Questions on finding integer/rational solutions of equations.
2
votes
4answers
132 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
34 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
92 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
163 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
124 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 ...
3
votes
3answers
120 views
How many solutions to prime = $a^3+b^3+c^3 - 3abc$
Let $a,b,c$ be integers.
Let $p$ be a given prime.
How to find the number of solutions to $p = a^3+b^3+c^3 - 3abc$ ?
Another question is ; let $w$ be a positive integer. Let $f(w)$ be the number of ...
2
votes
1answer
34 views
When is the complement of a diophantine set in the naturals also diophantine?
A diophantine polynomial is a (multivariable) polynomial with integer coefficients. If we write this polynomial as $p(x, y_1, \dots, y_n)$, then it defines the diophantine set $D_p = \{ x \in ...
1
vote
2answers
71 views
How many solutions to prime = $(d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$?
Let $a,b,c,d$ be integers $>-1$.
Let $p$ be a given prime.
How to find the number of solutions to $p = (d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$ ?
I assumed that this polynomial above does not ...
4
votes
3answers
148 views
Find $a, b, c, d \in \mathbb{Z}$ such that $2^a=3^b5^c+7^d$
Solve $2^a=3^b5^c+7^d$ over the positive integer.
I know $a$ is even because: $(-1)^a \equiv2^a = 3^b5^c+7^d \equiv1 \ (mod\ 3)$
0
votes
1answer
77 views
Number of integer solutions of an equation
How many integer solutions exist for the following equation with the given constraint:
Equation: $X_1 + X_2 + X_3 + X_4 = N$
Constraint: $1 \le X_1 \lt X_2 \lt X_3 \lt X_4 \le N$
I went as far as ...
5
votes
3answers
151 views
Quadratic Diophantine Equations
I note that the Diophantine equation, $x^2 + y^2 = z^2$, with $x, y, z \in \mathbb{N}$, has infinitely many solutions. Indeed, $(x, y, z) = (3,4,5)$ provides a solution, and for any $k \in \mathbb{N}$ ...
0
votes
0answers
42 views
Diophantine, elliptic analysis
We have seen the Erdös-Straus conjecture relating to the theory of elliptic curves.
How to study and analyze asymptotic estimates on Diophantine equation?.
How to use the theorem of ...
3
votes
2answers
101 views
Diophantine equation to characterize natural numbers
Let's consider
$$\Bbb N=\{0,1,2,3,\ldots\},$$
and, for each $k\in\{1,2,3,\ldots,\}$, let
$$o_k=2k-1$$
be the sequence of odd natural numbers.
Given that for each $m\in\Bbb N$, if $a$ is odd, the ...
0
votes
1answer
24 views
Find the lowest natural root
I wanna know, for the equation below, how to:
Prove if there is always a natural root $x$ that makes $y$ natural
Get the lowest natural $x$ that makes $y$ natural
$$
x^2+8x-y^2=4n-16\quad\forall
...
1
vote
2answers
91 views
Integer solutions
How many positive integer solutions are there to $x_1 + x_2 + x_3 + x_4 < 100$?
I haven't seen any problems with "less than", so I'm a bit thrown off. I'm not sure if my answer is correct, but ...
2
votes
2answers
88 views
How can I find the integer solutions to $x^2+x-2y^2=0$?
I enter this equation in Wolfram Alpha : $x^2+x-2y^2=0$
and it gave me something like this :
and I am wondering how this solution is found and how to know if a given equation would guarantee to ...
151
votes
13answers
17k views
Unusual 5th grade problem, how to solve it
Find a positive integer solution $(x,y,z,a,b)$ for which
$$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$
Is your answer the only solution? If so, show why.
I was ...
2
votes
0answers
83 views
Diophantus again; not to say Pell.
Is there a way to solve the second degree Diophantine equation in two variables $ax^{2} -ny^{2} = b$ $(1)$ where a and b are known and n is a parameter; all solutions x= f(n) and y = f(n) ? For ...
