Questions on finding integer/rational solutions of polynomial equations.

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25 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 ...
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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 ...
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1answer
45 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 ...
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2answers
28 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?
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3answers
121 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 ...
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2answers
55 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 ...
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1answer
54 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 = ...
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1answer
98 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 ...
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3answers
99 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 $l^2+m^2+n^2$ < ...
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0answers
22 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 ...
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0answers
33 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 ...
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2answers
66 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 ...
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3answers
87 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 ...
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2answers
127 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 ...
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2answers
189 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. ...
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6answers
294 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 ...
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1answer
117 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: $$1 + \sum_{j=1}^{n-1}\left(j \prod_{k=1}^j x_k\right) = \prod_{j=1}^n x_j$$
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3answers
71 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 $a!+b!+c!=3^d$ ?
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2answers
107 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? ...
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266 views

Every natural number is representable as $\sum_{k=1}^{n} \pm k^5$ … if somebody proves it for 240 integers

(This post is inspired by "Is every $\mathbb{N}$ representable as $\sum\limits_{k=1}^{n} \pm k^3$"? My question is at the end.) The problem of whether every natural number $N$ is, $$N=\sum_{k=1}^n ...
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1answer
132 views

Finding solutions to $y^2 = x^3 - 27$

I am trying to find integer solutions to this equation: $$ y^2 = x^3 - 27 $$ With the other problem I tried I was able to use unique factorization in $\mathbb Z [\sqrt{n}]$. I don't know how to get ...
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2answers
84 views

Characterize the solution of a Diophantine equation $x^2+py^2=z^2$ [closed]

Characterize the solution of a Diophantine equation $$x^2+py^2=z^2$$ where $p$ is a prime of the form $p\equiv 1 \pmod4$ and $(x,y,z)=1$. Consider every possible cases.
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1answer
44 views

Diophantine equation - II

Find all ordered pairs (x,y) of positive integers x, y such that $x^2+4y^2=(2xy−7)^2$. I get the ordered pair (3,2) as the only solution and I was wondering if there could be anything else. If ...
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3answers
41 views

Equation with three variables

I am confused as how to solve an equation with three squared variables to get its integer solutions? As: $$x^2+y^2+z^2=200$$ Thanks!
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2answers
93 views

For a Diophantine equation $x^2+py^2=z^2$ show that $z$ is necessarily odd.

For a Diophantine equation $x^2+py^2=z^2$ where $p$ is a prime of the form $p\equiv 1(mod4)$ and $(x,y,z)=1$. Show that $z$ is necessarily odd.
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0answers
34 views

System of linear diophantine modular inequalities

How can we best find a numerical solution to a system of $m\ge2$ linear diophantine modular inequalities $$\big((a^j x+b_j)\bmod n\big)<c\;\text{ for }1\le j\le m$$ where $x$ is the only unknown, ...
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1answer
48 views

Finding solutions of $z^2=x^2+y^2$ where $\gcd(z, y) =1$

Is there an easy way to find the solutions of $$z^2=x^2+y^2$$ where $\gcd(z,y)=1$? I apologize if this is a duplicate
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1answer
41 views

Diophantine equation : two products of linear factors differ by a constant

Recently, I was asked the following question by a friend : find all $a,b,c,a',b',c',k \in {\mathbb Z}$ with $k\neq 0$ such that the identity $$ (X-a)(X-b)(X-c)+k=(X-a')(X-b')(X-c') $$ holds in ...
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2answers
81 views

How find this diophantine equation $(3x-1)^2+2=(2y^2-4y)^2+y(2y-1)^2-6y$ integer solution

Find this following Diophantine equation all integer solution $$(3x-1)^2+2=(2y^2-4y)^2+y(2y-1)^2-6y$$ or $$9x^2-6x+3=4y^4-12y^3+12y^2-5y$$ Maybe this equation can be solved by using Pell equation ...
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1answer
64 views

Find all integers that make this expression rational

I came up with this difficult problem a while ago while solving another relatively easy problem. Find all integers m and n, such that $m^2 + n^2$ is a square, and such that ...
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1answer
52 views

Solve $y^2=x^4+x+2$ in $x,y\in \mathbb{Z}$

I have to solve $y^2=x^4+x+2$ in $x,y\in \mathbb{Z}$ and right now I am stuck. This is how far I came: A little manipulation yields $y^2-2=x(x+1)(x^2-x+1)$. $x=1$ and $y=\pm 2$ are solutions. Assume ...
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1answer
38 views

How to solve the diophantine equation $n^3-n-1=k^2-k+1$?

