Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

Filter by
Sorted by
Tagged with
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 ...

1
2 3 4 5
87