Questions tagged [diophantine-equations]
Use for questions about finding integer or rational solutions to polynomial equations.
4,311
questions
3
votes
3answers
51 views
solve in N the equation $x^2-2^y=2021$
I have this equation to solve $x^2-2^y=2021, x,y \in N$
I was thinking of seeing it as a Diophantine equation, but it doesn't seem very logical
2
votes
3answers
86 views
Find all integer solutions for $x^3+1=y^2$.
Find all integer solutions for $$x^3+1=y^2.$$
Attempt:
By guessing, I found five pairs of integer solutions for the equation: $(2, \pm 3)$, $(0, 1)$, $(-1, 0)$ and $(0, -1)$, but really I don't know ...
-2
votes
2answers
26 views
Value of m for which the function will give integers as an output. [closed]
$F(m)=(2m^3+2m)/(m^2+1)$ and $g(m)=(m^4+1)/(m^2+1)$ What are the values of $m$ other than $1$ for which solution of both function will be integers. Please tell if there is any formula to find so or ...
-2
votes
1answer
38 views
If $\frac{2m^3+2m}{m^2+1}$ and $\frac{m^4+1}{m^2+1}$ are integers then $m=?$ [closed]
$\frac{2m^3+2m}{m^2+1}$ and $\frac{m^4+1}{m^2+1}$
Find value of $m$ other than $1$ for which solution of both equations are in integers.
Please tell if there is any formula to find so or any technique....
0
votes
1answer
43 views
Show that for any positive integer $n>1$ there exist positive integers $x$ and $y$ such that $n^3+x^3=(x-1)^3+y^3$.
A quick check shows that $x=y=n+1$ work pretty good. Since,
$n^3+(n+1)^3=\{(n+1)-1\}^3+(n+1)^3.$
How to check whether there exist any other type of solutions or not? Please suggest.. Thanks in ...
1
vote
2answers
43 views
Find $a \in \mathbb N$ such that $x^2+ax-1 = y^2$ has a solution in positive integers
Question: Find all the positive integers $a$ such that $x^2+ax-1 = y^2$ has a solution in positive integers $(x,y)$.
Comments: It's easy to see that this equation rarely has a solution (in the sense ...
0
votes
0answers
17 views
An example of a polynomial function $f$ where the equation has no solution or has a finite number of solutions
I am asking about the following Diophantine equation:
$$32(d+1)²a²-cf⁴(b)=0$$
where $f$ is a polynomial function.
From this question (A Diophantine equation with infinitely many positive integer ...
0
votes
2answers
14 views
A Diophantine equation with infinitely many positive integer solutions
I am asking if the following Diophantine equation:
$$32(d+1)²a²-cb⁴=0$$
has infinitely many positive integer solutions.
2
votes
0answers
60 views
Fermat's last theorem for case of n=3
I'm aware that one can use a method of infinite descent, or even just refer to the more general case which has been proven by Andrew Wiles, but I was thinking about it the other day, and I remember ...
1
vote
0answers
38 views
Find all positive integers which are representable uniquely as $\frac{x^2+y}{xy+1}$ with $x,y$ positive integers.
$\textbf{Question:}$ Find all positive integers,which are representable uniquely as
$\frac{x^2+y}{xy+1}$
where x and y are positive integers.
I think this question maybe has something to do with ...
0
votes
2answers
69 views
Find all integer solutions for Mordell's Equation $x^2=y^3+k$, where $k=-35$.
I know that there are definitely solutions to this equation, two of which are $(\pm 36,11)$. However, I now need some guidance on where to begin finding the remaining solutions for this case of ...
-1
votes
1answer
65 views
Number Theory and Square Number Problem [closed]
Find the least positive integer $n >1$ such that the arithmetic mean of the first $n$ non zero perfect squares is again a perfect square.
Please help. Hope it gives me an idea of equations with ...
1
vote
0answers
58 views
For given positive integers $s$ and $t$, how many integer solutions are there to $x^2+7y^2=3^411^s23^t$?
Working in $Z[\sqrt{-7}]$, I know that I am trying to find how many $\alpha$ have norm equal to $3^411^s23^t$.
I have so far found examples of $\alpha$ with norms of 11 and 23 - these are $2 \pm \...
