Questions on finding integer/rational solutions of equations.
0
votes
1answer
100 views
Solutions of $a^2 = b^d -3^c$
The solutions of $a^2 = b^d -3^c$ are in the form $(a, b, c, d) = ((46)27^t, (13)9^t, 6t+4, 3)$. This is done by using calculator. As per my calculator, I have checked some terms, which are satisfied ...
23
votes
5answers
551 views
When is $1^5 + 2^5 + \ldots + n^5$ a square?
When is $1^5 + 2^5 + \ldots + n^5$ a square? I found that this happens sometimes: $n=13$ gives $1001^2$, $n=133$ gives $9712992^2$ and $n=1321$ gives $942162299^2$.
I feel that the ...
0
votes
2answers
79 views
Sylvester Theorem
Bonjour,
The equation $\binom{n}{k}=m^l$ has no entire solution for l$\ge$2 and 4$\le$k$\le$n-4.
Suppose that n$\ge$2k (since $\binom{n}{k}=\binom{n}{n-k}$).
According to the Sylvester theorem, the ...
8
votes
2answers
152 views
Solve $x^{y^2}=y^x$ which $x,y\in\mathbb{N}$.
Solve $x^{y^2}=y^x$ which $x,y\in\mathbb{N}$.
I can observe that $(x,y)$ can be $(1,1)$, then I don't know how to carry on. Please help. Thank you.
p.s.
I wonder if there's solution without ...
0
votes
1answer
88 views
Show the number of solutions
Show that the number of solutions of $x^2+y^2=m$, where $m=2^{\alpha}r$ and $r$ is odd, is given by $U(m)=4\sum_{u|r}(-1)^{\frac{u-1}{2}}=4\gamma(m)$, where $\gamma(m)$ denotes the number of positive ...
7
votes
2answers
363 views
When is $\left\lfloor \frac {7^n}{2^n} \right\rfloor \bmod {2^n} \ne 0\;$?
Is
$$\left\lfloor \frac {7^n}{2^n} \right\rfloor \bmod{2^n} \ne 0\;$$
always true when $n \ge 3$.
Baker's theorem on transcendental numbers that provide bounds for diophantine equations may be ...
3
votes
1answer
201 views
Diophantine equation (use class ideal group to solve)
Use ideal class group to find all integer solutions to the equation $$x^3=y^2+200$$
My approach: Observe that $\mathbb{Z}[\sqrt-2]$ is the field of integers in the ring $\mathbb{Q}(\sqrt -2).$ ...
1
vote
2answers
103 views
number of integral solutions for $x^2+y^2=5^k$
Prove that the equation $x^2+y^2=5^k$ has $4k+4$ integral solution.
Any ideas would be appreciated.
Thanks
0
votes
0answers
65 views
Pell type equation cum elliptic curve equation
I have seen this equation $y^3 - 3x^2 = p^m$ to determine the solutions. I know this is elliptic curve. I had some knowledge of elliptic curve. But, I was totally upset to determine the solutions of ...
1
vote
1answer
129 views
Pell equation of the special form
I have tried several times to solve the following equation and finally, I was failed to complete. Help me to find the solutions of Pell equation $y^2-2x^2 = p^m$, where $p$ is prime and $8|(p-1)$ or ...
0
votes
0answers
92 views
The equation $( m^2 + n^2 + q^2 )^2 = 36 ( u^2 + s^2 + t^2 )$
What is known about solutions in integers of the following equation ?
$$( m^2 + n^2 + q^2 )^2 = 36 ( u^2 + s^2 + t^2 )$$
I am asking this because I just recently have got these:
$$(a-b)^2 + ...
0
votes
1answer
30 views
Simultaneously smooth
I came across a problem recently which can be reduced to finding numbers $m$ such that $m$ and either $5m+1$ or $5m-1$ are $\{2,3,5\}$-smooth, i.e., of the form $2^a3^b5^c$ for nonnegative integers ...
1
vote
3answers
286 views
Are there any integer solutions to $a^3=b^2$?
I was wondering if there were any two integers $a$ and $b$ where $a^3=b^2$.
1
vote
1answer
140 views
Two Diophantine equations
What is known about solutions in integers of the equations; $x^4 - y^2 = z^6$
I got $x=4st(s^4 - t^4)$ , $z=4st(s^2 - t^2)$ , $y=(4st(s^2 - t^2))^2 (s^4 + t^4 - 6 (st)^2) $
and, the equation $x^2 - ...
16
votes
1answer
324 views
Integer solutions of $x! = y! + z!$
There was an interesting problem asked about triples $(x,y,z)$ which are solutions of
$$x! = y! + z!.$$
Here $(2,1,1)$ is a solution because $2! = 1! + 1!$, as are $(2,1,0)$ and $(2,0,1)$.
Now I ...
3
votes
1answer
46 views
Unique solution of a diophantine equation
Suppose $m_{1}^{h}+\cdots m_{k}^{h}=n_{1}^{h}+\cdots n_{k}^{h}$ for $h=1,\dots ,k$, where $0<m_{v}\leq q, 0<n_{v}\leq q, q$ positive integer. How do one show that the natural number $n_{v}$ must ...
