Questions on finding integer/rational solutions of polynomial equations.

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31 views

Help solving the following equation system:

Let $p, q$ be any two integers, and $a, s, t, n \in\mathbb{Z}$. How do I solve the following system for $s, t$: $$p - q^{n} = ta(n + 1)$$ $$p - (n + 1) a^{n} = sq$$ Please help.
7
votes
1answer
316 views

Ramanujan-Nagell Theorem Proof Question

I'm currently working through Stewart and Tall's Algebraic Number Theory. In particular, section 4.9 of this book provides a proof of the Ramanujan-Nagell Theorem, which states that the only integer ...
0
votes
1answer
43 views

Diophantine equation $y^6 + 3xy^4 - x^3y^3 - 6x^2 - 3xy - 12x = 9$

How to show that the diophantine equation $$y^6 + 3xy^4 - x^3y^3 - 6x^2 - 3xy - 12x = 9$$ has only the solutions $(x,y) = (-2, -3)$ and $(-2, 1)$?
1
vote
1answer
32 views

For which $a>0$ does the equation $x^2+y^2+z^2=a$ have a solution in $\mathbb{Q}_2$?

We want to check for which $a>0$ we have that the equation $x^2+y^2+z^2=a$ has a solution in $\mathbb{Q}_2$. $x^2+y^2+z^2=a$ has a solution in $\mathbb{Q}_2$ for $x \in \mathbb{Z}_2^{\star}, y ...
4
votes
0answers
115 views

Find all the solutions of $x^2+7=2^n$. [duplicate]

Checking for some small natural numbers $n$, I found out that $2^n-7$ is a perfect square for $n=3,4,5,7,15$. How can we find all of the numbers $n$ for which $2^n-7$ is a perfect square? What I ...
11
votes
2answers
125 views

When is $2^n -7$ a perfect square?

This came up while solving another ENT problem. I want to ask when is: $$2^n -7 \text{ where } n\geq 3$$ a perfect square? Specifically, I also wanted to know what would be the solutions when $n$ is ...
2
votes
1answer
83 views

Find all integer solutions of $x^4 + 2x^3 + 2x^2 + 2x +5 = y^2$

Find all integer solutions to $x^4 + 2x^3 + 2x^2 + 2x +5 = y^2$. I'm in a dead end. I've transformed the expression in the following state: $(x^2+1)(x+1)^2 = y^2 -4$ I couldn't see anyway in ...
0
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2answers
28 views

How do I solve $3(2^{x+2}-2^x) = 4a_1a_2a_3$

I encountered this problem but I'm not sure how to solve it since it has 4 unknowns. $$3(2^{x+2}-2^x) = 4a_1a_2a_3$$ What is known is that $x\in\mathbb{Z}$ and $a_1, a_2$ and $a_3$ are digits in a ...
0
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1answer
24 views

What are equivalent parametric equations?

What are equivalent parametric equations? Is there a fast method to prove that 2 parametric equations are non-equivalent?
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1answer
163 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 ...
0
votes
1answer
80 views

Diophantine equation exercise [duplicate]

Prove that the diophantine equation $x^4-2(y^2)=1$ has only 2 solutions. Any hint on how to start and what to do .. I do not have a lot of experience on non linear diophantine equations and do not ...
0
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0answers
13 views

To solve the system of Diophantine equations.

I decided to compile a single task and to record such a system. $$\left\{\begin{aligned}&xt+yw=az^2\\&xw-yt=br^2\end{aligned}\right.$$ $a,b - $ integers that are the problem. It is clear ...
2
votes
1answer
46 views

Pythagorean Quadruples Problem

What are all the solutions to $$2^{2x}+2^{2y}+1=n^2 $$ I tried using the parametrization of Pythagorean Quadruples, but it did not work quite well. There are $2$ parametrizations: ...
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3answers
17 views

Where am I going wrong in my linear Diophantine solution?

Let $-2x + -7y = 9$. We find integer solutions $x, y$. These solutions exist iff $\gcd(x, y) \mid 9$. So, $-7 = -2(4) + 1$ then $-2 = 1(-2)$ so the gcd is 1, and $1\mid9$. OK. In other words, ...
2
votes
1answer
28 views

Solving diophantine equation $6x+9y=1050$ where $x,y \in\mathbb{N}$

I have to solve this Diophantine equation: $6x+9y=1050$, where $x,y \in\mathbb{N}$. I am not sure as to how to solve this for only the whole numbers, but I think I'm doing it right. I used the ...
7
votes
2answers
83 views

Possible values of infinitely nested square root $n= \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x}…}}}$

If $$n= \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x}......}}}$$ Is it possible that $n$ is a integer for any $x=Z( \text{zahlen number})$.If yes .What is the value of $x$??
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2answers
15 views

