Questions on finding integer/rational solutions of polynomial equations.

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2
votes
2answers
53 views

Solving a quadratic Diophantine equation

I want to solve the following quadratic Diophantine equation: $$\frac{x(x-1)}{y(y-1)}=\frac{p}{q} \hspace{5 mm}, \hspace{5 mm}p\le q$$ For $p=1$ and $q=2$, it is easy to solve. Let $y=x+z$. Then ...
4
votes
2answers
85 views
+50

Pentagonal Numbers

I recently was passing some time on Project Euler, when I came across this question. It deals with finding Pentagonal Numbers $P_j$ and $P_k$ such that $P_j+P_k$ and $P_j-P_k$ are also pentagonal ...
1
vote
1answer
36 views

Diophantine equation = c? [on hold]

I'm used to solving the most basic equations not specifying the c, but now I have 7106x + 4320y = 6 And I don't know how to calculate the 6
11
votes
3answers
1k views
2
votes
1answer
92 views

Difficult sets of Equations, counting

Let $ m$ be the number of solutions in positive integers to the equation $ 4x+3y+2z=2009$, and let $ n$ be the number of solutions in positive integers to the equation $ 4x+3y+2z=2000$. Find the ...
0
votes
1answer
33 views

Diophantine equation by matrice?

I want to learn how solve simple ax+by=c with matrices (assuming that's the fasted method?), but it's difficult to find correct learning material. I've been through this process: 4386x + 89744y ...
3
votes
2answers
227 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 ...
0
votes
0answers
37 views

Find all integers $m,n$ for which $m^2+n^2$ is a square and $\sqrt{\frac{2m^2+2}{n^2+1}}$ is rational

This is a repost of my old question here. The question is as follows: Find all integers m and n, such that $m^2 + n^2$ is a square and $\sqrt{\frac{2(m^2+1)}{n^2+1}}$ is rational. I have made no ...
3
votes
0answers
35 views

Can the determinant of an integer matrix with $k$ given rows be the gcd of the determinants of the $k\times k$ minors of those rows?

I'm interested if the following is true: Let $n\geq k\geq1$ be integers, let $A\in\mathbb Z^{k\times n}$ and denote the $\binom nk$ $k\times k$ minors of $A$ by $A_1,\ldots,A_N$. Then the ...
5
votes
1answer
86 views

Thue equation $ x^4 - 6 x^3 y - x^2 y^2 + 6 x y^3 - y^4=-1 $

I need help solving the Thue equation $ x^4 - 6 x^3 y - x^2 y^2 + 6 x y^3 - y^4=-1 $. It can be written as $ x(x-y)(x+y)(x-6y) = (y-1)(y+1)( y^2 +1) $. From this I found 8 solutions ...
0
votes
1answer
22 views

Does there exist a finite set of homogeneous polynomials (+ property) whose unique solution is equivalent to a finite sequence of naturals?

Consider the set $\{2,3,5\}$ of natural numbers. Letting $p = 2, q = 3, r = 5$ we have: the polynomial equations: $$p + q = r, \\ p^2 + r = q^2 \\ q^3 - p = r^2 $$ Each is a homogeneous ...
50
votes
4answers
846 views

Finding triplets $(a,b,c)$ such that $\sqrt{abc}\in\mathbb N$ divides $(a-1)(b-1)(c-1)$

When I was playing with numbers, I found that there are many triplets of three positive integers $(a,b,c)$ such that $\color{red}{2\le} a\le b\le c$ $\sqrt{abc}\in\mathbb N$ $\sqrt{abc}$ divides ...
2
votes
2answers
84 views

If $(a+b)^n=\sum_{k=0}^{n}{n\choose k}a^{n-k}b^kc_k$, then $c_k=1$?

Be advised this is a real soft question: If $$(a+b)^n=\sum_{k=0}^{n}{n\choose k}a^{n-k}b^kc_k$$ Assuming $abc \neq 0$ must we have the following condition? $$c_k=1$$ for all $0 \leq k \leq n$ How do ...
2
votes
4answers
73 views

Is there an integer solution to $x^2+1978=y^2$

Is there an integer solution to $x^2+1978=y^2$? Don't know really how to approach this. Thanks
0
votes
0answers
14 views

Book recommendations for high order diophantine equations

I'm trying to approach a problem. Basically of this form f(w,x,y,z) = 0 where f is an octic diophantine equation. I'm trying to find solutions, or conditions for where a solution exists. Can you ...
7
votes
1answer
68 views

Can the determinant of an integer matrix with a given row be any multiple of the gcd of that row?

Let $n\geq2$ be an integer and let $a_1,\ldots,a_n\in\mathbb Z$ with $\gcd(a_1,\ldots,a_n)=1$. Does the equation ...
27
votes
5answers
473 views

Why does the diophantine equation $x^2+x+1=7^y$ have no integer solutions?