4
votes
1answer
74 views
Show that there are no integer solutions to $2x^{11}+3y^{11}=6z^{11}$
I have managed to show that $x$ must be a multiple of $3$ and $y$ must be even, which produces the equation
$$3^{10}s^{11}+2^{10}t^{11}=z^{11},$$
with $s=x/3$ and $t=y/2$. I have tried to approach ...
0
votes
0answers
29 views
Method(s) to turn a “single equation” problem into a “double equation” problem?
In the world of Diophantine analysis, there are single equation problems (e.g., “Find all solutions $(x,y)$ such that $x^3=y^2+1$”), double equation problems (e.g., “Find all numbers $p$ such that $p+...
0
votes
3answers
80 views
$\left(3z\right)^3\ne 3\left(x+y\right)\left(3z-x\right)\left(3z-y\right)$
Prove that
$$\left(3z\right)^3\ne 3\left(x+y\right)\left(3z-x\right)\left(3z-y\right)$$
Is true for $x$, $y$ and $z$ being positive integers, with $x$ and $y$ being co-prime and $3z<x<y$.
...
1
vote
1answer
50 views
Prove $3(x+y)(x+z)(y+z)\neq a$ cubic when $x,y,z$ are different co-primal positive integers
Prove $3(x+y)(x+z)(y+z)\neq a$ cubic when $x,y,z$ are different co-primal positive integers.
I believe you can look at the prime factors of $x,y$ and $z$. As for the equation to equal a cubic, the ...
1
vote
0answers
20 views
Properties of a mapping
Given an equation that I've been looking at of the form $$(2^n-1)(1+x_1+x_1^2+...+x_1^{2j_1})(1+x_2+x_2^2+...+x_2^{j_2})\cdot \cdot \cdot(1+x_m+x_m^2+...+x_m^{j_m})+1=2^n\cdot x_1^{j_1}\cdot x_2^{j_2}....
2
votes
4answers
125 views
A solution of Diophantine equation: $\big(x+y+z\big)^{3}=27x y z$ with $(x,y)∈Z$
A solution is:
$x=(r+s)^{3}(r-3s)^{3}\big(r^{4}-4r^{3}s+4r^{2}s^{2}+3s^{4}\big)^{3}$
$y=8r^{3}(2s-r)^{3}\big(r^{4}-4r^{3}s+4r^{2}s^{2}+3s^{4}\big)^{3}$
$z=(r^{2}-2r s+3s^{2})^{3}\big(r^{4}-4r^{3}s+...
1
vote
0answers
44 views
Iran Mathematics Olympiad Problem [duplicate]
If $x,y$ are Positive integers such that $3x^2+x=4y^2+y$ Prove that $x-y$ is a Perfect Square
My try:
We have $$3x^2+x-(4y^2+y)=0$$ a Quadratic in $x$
So $$x=\frac{-1+\sqrt{1+12(4y^2+y)}}{6}$$
...
4
votes
2answers
67 views
Find all possible positive integers $x$ and $y$ such that the equation: $(x+y)(x-y)=\frac{(y+1)(y-1)}{24}$ is satisfied.
My approach so far:
The given equation can be rewritten as: $x^2 -y^2=\frac{y^2 -1}{24}.$ This gives
$24x^2 +1=25y^2=(5y)^2.$ So $(24x^2+1)$ must also be a perfect square. This implies $x=0, 1$ is ...
3
votes
3answers
118 views
How to choose a special modulus to show that $6n^3 +3 = m^6$ has no solutions in the integers
I was stuck on a problem from Mathematical Circles: Russian Experience, which reads as follows:
Prove that the number $6n^3 + 3$ cannot be a perfect sixth power of an integer for any natural number ...
1
vote
1answer
96 views
Solving a system of equalities in 4 variables (but no numeric constant)
I have pairwise relatively prime positive integers $a$, $b$, $c$, and $d$ such that
$$
\frac{a-b}{4} = \frac{2a-9c}{7} = 27d-10a = 9c-2b = \frac{27d-10b}{41} = \frac{3d-5c}{4} \tag{$\star$}
$$
and
$$
...
0
votes
1answer
36 views
On a homogeneous Diophantine equation
I want to solve the diophantine equation $c_1x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 = 0$ (I), when $c_1+c_2+c_3+c_4=0$. I first consider $x_1 = t$ and $(c_2+c_3+c_4)t = c_2x_2+ c_3 x_3 + c_4 x_4$; then I ...