Finding $2m+1=2\alpha k+\alpha^2$ quickly

Given some positive integer $m$ I'm looking for all solutions $\alpha,k>0$ to $2m+1=2\alpha k+\alpha^2$ with $0<k^2<2m.$ Right now I'm finding these by looping over each of these possible $k$ ...
8
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2answers
329 views

How to find value of $x+y+z+u+v+w$

let $x,y,z,u,v,w$ be positive integer numbers,and such $$1949(xyzuvw+xyzu+xyzw+xyvw+xuvw+zuvw+xy+xu+xw+zu+zw+vw+1)=2004(yzvw+yzu+yzw+uvw+y+u+w)$$ Find this value of $$x+y+z+u+v+w=?$$ My try: maybe ...
0
votes
1answer
34 views

Sum of consecutive integers is a perfect square, perfect cube

Let $p,q,r,s,t$ be consecutive positive integers such that $q+r+s$ is a perfect square and $p+q+r+s+t$ is a perfect cube. Find the smallest possible value of $r$?
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2answers
46 views

Linear congruences $2X\equiv9\pmod{26},\pmod{25}$

May double that of a natural number let rest $9$ when divided by $26$? And when divided by $25$? I tried: $$2X\equiv9\pmod{26}$$ As $(26,2)=2$ and $2\nmid9$ then the congruence linear not ...
3
votes
1answer
49 views

Solving A Certain Diophantine Equation

I am stack on finding the solution of the diophantine equation: $d(2^{k+1}-1)-b^2(2^{k+1}-2)=1$. where $k\geq 1$ and $b^2>d$ for $b$ an odd composite integer. Is there a solution to this ...
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vote
2answers
69 views

The equation $x^4+y^4=z^2$ has no integer solution

The equation $$x^4+y^4=z^2$$ has no integer solution for $(x, y, z), x \cdot y \neq 0 , z >0$. We suppose that there is a solution $(x, y, z)$. We consider the set $$M=\{z \in \mathbb{N} | ...
3
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2answers
33 views

Is there are integer solutions for this equation: $ 65x-4y= 129$ [on hold]

My question is: Is there are integer solutions for this equation: $$ 65x-4y= 129$$
5
votes
4answers
779 views

Find all solutions of $1/x+1/y+1/z=1$, where $x$, $y$ and $z$ are positive integers

Find all solutions of $1/x+1/y+1/z=1$ , where $x,y,z$ are positive integers. Found ten solutions $(x,y,z)$ as ${(3,3,3),(2,4,4),(4,2,4),(4,4,2),(2,3,6),(2,6,3),(3,6,2),(3,2,6),(6,2,3),(6,3,2)}$. ...
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0answers
31 views

Polynomial/ Exponential diophantine equation

I am looking for the reference characterizing all the cases when $$an^2+bn+c=2^m$$ has infinitely many positive integer solutions (m,n). Thanks.
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1answer
54 views

The diophantine equation $a^7+b^7=7^c$

Determine all the triples of positive integers $a,b,c$ such that $a^7+b^7=7^c$.
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0answers
36 views

Transforming the cubic Pell-type equation for the tribonacci numbers

The Lucas and Fibonacci numbers solve the Pell equation, $$L_n^2-5F_n^2=4(-1)^n\tag1$$ The tribonacci numbers $z = T_n$ are positive integer solutions to the cubic Pell-type equation, $$27 x^3 - 36 ...
1
vote
1answer
68 views

The diophantine equation $y^2=x^3+7$ has no solutions.

In my lecture notes there is the following example: The diophantine equation $y^2=x^3+7$ has no solutions. Proof: If the equation would have a solution, let $(x_0, y_0)$, $y_0^2=x_0^3+7$, then ...
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2answers
201 views
1
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1answer
122 views

Linear Diophantine Equations: Integer Solutions $x,y$ exist for $ax+by=c$, but how do I find them by hand?

I'm trying to find which of $133x+203y=38$, $133x+203y=40$, $133x+203y=42$, and $133x+203y=44$ have integer solutions. I know that only the third equation suffices for these conditions because ...
1
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0answers
42 views

Gap:$\;\;L(90,28) := {\{n90 - m28 ∈ N, n, m ∈ N, n < 28\}}$

Which elements of the sets Gap:$$L(90,28) := {\{n90 - m28 ∈ N, n, m ∈ N, n < 28\}}$$ $$$$What would be a quick way to resolve?
2
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4answers
99 views

Find all the prime integer solutions to $q^2(p-1)=(p+1)(q+1)$

Let $p,q$ be prime numbers. Find all the integer solutions to: $$q^2(p-1) = (p+1)(q+1)$$ I am almost sure that $q=2$,$p=7$ is the only solution. Thus I assumed that $p$ and $q$ were both odd to ...
5
votes
3answers
708 views

Derivation of Pythagorean Triple General Solution Starting Point:

I was reading on proof wiki about the derivation of the general solution to the pythagorean triple diophantine equation: $$ x^2 + y^2 = z^2,. $$ where $x,y,z > 0$ are integers. I came across the ...
2
votes
1answer
89 views

$x^4-2x^3+x=y^4+3y^2+y$ in the set of integers

The task is to solve the equation $x^4-2x^3+x=y^4+3y^2+y$ in integers. I expect is has something to do with factorizing but have no concrete idea; any help? thx guys
0
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1answer
67 views

An effcient method of solving a Diophantine equation with 3 variables $Ax+By+Cz=D$?

