Questions on finding integer/rational solutions of polynomial equations.

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0
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1answer
55 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
2answers
62 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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0answers
31 views

Hensel's lemma in $n $ variables

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

Solve diophantine using modulus

Find all pairs of positive integers $(m, n)$ that satisfy, $mn + 3m - 8n = 59$ Using Modular arithmetic. Okay, this is a diophantine equation, where can I begin?
1
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0answers
67 views

Form of solutions Pell's equation

I'm studying a proof regarding Pell's equation. It has the form $y^2 - Dx^2 = 1$ with $D \in \mathbb{N}$. Namely that it has an infinite number of solutions if $D$ is not a perfect square. I already ...
1
vote
2answers
25 views

Diophantine equation got wrong

I am trying to understand Diophantine equation article in wiki. They say that in the given equation: $$ax + by = c$$ There will be such integers $x,y$ if and only if $c$ is a multiplier of greatest ...
2
votes
1answer
80 views

How many ordered triples $(a, b, c)$ exist?

How many ordered triples $(a, b, c)$ of positive integers exist with the property that $abc = 500$? Breaking it up, $500 = 2^2\cdot5^3$ $abc = 2^2 \cdot 5^3 = 2\cdot 2 \cdot 5 \cdot 5 \cdot ...
5
votes
2answers
78 views

How to find all integer solutions of $p^2+q^2=((2q+1)^2+q+1)^2+1$

$$p^2+q^2=((2q+1)^2+q+1)^2+1$$ How do I find integer solutions to this equation? I've already found $(p,q)=(11,1)$. How do I go about finding new ones?
0
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0answers
107 views

Diophantine equation 3-rd degree.

When I decided this Diophantine equation, it became clear. If the coefficients are expressed as follows. $$b(x^3+y^3)=az^3$$ Where $$b=q^2+3n^2$$ $$a=2(q^2-3n^2)$$ When you can represent the ...
0
votes
1answer
42 views

Solving $y^2 = 1263465 + 144x$ for integers $x,y$

I've thrown this equation up as part of some research I'm doing. $$y^2 = 1263465 + 144x$$ I was hoping there is a quick way to solve this without stepping through all the values. The value I'm ...
0
votes
1answer
30 views

Diophantine equation $^2$

Let $x,y,n \in \mathbb{N} $. The $n$ is given and then we would like to solve: $x^2 + y^2 = n^2$ Is it possible? If yes, how to do it? Thanks in advance.
0
votes
1answer
18 views

Help finding the common factor of $q^{n-1}-p$ and $kq-p$

let $p,q$ be 2 non-zero coprime integers,$n\in\mathbb{Z}>1$ and $k$ any integer. For what $k$ do $q^{n-1}-p$ and $kq-p$ have a common factor? So far, I have been able to come up only with the ...
2
votes
0answers
68 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
2answers
55 views

Diophantine equation with division

How can I find all the cases where y is positive integer in the next equation: $$\frac{ax + b}{c-x} = y$$ $a,b,c,x$ are not negative integers $a,b,x < c$ $ax + b = 0$ is a trivial solution
1
vote
1answer
62 views

Diophantine Equation: $a^3=a(b^2+c^2+d^2)+2bcd$

Let $a,b,c,d\in \mathbb{Z}$. Solve $$a^3=a(b^2+c^2+d^2)+2bcd$$ I've tried everything but I haven't been able to find a general solution. Note: We may assume $\gcd(a,b,c,d)=1$ because of homogeneity. ...
0
votes
2answers
39 views

On the Diophantine Equation $(x-h)^2+(y-k)^2=c$

I am just curious about the equation of the circle centered at (h,k) whose form is we know $(x-h)^2+(y-k)^2=r^2$. If we consider its solution over the set of integers then we have a Diophantine ...
0
votes
1answer
54 views

Solving Diophantine equation $1/x^2+1/y^2=1/z^2$

How can we find positive integers solutions $(x,y,z)$, where $\gcd(x,y,z)=1$ for the equation: $$1/x^2+1/y^2=1/z^2$$ Can we conclude that $x$ and $y$ are not coprimes for it to have solution?
2
votes
0answers
70 views

How to solve this equation $x^5 +4^y=2013^z$ in positive integers?

I think to solve the equation in positive integers. It appears in a contest and I don't remember where. I obtain that $x$ must be an odd number and further $x=1 \, mod\, 4$. Any hint is appreciated.
2
votes
1answer
114 views

Solutions to $y^2 = x^3 + k$?

As you know, the equation $y^2 = x^3 + k$ for $k = (4n-1)^3 - 4m^2$, with $m, n \in \mathbb{N}$ and no prime number that p is congruent to 1 modulo 4 count m, don't have any answer and its proof can ...
5
votes
0answers
135 views

Primes as sum of squares.

