4
votes
3answers
93 views

Find all positive integers satisfying $\frac{2^n+1}{n^2} =k $

Find all positive integers satisfying $$\frac{2^n+1}{n^2} =k $$ where $k$ is a integer. I can't just come up with a solution.
0
votes
1answer
28 views

Solution to Diophantine equation with constraint.

solve the following equation over $z_x,z_y$ \begin{align} &az_x=bz_y\\ &\text{s.t. } a,b,z_x,z_y \in \mathbb{Z} \text{ and } 1 \le z_x \le N \text{ and } 1 \le z_y \le N \end{align} How ...
4
votes
0answers
64 views

Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be ...
2
votes
1answer
55 views

Hard Simultaneous Diophantine Equations

Find all positive integers $a,b,c,d,e,f$ such that : $de^2=ab^2+1$ and $df^2=ac^2+1$. I tried subtracting them, it factors quite nicely. But after that, haven't a clue. I'm not sure if it's even ...
0
votes
2answers
79 views

Find all non-trivial triplets $(a,b,c)$ such that $ a+b^{d}\equiv0\pmod c $ for all $d>0$

Let $a,b,c$ be co-prime integers $>2$ . Find all non-trivial triplets $(a,b,c)$ such that $ a+b^{d}\equiv0\pmod c $ for all $d>0$.
3
votes
2answers
54 views

Which natural numbers can be represented as a sum of natural numbers raised to different powers?

Waring's problem asks about natural numbers that can be represented as a sum of natural numbers all raised to the same power $k$. I'm wondering which natural numbers can be represented as a sum of ...
10
votes
1answer
157 views

Cubic polynomial equal to a cube

I've been researching cubes and I'm trying to solve this Diophantine equation over the integers. $$ax^3 + bx^2 + cx + d = y^3$$where a, b, c, d are parameters for a given $n$. For example, for $n = ...
0
votes
0answers
43 views

Diophantine Equations involving cubes

I'm doing some number theory research and I came across these two Diophantine equations (created under my own transformations): $$y^3 = ax^3 + bx$$ (where $a$ and $b$ are parameters) $$z^3 = x^2 + ...
0
votes
0answers
35 views

Hard Diophantine: $ xy-\frac{(x+y)^2}{n}=n-4 $

Solve in positive integers $x,y$: $ xy-\frac{(x+y)^2}{n}=n-4 $ $n>4$ is a given positive integer. I cannot even solve in the case $n=5$. I have been able to find $x,y$ and construct $n$ using ...
3
votes
2answers
120 views

Solving $x^3+y^3=x^2y^2+1$ in non-negative integers

I wanted to solve $x^3+y^3=x^2y^2+1$ in non-negative integers. First I set $a=x+y$ and $b=xy$ to get $b^2+3ab+1=a^3$. View as a quadratic in $b$, the discriminant = $4a^3+9a^2-4$, which needs to be a ...
0
votes
1answer
51 views

Solve for x,y: $x^2+1=2y^2$

Solve for integers $x,y$ such that $x^2+1=2y^2$? I tried factoring as $(x-y)(x+y)=(y-1)(y+1)$ but couldn't continue from here, I would appreciate any help. Thanks!
2
votes
2answers
57 views

Diophantine: $px^2+2=y^2$ where $p\in \mathbb{P}$

Solve the Diophantine Equation: $px^2+2=y^2$, where $p$ is a prime number and $x,y$ integers. I tried this for ages but didn't get anywhere, but I don't know any advanced machinery since I am only in ...
5
votes
1answer
104 views

Fermat: Prove $a^4-b^4=c^2$ impossible

Prove by infinite descent that there do not exist integers $a,b,c$ pairwise coprime such that $a^4-b^4=c^2$.
8
votes
6answers
1k views

Are there finitely many Pythagorean triples whose smallest two numbers differ by 1?

Has it been shown whether there is a finite or infinite number of Pythagorean triples whose smallest two numbers differ by 1? In either case I’d appreciate a link to the proof. Edit: thank you all ...
5
votes
1answer
162 views

How to prove that $3ab(a+b)$ cannot be a cube?