0
votes
3answers
42 views
Finding K value to solve a problem using diophantine equation
I have to prove that any number that divided by 5 gives a remainder of 1 and divided by 7 a remainder of 2, also gives a remainder of 16 when divided by 35, using a diophantine equation.
So first I ...
0
votes
1answer
79 views
Nice sequences related to the Diophantine equation $d^{m+1} =a^{m}+ b^{m}+ c^{m}$
$$1, 3, 12, 32,...$$
Above is the sequence of the number of solutions, if there are, to the Diophantine equation :
$d^{m+1} =a^{m}+ b^{m}+ c^{m}$ for $m =2$, in positive integers where $a, b$ and ...
24
votes
1answer
407 views
Finding integer solutions for trigonometric equation $8\sin^2\left(\frac{(k+1)\pi}{n}\right)=n\sin\left(\frac{2\pi}{n}\right)$
I thought up the problem of finding a regular $n$-sided polygon that has a diagonal with lenght $d_k$ such that the area of the polygon equals ${d_k}^2$. By doing some easy trigonometry within the ...
0
votes
0answers
53 views
Diophantine Equation: $f(x)f(y) = f(z^2)$ where $f$ is quadratic
In the study of the Diophantine Equation $f(x)f(y) = f(z^2)$ where $f$ is quadratic, the computational proofs I have seen (for specific $f$) rely on Pell's Equation.
For example, if $f(t) = t^2+t+1$, ...
0
votes
2answers
146 views
Diophantine equation $a^m + b^m = c^n$ ($m, n$ coprime)
Arising from this recent question, and in particular the answer by Gerry Myerson, it occurs to me that, if $m$ and $n$ are coprime integers, non-trivial solutions can be found to any Diophantine ...
5
votes
5answers
192 views
Quintic diophantine equation
How can I find non trivial primitive integer solutions, to the Diophantine equation $$a^4+b^4+c^4=d^5$$
Can anyone find me solutions to this equation?
Or if possible a parametric equation that ...
-1
votes
2answers
78 views
Counting the number of positive prime solutions to third degree Thue equations equal a prime.
Counting the number n of positive (x,y) = (p,q), p and q primes, solutions to third degree Thue equations equal a prime r, and p,q,r <1000 i get with the Pari gp built-in functions thueinit() and ...
1
vote
1answer
94 views
Solutions to easy Diophantine $8pq +1 = a^{2}$, p and q primes
Show that $p = 3$ and $p = 5$ are the only primes with a maximal $3$ solutions each to $8pq + 1 = a^2$, where $p$ and $q$ are prime.
2
votes
3answers
228 views
Solving a system of Pell and Pell-like equations
Solve (Find all the solutions, if there are, or prove there are not) the system of 2 Pell or Pell-like simultanous equations over the positive integers :
$2b^{2}= a^{2} +1 = 3k^{2} + 2 $ with the 3 ...
2
votes
0answers
95 views
Diophantine equations/Diophantine Geometry
I am very knew to this site and I am eagerly waiting for solutions of:
(1) Let $x$ be an algebraic number with degree $n > 1$. Then there exists only finitely many rational numbers $p/q$ (in ...
0
votes
1answer
113 views
Solve another Diophantine equation with 2 variables and odd degree 5
See also the already solved question:
Solve a Diophantine equation with 2 variables and odd degree 5
Prove that there are no non trivial integer solutions to the equation $a^{5} -1 = 2b^{5}$
5
votes
1answer
75 views
Quadratic Diophantic equation
Hello :) i want to give a answer op the following question:
For which prime number $p$ can we give a solution of the diophantic equation given by $x^2-65y^2=p$.
I want to solve the question without ...
1
vote
1answer
99 views
How to prove that the equation $x^2-3y^2=17$ has no integer solutions?
How to prove that the equation $$x^2-3y^2=17$$
has no integer solutions? Can you help me?