If $p_{i}$ and $p_{j}$ are two primes of the form $4k+1$ , with $p_{j} > p_{i}$, show that if $p_{j} \neq$ sum of two squares $p_{i}$ is also not equal to sum of two squares. It is well ...
0
votes
2answers
36 views

Solve $x^3-ax=by$ If $\gcd(x,y)=1$

Solve the diophantine $x^3-ax=by$ If $\gcd(x,y)=1$. Any hint? My first impression is $\gcd(x,b)>1$ and $x^2=ky+a$ for some integer $m$. I conclude that as long as there exists a square integer ...
0
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0answers
37 views

Simultaneous Diophantine Equations

Is there an elegant way to determine whether there is more than one integer solution (not counting plus or minus the same value) in the diophantine equations $x^2 + m=y^2$ $x^2+n=z^2$ where $m$ and ...
0
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0answers
20 views

Sparse Diophantine Linear Equations System

Do you know any paper/algorithm which deals with Solving the general solution of a sparse Diophantine Linear Equations System?
3
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0answers
149 views

Some Diophantine problems for equal sums with high powers

Given rationals $R = a,b,c,d,e,f$. Define, $$F_n = a^n+b^n+c^n-(d^n+e^n+f^n)$$ If $F_\color{red}1=0$, is there a rational solution to $7F_3x^4+7F_5x^2+F_7 = 0$? Then for $k=1,2,8$, ...
4
votes
2answers
59 views

Polynomial Diophantine Equation

If $x$,$y$ $\in \mathbb Z$, find all the solutions of $$y^3=x^3+8x^2-6x+8$$ I have tried factorizing the equation but the polynomial on $\text{R.H.S.}$ doesn't have any integral roots. ...
0
votes
1answer
46 views

Prove this diophantine equation $b^2=a^3+ac^4$have no integer solution,

show that this diophantine equation: $$b^2=a^3+ac^4$$ has no soluton in non-zero integers [Hint: first show that $a$ must be a perfect square] This problem is from this PDF I know this ...
1
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0answers
48 views

An elliptic curve for the multigrade $\sum^8 a_n^k = \sum^8 b_n^k$ for $k=1,2,3,4,5,9$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ was found in 1967 by Lander et al. In 2010, Bremner and Delorme found it had the highly structured form, $$\small(u + 9)^k + (u ...
0
votes
1answer
47 views

Another look at the trinomial of the form: $ax^n+bx+c=0$

Has the trinomial of the form $ax^n+bx+c=0$ been fully studied for $n>2$? If so, please let me know of any reference or interesting findings. Thanks.
2
votes
3answers
80 views

Why are there no integer solutions to $m^2 - 33n + 1 = 0$?

How many solutions does the equation $m^2-33n+1=0$, where $m,n\in\mathbb Z$, have? The answer is no solutions exist. But why?
3
votes
3answers
82 views

Solving $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = m \in \mathbb{Z}$, $\frac{a}{c} + \frac{c}{b} + \frac{b}{a }= n \in \mathbb{Z}$

Whether non-zero integers $a, b, c$ with the property that $$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = m \in \mathbb{Z}$$ and $$\frac{a}{c} + \frac{c}{b} + \frac{b}{a }= n \in \mathbb{Z}$$ Calculate ...
1
vote
1answer
26 views

Solving $y^{ax}=x^2b$ over integers

Let $x,y,a,b \in \mathbb{Z}>1 $and $\gcd(x,b)=1,$ $y^{ax}=x^2b$, I cannot find any integral solution. What I have done so far: I assume there must be 2 coprime integers $c, d>1$ such that ...
1
vote
1answer
45 views

How to solve $x^2-4y=m^2$ where $m$ is given?

Respected all Kindly help me to solve the following diophantine equation. The equation is given by $x^2-4y=m^2$ where $m\in \mathbb Z$ is given. How to solve this equation in integers? I have read ...
0
votes
2answers
61 views

How to solve $(xy)^2+a(xy)+bx+cy+d=0$ in integers?

Respected all. We know that $x^2+y^2+2gx+2fy+c=0$ represents a circle and the parametric solution for it is $x=\cos t, y=\sin t$. But I was wondering what would happened for the following equation ...
1
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2answers
37 views

Solving diophantine equation in two variables

We need to find all positive integer solutions for the equation: $$ {x^2+6 x y+ 10 x+30 y -1470}= 0$$ How can we determine these solutions?
5
votes
3answers
131 views

How find this diophantine equation integer solution $a^3+b^3=(2ab+1)^2$

Find this following diophantine equation integer solution $$a^3+b^3=(2ab+1)^2$$ I think this equation only have two following solution $$(a,b)=(1,0),(0,1)$$ maybe this equation have no other ...
1
vote
2answers
57 views

Diophantine Equation With Varying Exponents

I am considering the following Diophantine Equation - the approach I tried became the study of too many different cases - so many that I left it and tried to find an easier way. I wonder if anyone ...
1
vote
1answer
55 views