This following Problem is from Pell equation chapters exercise Let $y>3$ positive integer numbers, show that following diophantine equation $$x^2+x+1=7^y\tag{1}$$ has no integer solutions. ...
0
votes
2answers
36 views

Prove that for any integers $x,y,z$ there exist $a,b,c$ such that $ax+by+cz=0$

It is rather obvious that for any 3 coprime integers $x,y,z$ there exist 3 non-zero integers $a,b,c$ such that: $$ax+by+cz=0$$ Any simple argument to prove it?
5
votes
0answers
84 views

Diophantine equation: $13^x+3=y^2$

$$13^x+3=y^2\iff \left(4+\sqrt{3}\right)^x\left(4-\sqrt{3}\right)^x=\left(y+\sqrt3\right)\left(y-\sqrt3\right)$$ $\gcd\left(y+\sqrt3, y-\sqrt3\right)=1$, therefore ...
13
votes
0answers
663 views
+100

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers ...
0
votes
1answer
24 views

Solutions to a quadratic diophantine modular equation

I wonder if solutions are known for this quadratic diophantine modular equation: x²=y² mod (p1 p2) where p1,p2 are given primes and x,y are integers and unknowns?
2
votes
1answer
101 views

How can I solve $x^2+2=y^3$ in $\mathbb{Z}$?

Prove that $\left \{ (x,y)\in\mathbb{Z}^2:x^2+2=y^3 \right \}\subseteq \left \{ (-5,3),(5,3) \right \}$.
2
votes
3answers
58 views

Solving a system of two equations

I have a system of equations: $$ \begin{cases} x\cdot y=6 \\ x^y+y^x=17 \end{cases} $$ I was able to guess that the pair $2,3$ satisfies the system, but my question is: how to solve such system of ...
3
votes
2answers
86 views

Integer solutions to $x^2 + dy^2 = c$

I'm trying to find all integer solutions of an equation $x^2 + dy^2 = c$ with $d,c \in \mathbb{Z}_{>0}$. I'm well aware of the methods that exists when $d \in \mathbb{Z}_{<0}$ or when $c$ is a ...
4
votes
2answers
97 views

Consecutive sets of consecutive numbers which add to the same total

I'm looking at examples of numbers that can be written as the sum of integers from $j$ to $k$ and from $k+1$ to $l$. For example $15$ which can be written as $4+5+6$ or $7+8$. Or $27 = 2+3+4+5+6+7 = ...
2
votes
2answers
197 views

Why is there a $p\in \mathbb{N}$ such that $mr - p < \frac{1}{10}$?

I am reading the following part of the paper of Denef : Let $R$ be a commutative ring with unity and let $D(x_1,\dots , x_n)$ be a relation in $R$. We say that $D (x_1,\dots , x_n)$ is diophantine ...
8
votes
3answers
694 views

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

Calculate all integer solutions $(x,y,z)\in\mathbb{Z}^3$ of the equation $x^2+y^2+z^2 = 2xyz$. My Attempt: We will calculate for $x,y,z>0$. Then, using the AM-GM Inequality, we have $$ ...
1
vote
1answer
91 views

How to solve special type of Diophantine equation

I am so excited to learn finding integer solutions of the equation $x^2 -y^5 = x-y$. I just found few solutions by plugging various integers in place of $x$ and $y$. But, I need a permanent method or ...
2
votes
2answers
150 views

Solving the equation $ x^2-7y^2=-3 $ over integers

I'd like to solve the following Pell equation: $$ x^2-7y^2=-3 $$ Where $x$ and $y$ are integers. I applied the usual procedure, which avoids continued fractions: The two minimal positive integer ...
2
votes
3answers
55 views

Nonlinear system Diophantus.

In the extant books of Diophantus, are considered in the system of equations. Of interest is the non-linear system of Diophantine equations. Some simple systems from his book manages to solve it. ...
0
votes
1answer
51 views

Why does this equivalence stand?

I am reading the proof of the following theorem: THEOREM A. Let $R$ be an integral domain of characteristic zero; then the diophantine problem for $R[T]$ with coefficients in ...
4
votes
1answer
82 views

Integer solutions to $x^2=2y^4+1$.

Find all integer solutions to $x^2=2y^4+1$. What I tried The only solutions I got are $(\pm 1 ,0)$, I rewrote the question as : is $a_{n}$ a perfect square for $n>0$ were $$a_0=0,\quad ...
10
votes
3answers
382 views

Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$

I am trying to find all solutions to (1) $y^3 = x^2 + x + 1$, where $x,y$ are integers $> 1$ I have attempted to do this using...I think they are called 'quadratic integers'. It would be ...
5
votes
1answer
426 views

Integer solutions of $x^4 + 16x^2y^2 + y^4 = z^2$

I come across this question very long ago. I just got one solution by my computer search. If any one know the other solutions and resolvability, please let me know. $$x^4 + 16x^2y^2 + y^4 = z^2$$ has ...
3
votes
2answers
208 views