1
vote
1answer
80 views
GCD to Linear Diophantine Equation without Euclid Algorithm
Is there a technique other than performing Euclid's algorithm in reverse that can elegantly show that if GCD$(a,b) = d$ then there exist integers $x$ and $y$ such that $ax + by = d$?
0
votes
2answers
144 views
Quadratic equations
Does anyone know how to find integer solutions of the quadratic equation
$$y^2+y+z=f$$
where $z$ is a fixed odd prime or $1$ and $f$ is a fixed odd prime greater than $3$?
This problem arose from ...
2
votes
1answer
108 views
How to find all solutions for Pell's equation $x^2 - Dy^2 = -1$ after the first $x_0$ and $y_0$?
How to find all solutions for Pell's equation $x^2 - Dy^2 = -1$ after the first $x_0$ and $y_0$?
for example if we have $x^2 - 2 y^2 = -1$ then the smallest integer solution for $(x,y) = (1,1)$
How ...
0
votes
0answers
93 views
integer solutions to $a^m+nx^2 = y^n$ with various conditions
I consider the following equation with conditions of obtaining solutions
$$a^m+nx^2 = y^n$$
This equation has solution when $a$ is an even prime and $x, y, m$ are positive integers with $(nx, y) = ...
14
votes
1answer
449 views
How to compute rational or integer points on elliptic curves
This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever ...
2
votes
3answers
129 views
What is the number of combinations of the solutions to $a+b+c=7$ in $\mathbb{N}$?
My professor gave me this problem:
Find the number of combinations of the integer solutions to the equation $a+b+c=7$ using combinatorics.
Thank you.
UPDATE
Positive solutions
5
votes
1answer
192 views
Finding the all integer solutions
How to Find the all integer solutions for:
$$x+y+z=3$$
$$x^3+y^3+z^3=3$$
0
votes
1answer
26 views
All solutions for a homogene equations
Suppose we have an equation of the form $ax+by=0$ with $a,b,c \in \mathbb{Z}$. For simplicity, $a \neq 0, b \neq 0$. Then, a single solution to this equation is $(x_0, y_0)=(-a, b)$. My book states ...
3
votes
0answers
74 views
how many natural numbers on a sphere
how many natural solutions are there to the following equation:
$$ \sum_{i=0}^k x_i^2 = n$$
where $n,k \in\ \Bbb{N}$
i well like to get a answer for every n and k, but could do with just $k=2,3$.
3
votes
2answers
278 views
Modification of 5th question from BMO'81
First of all I will introduce original problem (Question 5 from British Mathematical Olympiad).
You can find complete list of BMO'81 there BMO'81.
Find, with proof, the smallest possible value ...
0
votes
0answers
146 views
How to find integer solutions for $x^3 - (ay^2 +by+c)= 0 $?
How can I find integer solutions for $x^3 - (ay^2 +by+c)= 0 $?
An example for that is : "How many trigonometric numbers are also cube ?"
This produces the following formula $x^3 - (0.5y^2 + 0.5y)= 0 ...
2
votes
2answers
232 views
How to find integer solutions for $x^3 - y^2 = 0 $?
How can I find integer solutions for $x^3 - y^2 = 0 $ ?
In case that there are infinite number of solutions .How can we prove that ? and how to generate first few solutions ?
0
votes
1answer
132 views
How to find integer solutions for an ellipse equation?
How can I find the positive integer solutions to $x$ and $y$, given the integers $a$, $b$ and $c$ in the following ellipse equation in the form:
$\frac{x^2}{a^2} + \frac{y^2}{b^2}=c$
For example, ...
4
votes
1answer
129 views
How to find odd solutions only for Pell's equation $x^2 - Dy^2 = 1$?
How can I find only the odd solutions for Pell's equation: $$x^2 - Dy^2 = 1$$ Specifically where $x$ is odd (but $y$ may be even or odd).
Is there a way to generate the odd solutions to $x$, and can ...
0
votes
0answers
125 views
What is the sixth Martin quadruple?
Define a Martin quadruple {a,b,c,d} as a solution in non-zero integers to the system,
$a+b+c+d = x^2$
$a^2+b^2+c^2+d^2 = y^2$
$a^3+b^3+c^3+d^3 = z^3$
It can be shown that there are an infinite ...
1
vote
0answers
60 views
Relation between b and c such that $ b^2 + c^2 + b^2 c^2$ is a perfect square [duplicate]
Possible Duplicate:
On the equation $(a^2+1)(b^2+1)=c^2+1$
I came across a problem: What is relation between $b$ and $c$ such that $b^2 + c^2 + b^2 c^2$ is a perfect square?
After trying ...
7
votes
4answers
540 views
Integer solutions to $ x^2-y^2=33$
I'm currently trying to solve a programming question that requires me to calculate all the integer solutions of the following equation:
$x^2-y^2 = 33$
I've been looking for a solution on the ...