Are there solutions to FLT which are linearly independent over $\mathbb{Z}$

Specifically, I would like to know if there is some $R$, where $R$ is a ring with unity $\mathbb{Z} \subseteq R$ there are $x,y,z \in R$ and a prime $p \in \mathbb{Z}$ such that $x^p + y^p + z^p = ...
1
vote
1answer
105 views

Solving $a^5=a^3bc+b^2c$ in integers

Solving $a^5=a^3bc+b^2c$ in integers. I tried assuming there is a common divisor first of a,b,c, then a,b and 2 other pairs, but not sure how to arrive to a contradiction, trying some things right ...
3
votes
3answers
108 views

Diophantine equation $l^2+m^2+n^2=p^3+q^3$

I'm not familiar with Diophantine equations. I would like to solve the following equation: $$l^2+m^2+n^2=p^3+q^3$$ where $l,m,n,p,q\in\mathbb{N}$. I need a list of solutions where $l^2+m^2+n^2$ < ...
0
votes
0answers
28 views

Solving a simple diophantine equation

Solve the diophantine equation: $f(x) = 4x+10y=16$ We have that: $\gcd(4, 10) = 2 \implies f(x) \iff 2x+5y=8$ And: $5 = 2\cdot2 + 1 \iff 1 = 5 - 2\cdot2 = -2\cdot2 + 1\cdot5$ From which we can ...
2
votes
0answers
41 views

Properties of non-equivalent solutions to the generalized Pell equation

Given the Diophantine equation $$ r^2-ds^2 = x^2-dy^2 = q, $$ (where $q$ is a potentially unknown integer, and certainly need not be $1$), the two solutions $(r,s)$ and $(x,y)$ are called equivalent ...
2
votes
0answers
45 views

Number of integral solutions to a polynomial

Given a polynomial of $n$th order, represented by $$f(x)=a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-2}x^{2}+a_{n-1}x+a_{n}=0$$ Is it possible to find the number of integral solutions/roots to ...
2
votes
2answers
69 views

Consecutive cubes equal to a square $\frac{1}{8}ab(a^2+b^2-1) = y^2$, and Pythagorean triples

If we wish that the sum of $b$ consecutive cubes with initial cube $c=\tfrac{1}{2}(1+a-b)$ is equal to a square, then we have the rather simple equation, $$F_k=\tfrac{1}{8}ab(a^2+b^2-1) = y^2$$ It ...
3
votes
3answers
96 views

Pythagorean type diophantine equation.

How to find all solutions to $$ a^2+b^2+c^2+d^2=e^2+2$$ where all variables $a$ to $e$ are positive integers and $e^2 \equiv 1 \mod 8$ I tried using parameterization similar to ...
10
votes
2answers
130 views

Show $1+x+(x^2/2!)+ \cdots + (x^n/n!)=0$ has no rational solutions for all $n>1$.

Prove that the equation $$1+x+\frac{x^2}{2!}+ \cdots + \frac{x^n}{n!}=0$$ has no rational solutions for all $n>1$. Assume there is a rational solution $\frac{p}{q} \in \mathbb{Q}$ with ...
3
votes
2answers
200 views

Solving an equation for two primes

This is from contest preparation: Find all pairs of primes $(p, q)$ that satisfy $$p^q - q^p = p q^2 - 19$$. It looks simple, but I spent hours trying to solve it... and no luck so far. ...
7
votes
6answers
300 views

How prove this diophantine equation $(x^2-y)(y^2-x)=(x+y)^2$ have only three integer solution?

HAPPY NEW YEAR To Everyone! (Now Beijing time 00:00 (2015)) Let $x,y$ are integer numbers,and such $xy\neq 0$, Find this diophantine equation all solution $$(x^2-y)(y^2-x)=(x+y)^2$$ I ...
1
vote
1answer
120 views

Diophantine equation $1 + \sum_{j=1}^{n-1}\left(j \prod_{k=1}^j x_k\right) = \prod_{j=1}^n x_j$

What are the positive solutions $(x_1,x_2,\ldots,x_n)$ for the Diophantine equation: $$1 + \sum_{j=1}^{n-1}\left(j \prod_{k=1}^j x_k\right) = \prod_{j=1}^n x_j$$
4
votes
3answers
77 views

How to find all positive integers $a,b,c,d$ with $a\le\ b\le c$ such that $a!+b!+c!=3^d$ ?

How to find all positive integers $a,b,c,d$ with $a\le\ b\le c$ such that $a!+b!+c!=3^d$ ?
-1
votes
2answers
112 views

AMC Putnam 1986 № B2

There was one task on the competition http://kskedlaya.org/putnam-archive/ I'm not much will change. Is it possible to solve such a system of Diophantine equations? ...