Solutions of $x^2 + 119 = 15 \cdot 2^n$ without trial and error

I seen this equation at math.stack exchange The equation $x^2 + 119 = 15 \cdot 2^n$ has only six solutions. Those are $(1,3) ,(11, 4), (19, 5), (29, 6), (61, 8)$ and other one is I don't know. This ...
2
votes
6answers
126 views

No integer $x$ such that $(x-y)^3+ x^3 = (x+y)^3$

It seems there is no integer $x$ such that such that $(x-y)^3+ x^3 = (x+y)^3$ where $y$ is a non-zero integer. At least I can't find one. Am I right and if so, how can one show it?
2
votes
1answer
401 views

What are the necessary and sufficient conditions for a cubic equation to have integers roots

Let's start with Fermat equation with the lowest power, $x^3 + y^3 = z^3$. Now let's set $y = x + a, z = x + b$ with $b > a$ and $a,b$ integers. then the equation becomes $$x^3 + (3a-3b)x^2 + ...
1
vote
3answers
78 views

If $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$, must we have always $-\frac{a_0}{a_n} \in \mathbb{Z}$?

Let consider the polynomial with integer coefficients: $$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ If $f(x)=0$ and $x \in \mathbb{Z}$ with $a_n\neq 0$ If all the roots are integers, must we ...
10
votes
6answers
516 views

Finding a Pythagorean triple $a^2 + b^2 = c^2$ with $a+b+c=40$

Let's say you're asked to find a Pythagorean triple $a^2 + b^2 = c^2$ such that $a + b + c = 40$. The catch is that the question is asked at a job interview, and you weren't expecting questions about ...
1
vote
0answers
76 views

Solve $x^2 + 2 = y^3$ for integer $x$ and $y$ [duplicate]

I am asked to find all integers $x$ and $y$ which satisfy $x^2 + 2 = y^3$. I am given the hint that I should work in the unique factorization ring $\mathbb{Z}[\sqrt{-2}]$. So I could write the ...
0
votes
2answers
39 views

Diophantine Equations Question

The question that I am working is: Given the following diophantine equation: 53x + 12y = 2 determine the interger solutions (if any). The problem that I am facing is that I tried to find two ...
9
votes
1answer
395 views

On the equation $m^3-m^2+1 = n^2$

(i) How can I find all positive integers $m$ such that $m\equiv 4 \pmod 7$ and $m^3-m^2+1$ is a perfect square? (ii) Is there a method to solve this equation over positive integers: $$m^3-m^2+1 = ...
1
vote
6answers
101 views

Prove that the equation $\ 5x^4 + x − 3 = 0\ $ has no rational solutions.

I'm locked at $\ x\left(5x^3 + 1\right) = 3$. Not too sure where to go from there but I'm getting the feeling it's really really obvious..
1
vote
1answer
54 views

Please help understand how $ax^2+by-c=0$ is NP Complete

I found a statement that $ax^2+by-c=0$ is NP Complete. However I am unable to find any document showing the proof. There is a paper on few pay-walled sites but they are out of reach for me. The ...
8
votes
1answer
280 views

If $p$, $q$ are naturals, solve $p^3-q^5=(p+q)^2$.

In If $p,q$ are prime, solve $p^3-q^5=(p+q)^2$., the author asks to solve the equation $p^3-q^5=(p+q)^2$ for primes $p$ and $q$. A proof is given that $p=7, q=3$ is the only solution. In this ...
0
votes
1answer
52 views

Solving Diophantine Equation - odd Periods

I am trying to solve the Diophantine equation using continuous fraction . x ^ 2 - D * Y ^ 2 = 1 Keeping this document as reference http://library.msri.org/books/Book44/files/01lenstra.pdf In ...
6
votes
1answer
44 views

Find this this diophantine equation the number

Let $a,b$ be positive integer numbers. Find the number of pairs $(a,b)$ satisfying $$\dfrac{ab}{1998}=\sqrt{a^2+b^2}+a+b.$$
4
votes
0answers
89 views

Mordell Equation $y^2 = x^3 - 20$. [closed]

Prove that the only integral solutions to $y^2 = x^3 − 20$ are $(x, y) = (6, \pm14)$.
4
votes
1answer
56 views

Solving the Mordell Equation $y^2 = x^3 − 2$; what would be a general strategy?

I am looking at the solution provided in my lecture notes for solving this particular mordell equation: $$y^2 = x^3 − 2$$ which factors into: $$ (y- \sqrt {-2})(y+ \sqrt {-2}) = x^3 $$ In the ...
6
votes
2answers
86 views

Integral solutions to $56u^2 + 12 u + 1 = w^3$

I would like to find all integer solutions to $$56u^2 + 12 u + 1 = w^3.$$ My computer thinks the only integral point is $(0,1).$ This problem arises from Integer solutions of $x^3 = 7y^3 + 6 y^2+2 ...