I'm trying to make an efficient algorithm to find one of the solutions and how many solutions there are to the equation $$Ax+By+Cz=D$$ where $A,B,C,D\in \mathbb Z$ and the range for $x,y,z\in \mathbb ...
3
votes
1answer
109 views

Number Theory Question: $x^2-33y^3=10$ no solutions

I've been struggling to get my head around this for a while! Show that: $x^2 - 33y^3 = 10$ has no integral solutions
0
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1answer
54 views

Solving the equation $2x^3+3x^2+x-6n^2 = 0$

It came up when I was trying to solve the equality $\sum_{i = 1}^{x}i^2 =n^2$ for integers $x$ and $i$. I've reduced it to the equation $2x^3+3x^2+x-6n^2 = 0$, which I don't know how to tackle. Is ...
5
votes
1answer
178 views

Diophantine Equation in $\mathbb{Z}$

I would like to know how to solve $2x^2 - y^{14} = 1$ in integers. I've transformed it into $(y^7 - 1)^2 + (y^7 + 1)^2 = (2x)^2$ and I have stopped here.
5
votes
2answers
55 views

Diophantine equation not solvable in $\mathbb{Q}$, but in $\mathcal{O}_p$

I'm trying to think of an example of a diophantine equation which can be solved in $ \mathcal{O}_p$ (meaning it can be solved $\mod p^k$ for all $ k $) for all prime $ p $'s, but not in $\mathbb{Q}$ ...
-1
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1answer
52 views

On no. of solutions of product of positive integers equal to sum [on hold]

$n \ge 2$ be an integer , let $a(n)$ be the no. of solutions in positive integers of $x_1+x_2+...+x_n=x_1x_2...x_n ; x_1 \le x_2 \le ... \le x_n$ , then is it true that $a(n+1)=1 \implies n$ is ...
1
vote
1answer
33 views

Solving the Diophantine equation $ax + by = c$ using Maple [closed]

I wrote a program in Maple called EEAsolve (I'm not sure how I can show everybody the code), and what it does is takes 3 parameters from $ax + by = c$: $a$, $b$, and $c$. When I run the program with ...
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0answers
28 views

Hensel's lemma in $n $ variables

I'm trying to find a proof for the following formulation of Hensel's lemma: $$\text{Let } f \in \mathbb{Z}[x_1, \dots, x_n], a = (a_1, \dots, a_n) \text{ be such that (with } p \text{ prime)}$$ $$ ...
0
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2answers
44 views

find a parametric solutions for a special equation

Let $a,b,c$ be rational.Find a rational parametric solutions for $a,b$ and $c$ so that simultaneously $$a^2-c^2=\square$$ and $$b^2-c^2=\square$$
6
votes
1answer
106 views

Solve $3x^2-y^2=2$ for Integers

If $x$ and $y$ are integers, then solve (using elementary methods) $$3x^2-y^2=2$$ I tried the following If $y$ is even, then $4|y^2$ and hence $2|y^2+2$ (and $4$ doesn't divide it), but ...
16
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2answers
1k views

Prove that $x^3 + y^3 = z^3$ has no integer solutions as briefly as possible

Can someone provide the proof of the special case of Fermat's Last Theorem for $n=3$, i.e., that $$ x^3 + y^3 = z^3, $$ has no positive integer solutions, as briefly as possible? I have seen some ...
1
vote
2answers
59 views

Perfect square given by ${r}^{2}+u\,r+v$, where r is variable and u,v are constants

I am looking for solution for a problem of finding a perfect square given by $$ {r}^{2}+u\,r+v $$ where $u > 0$ and $v > 0$ are integer constants and expected $r >= 0$. The closest I was ...
2
votes
0answers
64 views

The number of solution of a Diophantine equation

If we fixe $n\in \mathbb{N}$. I was wondring if there is an estimation of the number of the integer solutions of the equation : $$x_1^2+x_2^2+\cdots+x_n^2=n^3 $$ where $x_i>0$ for all ...
1
vote
0answers
40 views

The Diophantine equation $ax+by = p^z$.

Let $a,b \in \mathbb{Z}$ and let $p$ be a prime. Let $d = \gcd(a,b)$, and let $s$ and $t$ be integers such that $d = as+bt$. The solutions $(x,y,z) \in \mathbb{Z} \times \mathbb{Z} \times ...