Questions on finding integer/rational solutions of polynomial equations.

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1answer
16 views

Figuring out variable pairs in an inequality

Let $x$ and $y$ be positive integers such that $45 < 8x + 5y < 60$ How many $(x,y)$ pairs can be found? (Ans: 16) Of course, there is a way to write it one by one. On the other hand ...
0
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1answer
22 views

A ternary quadratic non-homogeneous diophantine equation in $\mathbb Z[t]$

I am interested in the diophantine equation in $\mathbb Z[t]$: $$6Z^2 + 5((t + 1)X + tY − 1)Z +((t + 1)X + tY − 1)^2+ XY = 0$$ (the unknown variables are $X,Y,Z$) Can one determine ALL the solution ...
0
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2answers
52 views

Fermat's Last Theorem - Variation with arithmetically descending exponents

Are there solution(s) to the following variant of Fermat's Last Theorem in the positive integers? $$ a^n + b^{n-i} = c^{n-2i} $$ I haven't been able to identify any trivial solutions. To my ...
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2answers
33 views

Lucas's proof of a special case of Beal's conjecture

While studying the properties of a certain elliptic curve, I came across the equation $x^4+y^4=z^3$. There is no solution of this equation in relatively prime integers, and this is a special case of ...
9
votes
4answers
152 views

Diophantine equation $(x+y)(x+y+1) - kxy = 0$

The following came up in my solution to this question, but buried in the comments, so maybe it's worth a question of its own. Consider the Diophantine equation $$ (x+y)(x+y+1) - kxy = 0$$ For $k=5$ ...
4
votes
2answers
82 views

Rational solutions to $x^4+y^4=cz^2$

Suppose $c\neq 1$ is a squarefree number, and consider the curve $x^4+y^4=cz^2$. How can I find rational points on this curve? What I really want to know is how to transform this into an elliptic ...
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0answers
18 views

Quadratic Diophantine equation

How to find the integral solutions of $$ \ x^2 + y^2 =2z^2 $$ such that x,y are distinct and $$ z^2 < 2(min(x,y))^2 $$ This can be reduced to $$ a^2 + b^2 = 2 $$ such that a,b are rational ...
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0answers
59 views

A diophantine problem with big numbers!

Find all pairs of positive integers $(a, b)$ such that $a^2 b^2 +300 \mid a^2(300 b^2 -a)$ and $300 b^2 -a>0$. I've tried so many different ways, I only concluded that $a<300$.
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1answer
36 views

Recurrence relation between solutions of a quadratic Diophantine equation

I have the fundamental solution of the following Diophantine equation: $$\frac{x(x-1)}{y(y-1)}=\frac{m}{n} \hspace{5 mm}, \hspace{5 mm} m \le n$$ $$nx^2-my^2-nx+my=0$$ Is it possible to derive a ...
1
vote
1answer
22 views

Largest Number that cannot be expressed as 6nm +- n +- m

I'm looking to find out if there is a largest integer that cannot be written as $6nm \pm n \pm m$ for $n,m$ elements of the natural numbers. For example, there are no values of $n,$m for which $6nm ...
8
votes
1answer
66 views

Primes of the form $x^2+n\cdot y^2$, given $n$?

In an attempt to get to grips with algebra for a course I intend to follow, I was working through a bunch of exercise sheets. A series of questions got me wondering: Given an integer $n$, is there ...
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
2
votes
2answers
64 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 ...
0
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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 ...
11
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3answers
1k views

Can 720! be written as the difference of two positive integer powers of 3?

Does the equation: $$3^x-3^y=720!$$ have any positive integer solution?
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0answers
38 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 ...
0
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1answer
23 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 ...
3
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0answers
37 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 ...
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4answers
76 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
2
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2answers
87 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 ...
5
votes
1answer
88 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 ...
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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 ...
2
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1answer
105 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 ...
7
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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 ...
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?
6
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2answers
117 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 ...
5
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0answers
94 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 ...
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 ...
0
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1answer
25 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
102 views

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

Prove that $\left \{ (x,y)\in\mathbb{Z}^2:x^2+2=y^3 \right \}\subseteq \left \{ (-5,3),(5,3) \right \}$.
4
votes
2answers
101 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 = ...
3
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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 ...
1
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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
152 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 ...
27
votes
5answers
493 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. ...
4
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1answer
84 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 ...
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 ...
2
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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. ...
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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 ...
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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 ...
0
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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 ...
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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..
2
votes
6answers
127 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?
1
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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 ...
6
votes
1answer
45 views

Find this this diophantine equation the number

Let $a,b$ be positive integer numbers. Find the number of pairs $(a,b)$ satisfying $$\dfrac{ab}{1998}=\sqrt{a^2+b^2}+a+b.$$
0
votes
1answer
55 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 ...
0
votes
1answer
19 views

Unique solution to $T=a_1b_1 + a_2b_2 … a_{10}b_{10}$ for some set of $a_i$?

If I have 10 types of objects of unique weights, and I know the sum of their weights and the total number of objects, and I know for each type of object, there is somewhere from 0 to 1000 of that that ...
4
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0answers
59 views

Structure of first-coordinate-projection of set of solutions of “elliptic” diophantine equation $xy(6-(x+y))=6$

Say that a rational number $a$ is good iff there is a rational number $b$ with $ab(6-a-b)=6$, or equivalently iff $a^4 - 12a^3 + 36a^2 - 24a$ is the square of a rational number. Denote by $G$ the set ...
2
votes
1answer
30 views

The diophantine problem for $R[T]$ is solvable iff the diophantine problem for $R$ is solvable

One part of the paper that I am reading is the following: Let $R$ be a commutative ring with unity and let $R'$ be a subring of $R$. We say that the diophantine problem for $R$ with coefficients ...
3
votes
3answers
85 views

When is the difference of two consecutive positive cubes a perfect square?

Are there only finitely many solutions in positive integers $m,n$ to the equation $$(m+1)^3-m^3=n^2\; ? $$