0
votes
0answers
39 views
Diophantine equations that involve Gregory coefficients: a computational exercise
In this post, for integers $k\geq 1$, we denote the Gregory coefficients as $G_k$. Wikipedia has an article for Gregory coefficients, are known as reciprocal logarithmic numbers (I add this as ...
1
vote
0answers
82 views
Generalized Fermat equation for signature [4,5,7]
The generalized Fermat equation has been solved for many signatures. But, I can't find a determination that the signature $[p,q,r]=[4,5,7]$ has no solutions. Is this signature
still an open problem?
...
0
votes
2answers
37 views
Method to solve factored quadratic diophantine equations?
Is there a method that can solve all quadratic diophantine equations of the following type
$$X (X + a) = Y (Y + b)$$
where $a,b$ are given integers?
1
vote
0answers
181 views
proof that no non negative odd integer solution of this type of diophantine equation exists given even non negative integer solution
Given the equation $$ (2^n-1)(1+x+x^2+...x^{2k})(1+y+y^2+...+y^{2j})=2^nx^{2k}y^{2j}-1 $$ a non negative integer solution exists of the form $(2^n,2^{2kn+n}$). How can I (try to) show that (1) this ...
1
vote
1answer
49 views
find integer solutions under square root [duplicate]
I have a equation $y = \sqrt{5x^2+2x+1}$ and I'm trying to generate integer solutions. I've tried Vieta jumping but it failed.
So I generated by brute force few solutions and find these: x=2, 15, ...
1
vote
1answer
177 views
Can my short proof of Fermat´s Last Theorem be true or is there a serious flaw?
A proof of Fermat´s Last Theorem using only Gauss´s Lemma for the roots of monic polynomials with integer coefficients.**
Writing the Fermat equation
$$a^n + b^n - c^n = (c-p)^n + (c-q)^n - c^n = 0$$...
2
votes
4answers
117 views
Are there infinitely many primitive Pythagorean $n$-tuples?
Let's define a Pythagorean $n$-tuple ($n \geq 3$) as a tuple of distinct natural numbers $(x_1, ... , x_{n-1}, y)$, such that $x_1^{n-1} + ... + x_{n-1}^{n-1} = y^{n-1}$. Let's call a Pythagorean $n$-...
0
votes
1answer
26 views
Fundamental Discriminant
Let $d=2\bmod4$ and $D=d^3-1$. We will show that the elliptic curve $E_D: y^2=x^3+D$ has no integer solutions $x,y$.
At first we show that $x$ is odd which is simple. Then we have the following ...
0
votes
5answers
60 views
How to find infinitely many positive integer solutions to $x^2+y^2+z^2=w^2$
How to find infinitely many positive integer solutions to this equation?
$$x^2+y^2+z^2=w^2$$
Why my answer makes no sense:
$x=n$
$y=(n+1)$
$z=\sqrt{ 2n(n+1) }$
then $w^2=(n+(n+1))^2$ for some ...
6
votes
5answers
189 views
Does $a^2 + b^2 = 2 c^2$ have any integer solution?
Does the equation $a^2 + b^2 = 2 c^2$ have any integer solution with $|a| \neq |b|$?
I think not, because of these equations for pythagorean triplets:
$$\left(\frac{a}{c}\right)^2+\left(\frac{b}{c}\...
1
vote
1answer
33 views
Diophantine equation (three variables)
Show that the equation $$x^3+2y^2+4z=n$$ has an integer solution $(x,y,z)$ for all integers $n$.
This seems to be an Diophantine equation with three variables. How can I restrict $n$ in order to find ...
2
votes
1answer
43 views
Cubic Discriminant Uses
The discriminant for cubic equations is -
$Δ\:=b^22\:^2−4ac^3−4b^3d−27a^2d^2+18abcd$
And I am aware that you can determine the number of roots a cubic has using method shown below -
$Δ\:>0$ ...
2
votes
2answers
139 views
What is the solution of $x^3+y^3+z^3=429$ in integers?
I have tried to solve $x^3+y^3+z^3=429$ using mathematica (Reduce[x^3+y^3+z^3 == 429 && x > 0 && y > 0,&& z > 0, {x, y,z}, Integers] ) and wolfram alpha I can't come up to ...
0
votes
1answer
70 views
Proving no solution exists / find solutions for $A^A + B^B = C^{A+B}$ Diophantine equation.
