - y^4 = z^2$ have no solutions.
I cant think of a method of reducing ...
2
votes
2answers
69 views
Twice a triangle is triangle
The question is to prove that there are infinitely many triangular numbers $T_n$ where $2 \times T_n$ is also a triangular number, and give the first few as an example.
My attempt:
$$2 \cdot {x(x+1) ...
3
votes
2answers
68 views
Factorials and Arithmetic Progression.
Are there sets of factorials $(a_1!,a_2!,a_3!,\dots,a_n!)$, such that they exist in Arithmetic progression.
$n$ is a natural number
I don't see any such examples(Except for $n=2$). And I don't see ...
0
votes
0answers
24 views
Sphere containment problem inside a rational convex polytope of general dimensions.
Given a positive number $r$ and a rational convex polytope (bounded polyhedra) described by its set of half-planes (system of linear inequalities: $A\cdot x \leq b$, where $A\in\mathbb{R}^{m\times ...
3
votes
2answers
73 views
Is it possible to solve for two unknowns from one equation?
Is it possible to solve for two unknowns using only one equation?
For example:
$x+3y=32$
Where $x$ and $y$ are integers.
Thanks :)
13
votes
2answers
230 views
How did Letac solve $x_1^k + x_2^k + \dots +x_9^k = 0$ for $k = 1, 3, 5, 7$ in 1942?
It's quite easy to find integer solutions to,
$$x_0^k + x_1^k + \dots +x_9^k = 0$$
for $k = 1, 3, 5, 7$. One I found is, if $x^2-10y^2 = 9$, then,
$$1 + 5^k + (3+2y)^k + (3-2y)^k + (-3+3y)^k + ...
6
votes
3answers
125 views
How many solutions are possible to this equation?
Given
$$A+2B+3C=N
$$
where $N$ is a given positive integer.
$A ,B,C\in\mathbb{N}$ vary from $0$ to $\infty$.
How many solutions will be there to this equation?
16
votes
7answers
1k views
Pythagorean triplets $x^2+y^2 = z^3$
I need to prove that the equation $x^2 + y^2 = z^3$ has infinitely many solutions for positive $x, y$ and $z$.
I got to as far as $4^3 = 8^2$ but that seems to be of no help.
Can some one help me ...
4
votes
2answers
85 views
The rational points on the curve: $y^2=ax^4+bx^2+c$.
I wonder how to find the rational points on the curve: $y^2=ax^4+bx^2+c$.
Is there infinite rational points on this curve?
For example:$y^2=x^4+3x^2+1.$If we set $y=x^2+k$,then $2kx^2+k^2=3x^2+1$, ...
1
vote
0answers
87 views
Week of the problem on Diophantine equation
S.E board!
This is a Diophantine equations problem, which is so interesting one can do by plugging the suitable values in unknown. When it comes for finding set of all solutions is may be tough. I ...
0
votes
0answers
30 views
How do I find the set of integers solving a system of equations that contain outliers?
I have a system of $s$ equations that should (but won't) all equal some real unknown scalar value, $x$:
$x = v_1*k_1 + a_1*k_1*m = v_2*k_2 + a_2*k_2*m = ... = v_s*k_s + a_s*k_s*m$
where,
$k_i$ are ...
15
votes
2answers
273 views
$(x-a)(x-b)(x-c)(x-d)=ex$
We can verify that $x=125,162,343$ are the roots of equation $(x-105)(x-210)(x-315)=2584x$.
My question is,Could you find five positive integers $a,b,c,d,e$, which $(x-a)(x-b)(x-c)(x-d)=ex$ has four ...
10
votes
2answers
301 views
Solve $y^2= x^3 − 33$ in integers
This is not homework, could someone provide a nice clear proof as I have been struggling with this for some time.
Solve the equation $y^2= x^3 − 33$; $x, y \in \mathbb{Z}$
1
vote
2answers
48 views
Diophantine equation on an example
I do have one task here, that could be solved my guessing the numbers. But the seminars leader said, also Diophantine equation would lead to solution. Has anyone an idea how it works? And could you ...
2
votes
5answers
99 views
Solutions to two Bézout equations solve a third one
Let $p$ and $q$ be relatively prime integers, and let $a, b, c, d$ be the minimal solutions of
$$\begin{align}
ap - bq & = 1 \\
cq - dp & = 1.
\end{align}
$$
Then I want to show that $$ac - ...
7
votes
2answers
94 views
Diophantine Quintuple?
I have come across the following set of numbers: $\{1, 3, 8, 120\}$
These are positive integers where the product of any two of the numbers equal to a number that is one less than a square number. ...
2
votes
2answers
83 views
Solve for system of diophantine equations
$\cases{x+1=a^2 \cr x^3-x^2+1=b^2}$
I just can found a trivial solution $x=0$. Is there any other ?
1
vote
2answers
124 views
Solve for diophantine equation $x^n + y^n + z^n =1$ [closed]
Solve for diophantine equation
$x^n + y^n + z^n =1$
$x^n+y^n+z^n=2$
Is this equation solve-able ?
12
votes
2answers
176 views
Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube
Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube.
I found $x=0$, any others ?