Consider the diophantine equation: $ 3ab(a+b)=c^3 $ where $a,b,c$ are non-zero integers,how do you prove that this equation has no integral solutions?
0
votes
6answers
105 views

Solve $f^2-e^2=d^2-c^2-b^2+a^2$

I'm looking for a solution or some clarifications for this equation: $f^2-e^2=d^2-c^2-b^2+a^2$ with $f>e$, $d>c$, $b>a$ and $f, e, d, c, b, a$ natural numbers. Regards
2
votes
1answer
76 views

Exponential Diophantine: $2^{3x}+17=y^2$

Is there a way of solving the following equation, in integers $(x,y)$, by hand? : $2^{3x}+17=y^2$. You can also try: $2^{2x}+17=y^2$ or more generally $2^x+17=y^2$; each of these has at least 1 ...
1
vote
1answer
63 views

Solution to the Diophantine equation $x^4+y^4=2z^2$ [duplicate]

Does there exists a nontrivial positive integer solution with $x\ne y,$ of $$x^4+y^4=2z^2.$$
-1
votes
0answers
37 views

Sparsity of Linear Diophantine Equations

If you are looking for integer solutions to the system. $$Ax=b$$ where $A$ is an integer matrix and $b$ is integer vector, then you can construct the solution space integer matrix $B$ and integer ...
2
votes
2answers
73 views

Diophantine equation: $n^p+3^p=k^2$

Find all solutions to the Diophantine equation $n^p+3^p=k^2$, where $p\in \mathbb{P}$ and $n,k$ positive integers. I have tried everything, from mods to bounding to LTE; nothing seems to work on ...
-2
votes
2answers
310 views

Is this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions?

Is this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions? Suppose $\ a^3 + b^3 = c^3,\ a,b,c \in \mathbb Z^*,\ $then: $c^3 - b ^ 3 = (c - b)((c - b) ^ 2 + 3cb) = ...
2
votes
3answers
58 views

Diophantine Equatiοn $x^3=2^y+15$

I would like some help with the diophantine equation $x^3=2^y+15$ I have tried working with last digits and modular arithmetic but that hasn't got me anywhere.
3
votes
3answers
130 views

$x^2+1$ is almost always square free

It seems like $x^2+1$ is almost always square free. Any research or heuristics why? I tried breaking the problem into solving $$x^2-ky^2=1$$ For various $k$, and I conjecture that for every $k$ there ...
1
vote
2answers
223 views

Solutions to the Mordell Equation modulo $p$

It is well known that the Mordell Equation $x^2 = y^3 + k$ has finitely many solutions, but has solutions modulo $n$ for all $n$. One proof of this involves using the Weil Bound to show that $x^2 = ...
1
vote
2answers
47 views

Diophantine solution to a fraction

How can we find solutions to the following equation: $$ y=\dfrac{x^2-1085}{14718-2x}$$ where $x,\ y$ are integers.
3
votes
4answers
128 views

Diophantine equation abc + abd + acd + bcd= 1

Is there a reference which classifies or at least gives an infinite family of integer solutions to the above equation? A solution to the problem would also be great obviously.
0
votes
1answer
27 views

integral point on conics

Suppose we have a conic $ax^2 + bxy + cy^2 + dx + ey + f = 0$ where $a,b,c,d,e,f \in \mathbb{Q}$. Is there a way of computing the integer points on this curve. Since it is affine an not projective we ...
0
votes
0answers
32 views

Solving a general Diophantine Equation

For "normal" equations in one variable we have several techniques for solving equations, such as $\sin(5x) = 5\pi\cos(5x)$ or $\ln(x + 2) = 4$. However, imagine we have the following equations: ...
1
vote
0answers
58 views

How to solve the diophantine equation $x^y = y^x $? [duplicate]

I know this might be an obvious question as we all know that the answers (beside $(x,y)=(1,1)$) are $(x,y)=(2,4)$ but the problem is, how is this exactly solved? Tags might be inaccurate so feel free ...
2
votes
2answers
88 views

Integer solutions of the equation: $x^2+y^2+z^2=kxyz$

Given the equation: $$x^2+y^2+z^2=kxyz$$ with: $(k,x,y,z)\in\mathbb{N}$, the only solution for $k=2$ is: $x=0,y=0,z=0$. For what values of $k$ the equations has solutions in which $x,y,z$ are ...
7
votes
2answers
267 views

Is $7^{8}+8^{9}+9^{7}+1$ a prime? (no computer usage allowed)

Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer. I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.
0
votes
2answers
34 views

Finding Solutions to a Diophantine Equation with Factorials

How many ordered pairs of positive integers $(a, b)$ are there such that $a!+\dfrac{b!}{a!}$ is a perfect square? Is the number of solutions finite? Source: Ran into it on Facebook. I have plugged ...
1
vote
1answer
66 views

Has anyone solved this general Diophantine Equation?

I know that Pythagorean triples have been parameterized, I also know that Andrew Wiles has proved that there are no distinct integer solutions for $ a^n + b^n = c^n$, when $ n \ge 3 $. However we may ...
2
votes
1answer
93 views

Solutions to a diophantine equation

I tried to find integer solutions to the following diophantine equation $$x^3 - 3y^3 + 5z^3 - 3xy^2 + 3x^2y + 9xz^2 + 7x^2z + 3yz^2 - 3y^2z + xyz = 0$$ but was unable to do so. I suspect that there ...
0
votes
0answers
29 views

Differential Diophantine Equations?