My first idea were some factorization-based solution. For example, adding 1 to both sides, and then: $$n^3-n-1=k^2-k+1$$ $$n^3-n=k^2-k+2$$ $$(n-1)n(n+1)=k^2-k+2$$ ...but I don't have idea, what to ...
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0answers
46 views

Solve the Diophantine equation $y^3=4x^2+4x+5$ in $x,y\in\mathbb{Z}$

I have to solve the Diophantine equation $y^3=4x^2+4x+5$ where $x,y\in\mathbb{Z}$ and I have been thinking now for a long time and I have really no clue how to do this. The only hint given in the ...
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1answer
55 views

Solve $y^2=x^3-4$ in $x,y\in \mathbb{Z}$

I am having trouble solving the diophantine equation given in the title. This is how far I came: We can factor in $\mathbb{Z}[i]$ $y^2+4=x^3\Rightarrow (y+2i)(y-2i)=x^3$. I want to show now that ...
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1answer
47 views

Diophantine equation involving factorial …

Question . Find all positive integer solutions to the equation below , $$(n-1)!+1=n^m$$ (i)observe that $n>1$ and $n$ is a prime number (if not we can choose a prime number $p<n$ such that $p|n$ ...
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Neat Diophantine Equation Question

After some fairly tedious work including studying multiple different cases separately, I have found all the solutions to $$a^n+1=b^2 $$ where $a$, $b$, $n$ can take on the value of any integer, be it ...
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1answer
125 views

When can $n^k+k$ be a perfect square?

For what positive integers $k$ does there exist a positive integer $n$ such that $n^k+k$ is a perfect square? Certainly for all $k$ such that $k+1$ is a perfect square, since we can substitute $n=1$. ...
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1answer
102 views

Congruence properties of $a^5+b^5+c^5+d^5+e^5=0$?

It is known that given a solution to, $$a^4+b^4+c^4 = d^4\tag1$$ then either $-c+d,\;c+d$ is always divisible by $2^{10}$. For example, $$95800^4+414560^4+217519^4=422481^4$$ then ...
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1answer
104 views

Is there a solution to $a^4+(a+d)^4+(a+2d)^4+(a+3d)^4+\dots = z^4$?

Is there a, $$a^4+(a+d)^4+(a+2d)^4+(a+3d)^4+\dots = z^4\tag1$$ in non-zero integers? One can be familiar with, $$31^3+33^3+35^3+37^3+39^3+41^3 = 66^3\tag2$$ I found that, ...
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1answer
87 views

Simple Question On Relationship Between Cubes And Squares

I'm new to this number theory business, not to mention terribly naive. I wonder whether someone could explain the technique (assuming there is one) to show whether the expression $12C - 3$ (where ...
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1answer
66 views

Find the integral solutions to$ x^2+y^2+z^2=x^2y^2$

I am unfamiliar with this type of problem. How does one solve this and under what category of math does this fall under. Find the integral solutions for $x^2+y^2+z^2=x^2y^2$
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1answer
255 views

How to solve $y^2=3x^4+3x^2+1$ for integers.

If $x,y \in \mathbb Z$ , then find all the solutions of $$y^2=3x^4+3x^2+1$$ I was asked this question by my friend who said that he encountered this while solving another problem. I have ...
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2answers
43 views

How do I prove that $x^s=(-1)^k \sum_{k=0}^{(r-1)/2}\binom{r}{2k}p^{s(r-2k)}$ has no solutions?

I have been struggling to prove that the following diophantine equation has no integral solutions if $r$ is odd, $s,p>1$ $$x^s=(-1)^k \sum_{k=0}^{(r-1)/2}\binom{r}{2k}p^{s(r-2k)}$$ Any hint on how ...
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0answers
28 views

Unique Solution To The Diophantine Equation

Show that the following Diophantine equation has a unique solution in positive integers $x^n+y^n=(x+y)^m$ with $x>y, m>1,n>1$. This could be solved by a direct use of Zsigmondy's theorem. ...
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1answer
46 views

Diophantin equation $a^3+b^3=c^3+5$

When trying to solve another equation, I came up with this equation: $$a^3 + b^3 = c^3 + 5, \space\space (a,b,c)\in\mathbb{Z}^3$$ It seems that it doesn't have any solutions. I tried to prove this. ...
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2answers
54 views

Diophantine equation $x^2 -y^2 = n$

Is there a method to find how much integer solutions $(x,y)$ has the diophantine equation $$x^2-y^2=n,$$ for a given $n \in \mathbb{Z}$?
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1answer
99 views

$f^2 - Dg^2 \ = \ 1 \quad \text{ with } \quad f, g \in K[X]$ not solvable?

could you help me with the following assignment? Let $K$ be a field with characteristic $0$ and $D \in K[X] \setminus K$. We write $rad(f)$ for the radical of a polynomial, the product of all monic ...
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2answers
37 views

Systems of Diophantine Equations

Find all ordered 4-tuples of integers $(a,b,c,d)$ that satisfy: $$a^n+b^n=c^n+d^n$$ for ALL positive integers $n$. Trivial solutions are $(k,p,k,p)$ and $(k,p,p,k)$ for any integers $k$ and $p$. ...
2
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1answer
36 views

Find Solutions To Some Diophantine Equations

I would like to find the solutions to $$i)\qquad a(a+b+c)=bc \\ii)\qquad a(a+b+c)=2bc \\iii)\qquad a(a+b+c)=3bc$$ for $0< a \le b \le c$ and of course: $\textrm{gcd}(a,b,c) = 1$ (since those are ...
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2answers
59 views

Proving there are infinetly many integer solutions to $ x^2 - 3y^2 = 1 $

I am trying to show that there are infinitely many solutions to the following diophantine equation: $$x^2 - 3y^2 = 1$$ But I don't really know where to start. I hear there are numerical ways to ...