1
vote
1answer
182 views
Solving a quadratic diophantine equation in two variables
I have an equation in the following form:
$$6mn+m+n=x$$
$$m,n,x\in\Bbb Z; \qquad0 < m,n$$
If I were given a value for $x$, how would I go about finding solutions to this equality for $a$ and $b$ ...
7
votes
4answers
176 views
Solve $a^3-5a+7=3^b$ over the positive integer
Solve $a^3-5a+7=3^b$ over the positive integer
I don't know how to solve such equation, please help me. Thanks
2
votes
3answers
75 views
Error in finding Solution for positive Diophantine equation
I need to find minimum value of c above which there always exists a non-negative solution for the equation
$$4x + 7y = c$$
I tried using Diophantine equation but I am not able to find the mistake in ...
0
votes
2answers
46 views
Finding $m+n$ from $m+n+mn+1=91$
If $m,n$ are natural numbers such that $m+n+mn+1=91$ .Then how to find $m+n$
3
votes
1answer
39 views
How many solutions to the diophantine equation:
$a + b + c + d = 22$
where
$\{a,b,c,d\}$ are distinct integers,
and
for each $x \in \{a,b,c,d\}, 1 \le x \le 9$.
Is there an elegant solution?
0
votes
1answer
70 views
Find $x,y,z \in Z$ , $1<x<y<z$ and $xyz - 1 = t(x-1)(y-1)(z-1)$
Find $x,y,z \in \Bbb Z, 1<x<y<z$ and $xyz - 1 = t(x-1)(y-1)(z-1)$.
Help me!
11
votes
2answers
206 views
How to find a “better description” (e.g. recurrence relation) for this sequence?
My solution to a problem in Project Euler required to solve this subproblem: find values of $k\in\mathrm{N}$ such that $3k^2+4$ is a perfect square.
As I was writting a computer program, I just tried ...
0
votes
1answer
42 views
Integer solution for $n_1 k_1 + n_2 k_2 + n_3 k_3 = 1$
For given integers $k_1,k_2,k_3$ is there an integer solution for the following equation: $$n_1 k_1 + n_2 k_2 + n_3 k_3 = 1$$
2
votes
4answers
127 views
is there any number pattern in the sum of square of two nos. and cube of 2 nos.
I wish to know the numbers which can be written in the form of sum of squares of two numbers and cube of two numbers and is there any pattern in it?
0
votes
0answers
32 views
Solving system of equations with mixed variable types
I'm looking for solutions to the non-linear system of equations
$$
n_1x + (n_1 - 1)y = a_1 \\
n_2x + (n_2 - 1)y = a_2 \\
n_3x + (n_3 - 1)y = a_3 \\
n_4x + (n_4 - 1)y = a_4
$$
where $x$ and $y$ are ...
0
votes
2answers
103 views
Non Linear Diophantine Equation in Three Variables
Find all positive integer solution to $abc-2=a+b+c$.
3
votes
1answer
60 views
Find $ k \in \mathbb{N}$ such that $x^3+y^3+z^3=kx^2y^2z^2$ have positive integer root
Find $k \in \mathbb{N}$ such that $x^3+y^3+z^3=kx^2y^2z^2$ have positive integer roots
I know a similar problem $x^3 + y^3 + z^3 = nxyz$
but I still can't solve my problem
2
votes
2answers
91 views
Set of natural number solutions to $x^2+y^2=z^2$
I know that there are infinitely many solutions to the equation $x^2+y^2=z^2$
$x,y,z\in N $
but if we restrict the numbers to {1,2,3,4...n}, then how many triplets (x,y,z) exist?
Asymptotical ...
5
votes
1answer
162 views
$x^2+y^2=z^2(1+xy)$ prove $z=\min \{x;y;z\}$ (with $x,y,z \in \mathbb{Z^+}$)
$x,y,z \in \mathbb{Z^+}$ such that $x^2+y^2=z^2(1+xy)$. Prove $z=\min \{x;y;z\}$
$$x^2+y^2=z^2(1+xy) \iff xy = \frac{x^2+y^2} {z^2} - 1$$. Assum $z>y \implies xy < x^2/z^2$, we have $xy \in Z ...
0
votes
4answers
77 views
Diophantine equation of second degree
How to solve this diophantine equation of second degree?
Solution, references, anything. I will be very grateful.
$x^2+y^2+z^2=2t^2$
Thank you.
2
votes
1answer
123 views
Diophantine equations - Perfect square and Perfect cube related
Solve following Diophantine equations:
$1) \ a^3-a^2+8=b^2$
2) $a, \ b,\ c \in \mathbb{Z^+}$$$\frac{a^3}{(b+3)(c+3)} + \frac{b^3}{(c+3)(a+3)} + \frac{c^3}{(a+3)(b+3)} = 7$$
3) $a^3-8=b^2$
In ...
1
vote
1answer
78 views
Like Diophantine equation
The equation $x^n - ny^x-nxy$ = $0$ has solution set $(n, x, y) = (1, 1, \frac12), (2, 1, \frac14), (3, 1, \frac16), \ldots$
I would like to know/learn the following (Kindly discuss)
1) If we ...
2
votes
1answer
55 views
How many solutions to prime = $2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$
Let $a,b,c$ be integers, no sign restriction.
Let $p$ be a given prime.
How to find the number of solutions to $p = 2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$ ?
Note, from Heron's ...