So this is both a question on its own as well as a request for where I can find information on a given topic. Consider Differential Equations in two variables of the form: $$P(Z,Z', Z'' ... Z^{[n]}, ...
1
vote
3answers
80 views

How to solve $b^2-a^2=d^2-c^2$

I'm looking for how to solve the equation $b^2-a^2=d^2-c^2$ where $a,b,c,d$ are naturals and $d>c>b>a>0$ , an algorithm would be appreciated Regards
0
votes
1answer
34 views

The sum of the cubes and the amount of combinations.

Quite simply turned out to solve this Diophantine equation, when he made the assumption that the solutions of these equations symmetric. So given this equation: ...
0
votes
2answers
41 views

$A^7 \not\equiv A(\mod 13) \Rightarrow A^{78} + 1 \equiv 0 (\mod 169)$

Let variable $A$ is integer and $A^7 \not\equiv A(\mod 13)$. Prove that $A^{78} + 1 \equiv 0 (\mod 169)$ Could someone explain, how to solve this type of problems? Any help would be greatly ...
1
vote
3answers
65 views

How to show $n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$ has no nonzero integer solutions?

How do we prove that $$n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$$ has no nonzero integer solutions? I know two ways to prove this by taking a geometric interpretation but I don't want such a version. How ...
0
votes
3answers
58 views

Solving different types of Diophantine equation [closed]

In each of the following three equations I need help in finding all solutions in positive integers : i) $\dfrac 1x+\dfrac 2y-\dfrac3z=1 $ ii) $\dfrac 1{x^2}+\dfrac 2{y^2}+\dfrac 3{z^2}=\dfrac 23$ ...
3
votes
1answer
85 views

motivation for talking about torsion points on an elliptic curve

Let's consider elliptic curves over a fixed field $k$. I understand that viewing the set of $k$-rational as an abelian group is interesting and useful, but I am confused about why the torsion points ...
3
votes
0answers
52 views

Solution of a equation in natural number nvolving reciprocal of prime

Let $p$ be a prime and $n$ a natural number . Solve in $\mathbb{N}$ the equation $$\sum_{k=1}^{n}\frac{1}{x^k_k}=\frac{1}{p}$$
0
votes
1answer
71 views

How prove this diophantine equation $3^x-2^y=k$ have finitely many integral solutions

For any $\forall k\in N^{+}$ show that the diophantine equation $3^x-2^y=k$ have finitely many integral solutions. My try: if $k=2m$,then $$3^x=2^y+k=2^y+2m$$ It is clear there is no integer ...
1
vote
0answers
19 views

Diophantine equations,is that what I have done right?

I have solved the following diophantine equations: $14x+35y=93$ $56x+72y=40$ That's what I have tried: $gcd(35,14)=7$ , but $7 \nmid 93,$ so the first diophantine equation has no solution. ...
2
votes
1answer
40 views

Why are these the solutions of the diophantine equation?

According to my notes: $$ \begin{align} & \text{ Let } a,b \text{ not both } 0. \\ & ax+by \text{ has a solution iff } (a,b) \mid c \\ & \text{ If } d:=(a,b) \mid c \text{ and } a=d \cdot ...
0
votes
0answers
12 views

Polynomial Roots of Bivariates

I've got a few polynomials that I am trying to get some results for (shown below). They come from the characteristic equation of a matrix. I have two variables in the polynomials, $\eta$ (which is ...
3
votes
1answer
56 views

Is it true that $f(x,y)=\dfrac{x^2+y^2}{xy-t}$ has only finitely many distinct integer values with $x,y$ positive integers?

Prove or disprove that if $t$ is a positive integer, $$f(x,y)=\dfrac{x^2+y^2}{xy-t},$$ then $f(x,y)$ has only finitely many distinct integer values with $x,y$ positive integers. In other words, ...
-2
votes
1answer
68 views

Project Euler - task №390.

When this task was not clear what the equation to be solved. This equation? $x^2y^2+z^2y^2+x^2z^2=r^2$ in integers. It is not clear, because this equation is quite simple and I do not think that ...
0
votes
0answers
50 views

System of quadratic diophantine equations 2

I am looking for a way to simultaneously transform the following four expressions into perfect squares, $1+x_1^2, 1+x_2^2, 1+x_3^2, x_1^2+x_2^2+x_3^2$, i.e. I want to find a rational parametrization ...
1
vote
4answers
157 views

Solutions to the diophantine equation: $2a^2 + 2b^2- c^2- d^2 = 0$

As suggested on Mathoverflow (http://mathoverflow.net/questions/168536/solutions-to-the-diophantine-equation-2a2-2b2-c2-d2-0) I am transfering this question to math-stackexchange: I am looking for ...