6
votes
2answers
315 views
Show $15x^{2} - 7y^{2} = 9$ has no integer solutions
I'm trying to show the quadratic binary has no integer solution. I've used the following process to transform it into a Pell's equation of the form $x^{2} - Dy^{2} = M$
If there is a solution, then ...
0
votes
1answer
129 views
Looking for a general and complete solution to the Diophantine $a^2 -2b^4 = -1$
The general and probably complete solution to $a^2+(b-1)^2 = (b)^2 is (2v+1)^2+(2v(v+1))^2 = (v^2+(v+1)^2)^2$ We get the triples $(a,b-1,b) = (3,4,5), (5,12,13), ...
3
votes
1answer
218 views
Solve a Diophantine equation with 2 variables and odd degree 5
Solve in integers the equation: $a^{5} +1 = 2b^{5}$
0
votes
2answers
148 views
Diophantine power equation of degree 4 and 5 and 2 variables
Prove that $a^4 + 1 = 2b^4$ and $a^4 - 1 = 2b^4$ have no solutions in integers. Same with $a^5$ and $b^5$.
5
votes
4answers
216 views
How do you find solutions to $2 x^2 +3 x +1 = y^2$ using integers for $x,y$
I am trying to solve an equation in integers to give a square number.
$$2 x^2 + 3 x +1 = y^2$$
while also satisfying $x=k^2 * n$ where $n$ is a very large integer given to us and $k$ can be any ...
4
votes
1answer
104 views
Whether a prime number $ p $ can be written in the form $ 3A + 2B $, where $ A,B \in \mathbb{N} $.
I would like to know whether or not a prime number $ p $ can be written in the form
$$
p = 3A + 2B,
$$
where $ A $ and $ B $ are positive integers.
1
vote
1answer
204 views
How to solve this Diophantine equation?
This is an exam question from Number theory (especially of quadratic field extensions):
For which prime number $p$ can we solve the Diophantine equation $x^2-31y^2=-p$. Find also a solution for ...
4
votes
3answers
80 views
solution for equation
For $a^2+b^2=c^2$ such that $a, b, c \in \mathbb{Z}$
Do we know whether the solution is finite or infinite for $a, b, c \in \mathbb{Z}$?
We know $a=3, b=4, c=5$ is one of the solutions.
2
votes
1answer
100 views
Solutions of Diophantine equations in Natural numbers
The one of solution of $x^4 - 2y^2 = -1$ is $x = 1$ and $y = 1$. However, the solution $(1, 1)$ of $x^4 - 2y^2 = 1$ is failed. We know $x = 1$ and $y = 1$ is small integers and we can check by trail ...
4
votes
2answers
137 views
Does this equation have integer solutions
Let $g\geq 2$ be an integer. (It will be the genus of some curve.)
Are there positive integers $d$ and $e$ such that the equality
$$ (e-2)(e-1) = 2d(g-1)+2$$ holds?
-1
votes
2answers
128 views
Diophantine equations special problem
Suppose $r$ is a rational number and for $k > 2$, consider $0\leqslant a_1< a_2<\cdots \leqslant a_k$. Also, for $n > 2$ and assume that we are not interesting the case of $n = 4 = k$, ...
1
vote
3answers
124 views
Finding the sum of all solutions
$2x + 3y = n$ has exactly $2011$ non-negative integral solutions. Determine the SUM of the possible values of $n$.
0
votes
0answers
71 views
Geometric interpretation of diophantine equations
Is there exist any geometric interpretation of diophantine equations?
( I am thinking about solving diophantine equations by geometrical methods).
For example I want to solve this:
$ 2x^6+y^7 = ...
0
votes
0answers
71 views
class number, divisibility, diophantine equation.
How to generalize or prove the following on class numbers of quadratic fields.
Let $n$ be a positive integer then $h(-4n) = 1$ if and only if $n = 1, 2, 3, 4$ or $7$.
If $D$ is Gauss discriminant ...