1
vote
3answers
78 views
Sine and Cosine equation ( diophantine )
$\cos(\frac{1}{ab} \pi) = \sin(\frac{a}{b} \pi)$
Let $a$ and $b$ be positive integers. What is the full set of solutions?
An example is $a = 2$ and $b = 5$.
I assume the best method is to take ...
1
vote
1answer
85 views
Equations $n^{am+bn}=m^{cn+dm}$
I will be very grateful, if someone show me, how to solve such equations.
Example 1.
$$
n^{m+2n}=m^{4n}
$$
n,m - positive integers. Thanks a lot.
0
votes
0answers
50 views
Problem solving n variable Linear Diophantine
I am searching a math library where n variable Linear Diophantine equation can be solved. If library with source code in C++,C# is available , please let us know.
1
vote
1answer
102 views
Proof- set uniqueness
Moderator Note: This question is from a contest which ended 22 October 2012.
Suppose that for $1\leq y\leq x$, and $x\geq 3$,
$$\Gamma_{x,y}=\left\{\left\lfloor\frac{2^x-1}{2^{y-1}}n - 2^{x-y} ...
1
vote
1answer
67 views
Solving $a + b x = c y$ in the integer domain for general $a$
I have the following equation:
$\frac{a + b x}{c} \in \mathbb{N}$ where $a,b,c,x \in \mathbb{N}$.
and I want to find all x that satisfy these requirements. This should be the same as:
$a + b x = c ...
2
votes
1answer
66 views
Elements of order two in $GL_2(\mathbb{Z})$
I would like to determine the elements of order two in $GL_2(\mathbb{Z})$. I reduced the problem to solve the diophantine equation $a^2+bc=1$ with $a,b,c \in \mathbb{Z}$, but I have no idea to solve ...
1
vote
2answers
76 views
Integer solutions of $p^2 + xp - 6y = \pm1$
Given a prime $p$, how can we find positive integer solutions $(x,y)$ of the equation:
$$p^2 + xp - 6y = \pm1$$
1
vote
0answers
66 views
Finding integer coordinates on a sphere's surface [duplicate]
Possible Duplicate:
Integer coordinate set of points that is a member of sphere surface
Assume $C$ is a sphere with radius $r$ and center in the origin (0,0,0). How can we find the set of ...
3
votes
1answer
162 views
All pairs (x,y) that satisfy the equation $xy+(x^3+y^3)/3=2007$
How we can find the all pairs $(x,y)$ from the integers numbers ,that satisfy the equation :
$$xy+\frac{x^3+y^3}{3} =2007$$
3
votes
1answer
115 views
Diophantine equation $2y^4-2y^2 +1=z^2$
How to solve the diophantine equation $2y^4-2y^2 +1=z^2$, where $(y,z) \in \mathbb{N}^2$ ?
Thanks,
W
6
votes
1answer
266 views
The nonexistence of the Collatz-“1-cycle” by an elementary proof - am I missing something?
The so-called "1-cycle" in the Collatz-problem was already disproved by Ray Steiner 1977. However, he used transcendental number theory to achieve that, and Lagarias commented, it is surprising that ...
5
votes
5answers
249 views
Favorite problems that lead to interesting diophantine equations?
I am looking for interesting problems (in number theory, or otherwise) that lead to interesting diophantine equations. The solution to the problem may be known, or it may be open... I just care for ...
7
votes
3answers
275 views
Number of integer solutions of $2n^2=a^2+b^2$
For a given integer $n$, how many positive integer $(a,b)$ pairs exist which satisfy $2n^2=a^2+b^2$?
In particular, I'm looking for all $n$s where there are exactly 105 solutions. (One solution is ...
2
votes
0answers
31 views
Solving $key=(\sum_{K=0}^n\frac{1}{a^K})\mod m$ with High limits
I was solving this equation:-
$$key=(\sum_{K=0}^n\frac{1}{a^K})\mod m$$
Given
$$ 1,000,000,000 < a, n, m \; < 5,000,000,000 $$
$$ a, m \; are \;coprime $$
I solved it bruteforcely but it ...
4
votes
1answer
214 views
Riemann hypothesis and diophantine equation
I read that showing Riemann hypothesis is true was equivalent to showing a particular diophantine equation doesn't have any solution.
Is there an explicit example of such a diophantine equation?
5
votes
1answer
158 views
Diophantine equation $a^3+a=b^2+1$
I have this Diophantine:
$$ a^3+a=b^2+1 $$
I found $a=2$, $b=3$ works.
Also $a=13$ , $b=47$ works.
How can I find all the integer solutions?
7
votes
0answers
371 views
How many integer solutions to a diophantine equation
Starting with the equation:
$\frac{1}{a}+\frac{1}{b}=\frac{p}{10^n}$,
I reached the equation:
$10^{n-log(p)} = \frac{ab}{a+b}$.
Now given the positive integer $n$, for what integer values of $p$ ...
