Questions on finding integer/rational solutions of polynomial equations.

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2
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0answers
42 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 ...
2
votes
1answer
44 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 ...
1
vote
1answer
26 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 ...
1
vote
0answers
37 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 ...
1
vote
1answer
49 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 ...
3
votes
1answer
34 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 $p|n$ ...
-1
votes
2answers
18 views

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 ...
6
votes
0answers
53 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 $n=1$. ...
5
votes
0answers
28 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 ...
1
vote
0answers
42 views

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

One can be familiar with, $$31^3+33^3+35^3+37^3+39^3+41^3 = 66^3\tag{1}$$ I found, $$29^4+31^4+33^4+35^4+\dots+155^4 = 96104^2\tag2$$ which has 64 addends. The equation, ...
7
votes
1answer
82 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 ...
0
votes
1answer
59 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 $x^2+y^2+z^2=x^2y^2$
0
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0answers
27 views

For which values of $n$ does $x^n+y^n=i$ has a zeros in $\mathbb{R}$? [on hold]

$ x $ and $y$ are real numbers and $i$ : is unit imaginary part . 1-for which values of $n$ does $x^n+y^n=i$ has a zeros in $\mathbb{R}$ ? 2-what are the possible geometrics forms of $x^n+y^n=i$ ...
8
votes
1answer
190 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 ...
0
votes
2answers
41 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 ...
1
vote
0answers
25 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. ...
2
votes
1answer
40 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. ...
2
votes
2answers
46 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 $n \in \mathbb{Z}$?
0
votes
1answer
97 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 ...
1
vote
2answers
36 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
votes
1answer
27 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 ...
1
vote
2answers
56 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 ...
0
votes
0answers
34 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 $(-2, 1)$?
0
votes
1answer
51 views

Find all $x,y\in\mathbb{Z}^+$ such that $2014^x+11^x=y^2$

Find all $x,y\in\mathbb{Z}^+$ such that $$2014^x+11^x=y^2$$ In my book it says that only solution is $(x,y)=(1,45)$, but solution is very complicated. They proved that $(x,y)=(1,45)$ is only solution ...
0
votes
1answer
25 views

Existance of solution of $Ax < b$

How to check if the inequality $Ax < b$ admits at least one solution. Entries of $A$, $x$ and $b$ are taken in $\mathbb{Z}$
0
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0answers
8 views

Can inequality $-1<(x-\tfrac{1}{2})^2 - 3 (y-\tfrac{1}{2})^2 < 1$ be solved with continued fractions?

It's known at Pell's equation $x^2 - 3 y^2 = 1$ can be solved using the periodic continued fraction expansion of $\sqrt{3}= [1;\overline{1,2}]$. Eventually we get convergents $\tfrac{p}{q} \approx ...
0
votes
1answer
21 views

One solution of a diophantine system

How to find one solution of $Ax = b$, where $A$ is a $(m, n)$ matrix and $x$ a vector of size $(n, 1)$. $A$, $x$ and $b$ are matrices of integers entries. How to check whether is a solution exists?
8
votes
3answers
115 views

$x^n + y^n = c$ has finitely many integral solutions?

Assume $n > 1$ and $n$ is odd because it's easy if $n$ is even. Please help prove this. $x^n + y^n = c$ has finitely many integral solutions if $c \neq 0$? Thank you all for replying. I think ...
0
votes
0answers
62 views

Parametric solutions to the Pell equation $x^2-dy^2=-4$?

I'm looking for identities for the fundamental solutions of, $$x^2-dy^2 = -4\tag{1}$$ The only one I know is, $$\begin{aligned} x \,&= m + 3 n + 2 m n^2 + n^3 + m n^4\\ d \,&= 4 + m^2 + 6 m ...
3
votes
0answers
28 views

For fixed $m$, find all positive integer solutions to $a^m+b^m = p^n$ where $p$ is prime

The problem: For fixed $m$, find all solutions to $a^m+b^m = p^n$ where $p$ is prime, all variables are positive integers, $a \le b$, and $m \ge 2$. This is a generalization of this question: ...
1
vote
0answers
25 views

Diophantine solutions to $x^y-y^x=1$ [duplicate]

$x^y-y^x=1$ for $x,y\in\Bbb Z$ and $x,y>1$ is clearly a special case of very well known Catalan's conjecture (now resolved). It seems to be very limited special case, but I was told by someone ...
0
votes
0answers
27 views

How to solve a Diophantine equation?