Given $A,B,C \in \mathbb{N}$ and $A,B,C > 1$ Prove that no solution exists / find existing solutions for:
$$A^A + B^B = C^{A+B}$$
I have 0 clues on where to start.
I think it is known ...
0
votes
4answers
90 views
Diophantine cubic equations
Specifically I mean the equation:
$$x^3+y^3=u^3+v^3+w^3.$$
It is relatively easy to find a solution (in this case) by guessing:
$$7^3+8^3=1^3+5^3+9^3.$$
However, I don't know whether that solution is ...
2
votes
1answer
81 views
Diophantine equation with power
Find all the integer solutions of:
$2^{n+1} + 41 = m^2$
I am stuck, and I am not sure if I am going on the right path..
adding 8 to both sides:
$2^{n+1} + 41 + 8 = m^2 + 8$
$2^{n+1} + 49 = ...
0
votes
0answers
37 views
If the equation $3x^2 − 5y^2 ≡ 1$ mod m has solutions for some m, does the diophantine equation $3x^2 − 5y^2 ≡ 1$ has solutions?
I am trying to figure out if the below statement is true or false:
If the equation $3x^2 − 5y^2 ≡ 1$ mod m has solutions for some m, then the diophantine equation $3x^2 − 5y^2 ≡ 1$ has solutions.
...
2
votes
1answer
116 views
Using a solution to show no others are possible in positive odd integer
If I have an equation say $$3(1+x+x^2)(1+y+y^2)(1+z+z^2)+1=4x^2y^2z^2 \quad (1)$$ and I know a non negative integer solution $x=4, y=64, z=262144$ then no odd positive integer solutions can possibly ...
2
votes
1answer
107 views
Can 4+(2k)! ever be a perfect square over the integers?
Is there any pair of natural numbers $\{ k, m \}$ satisfying: $4+(2k)! = m^2$?
I tried simplifying this into
$$(2^k)(k)!(2k-1)!! = (m-2)(m+2) ,$$
where !! denotes the double factorial, i.e., $1 \...
0
votes
1answer
70 views
Should I waste my time with wolfram alpha to solve such problem like $x^3+y^3+z^3=390$?
I'm sorry to ask this question probably it is not suitable here. However I read some papers in number theory related to the solution of $x^3+y^3+z^3=n$ for some known solved problem about ...
1
vote
2answers
115 views
diophantine question parity and sign
If I have an equation say $$3(1+x+x^2)(1+y+y^2)=4x^2y^2-1 \quad (1)$$ and I rewrite it as
$$4+3(x+y+xy+x^2+y^2+xy^2+x^2y)-x^2y^2=0$$ and I find a positive integer solution where both x and y are even,...
0
votes
1answer
33 views
generating solutions to diophantine equation
If I have a solution to say
$$4x_1+3x_2+3x_3+3x_4...+3x_n=4y$$ where $x_i ,y$ are non negative integers can I generate other solutions from my initial solution? And if I can would all solutions ...
0
votes
0answers
49 views
What are enough conditions for $x,y,z$ to have $x^3+y^3+z^3$ a perfect square?
In my question here I want to know if there are enough conditions for $x, y,z$ to have $x^3+y^3+z^3$ a perfect square , The necessary condition is clear that $x,y,z$ must be different than $4$ or $5$ ...
0
votes
1answer
17 views
Criteria for checking if points are the vertices of a hypercube
I asked a question over at Code Golf Stack Exchange which essentially asked folks to write a program to determine if a collection of $2^n$ points in $\mathbb{Z}^m$ is the vertex set of some $n$-...
1
vote
1answer
54 views
Solutions to Diophantine equation $\frac{1}{n}+\frac{1}{p}=\frac{1}{N}$
For each prime $p$ there seems to be a uniqe solution $n=(p-1)p$ to the Diophantic equation
$\frac{1}{n}+\frac{1}{p}=\frac{1}{N}$.
Is that right and if so, how to prove the unicity?
In spite of my ...
1
vote
2answers
68 views
Why $c$ closed to $-2\times10^n$ in $(1-c^2)^3+(c^3+10^nc^2-1)^3+(10^n c^2-1)^3=n$ for $n >1$?
I have tried many times to evaluate $(1-c^2)^3+(c^3+10^nc^2-1)^3+(10^n c^2-1)^3=n$ for $n >1$ as polynomial for some values of integer $n$ which are greater than $1$ for the solution of the titled ...