Has the equation $$-x^6 + 3x^4y + 9x^4 - 7x^3 - 3x^2y^2 - 9x^2y - 18x^2 - 3xy^2 - 18xy + 27x + y^3 + 18y^2 + 54 = 0$$ more integer solutions than $(x,y)\in \{(-3,0),(3,6),(-2,-4)\}$? Is there any ...
2
votes
2answers
121 views

Diophantine equation with bounded variables

I was reading the follow paper about cryptography: Fully Homomorphic Encryption over the Integers, but I faced a proof of Lemma which I didn't undestand: the Lemma A.1. In short, there is a ...
17
votes
3answers
2k views

Monstrous Diophantine Equation

If $x,y\in\mathbb{Z}^+$, then find all the integral solutions to: $$x^6-y^6+3x^4y-3y^4x+y^3+3x^2+3x+1=0$$ I tried solving this question for an hour but still couldn't get it. I tried mod ...
3
votes
0answers
60 views

A Tale of Two Quadratic Identities (Pell-like)

Question is at the end. Let all variables be integers. For some constants $a,b,c,d$, assume we have initial solution {$m,n$} to, $$a m^2 + b m n + c n^2 = d\tag{1}$$ Identity 1: $$a x^2 + b x y + ...
0
votes
0answers
24 views

Given $p=(m+n)/(u+v)$, express $p$ in terms of $m/u$ and $n/v$

Given $p=(m+n)/(u+v)$, express $p$ in terms of $m/u$ and $n/v$. My attempt; dividing numerator and denominator by $uv$ we obtain $p=((m/u)*1/v+(n/v)*1/u)/(1/u+1/v)$ but am stuck here
0
votes
1answer
27 views

Equality of Integer Powers

I know the following can be proven with logarithms, but I was hoping it could be proven without them. However, I've tried for several days, and I couldn't think of a proof. So I'd like to see what you ...
3
votes
1answer
83 views

Number theory: $x^y + 1 = y^x$

Today a friend told me the equality: $2^3 + 1 = 3^2$, and i wondered if there exist more solutions to the general problem $$x^y + 1 = y^x$$ where $x$ and $y$ are integers. Some research led me to the ...
1
vote
3answers
102 views

Find all solutions in positive integers of the diophantine equation $w^2+x^2+y^2=z^2$

It's an exercises of the text book Elementary Number Theory and It's Applications 6th Edition by Kenneth H.Rosen. I wanted to solve it using the method in solving the diophantine equation ...
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3answers
21 views

Let $ a$ be a positive integer. Show that $\text{gcd}(a,a-1) = 1$. use the result of par t $ a)$ to solve the Diophantine equation $ a+b=ab$

Not sure if I did part a right, not sure how to complete part $b)$ $a)$ Let $a$ be a positive integer. Show that $\text{gcd}(a,a-1) = 1$. Proof by contradiction suppose $\text{gcd}(n, n-1) = p > ...
5
votes
3answers
64 views

Eisenstein integers and applications to Diophantine equations

Solve the equation $7\times 13\times 19=a^2-ab+b^2$ for integers $a>b>0$. How many are there such solutions $(a,b)$? I know that $a^2-ab+b^2$ is the norm of the Eisentein integer $z=a+b\omega$, ...
0
votes
2answers
20 views

Two equivalent equation

Recently, one of my friends have told me that the following two equations are equivalent on the basis of the number of solutions. I checked the number of solutions to the two equations and found ...
1
vote
0answers
37 views

Theorem for Equal Sums of Like Powers $x_1^8+x_2^8+x_3^8+\dots$

Kindly see the question at the end of post. Solutions to the system of three equations, $$\begin{aligned} a^2+b^2+c^2+d^2\, &= e^2+f^2+g^2+h^2\\ a^4+b^4+c^4+d^4\, &= e^4+f^4+g^4+h^4\\ abcd\, ...
2
votes
3answers
89 views

Sinha's Theorem for Equal Sums of Like Powers $x_1^7+x_2^7+x_3^7+\dots$

Sinha’s theorem can be stated as, excluding the trivial case $c = 0$, if, $$(a+3c)^k + (b+3c)^k + (a+b-2c)^k = (c+d)^k + (c+e)^k + (-2c+d+e)^k\tag{1} $$ for $\color{blue}{\text{both}}$ $k = 2,4$ ...
1
vote
2answers
28 views

how many positive integer solutions to the following equation?

$a^2 + b^2 + 25 = ab + 5a + 5b$ I have tried looking for a factorisation that could solve this question but couldn't find anything useful - found $(a+b+5)^2$ - don't know if this is useful The ...
1
vote
1answer
26 views

How to show there are infinite solution to a given Pell's equation?

I was asked to prove the Pell's equation $$x^2-7y^2=1$$ has infinitely many solution. Here is what I did By using Brahmagupta method we can generate infinitely many integer solutions. Is that ...
0
votes
0answers
36 views

Finding set of integer pairs for which two integer polynomials intersect

I am wondering if there is a theorem in number theory that addresses the following issue: Suppose we have two polynomials, f and g, with integer coefficients. Is there a general way to find elements ...
0
votes
0answers
16 views

Number of solutions of a diophantine inequality

In my problem i am looking for the number of the nonnegative integer pairs $(x,y)$ which satisfy the inqueality $x+my\leq n$, where $m$ and $n$ are coprime integers. The answer in the book is given ...
4
votes
1answer
58 views

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

I want to solve equation $x^4 - 2x^3 + x = y^4 + 3y^2 + y$ in integers. The task comes from the LXVI Polish Mathematical Olympiad. Series with this task ended twenty days ago, so it is legal to talk ...
7
votes
0answers
166 views

Number of ways to express a binary number in a certain way

So I'm working on a problem where I get to a point where I have to count the number of solutions to an equation or at least find a decent upper bound to be used in an estimate I need later. The ...