Questions on finding integer/rational solutions of polynomial equations.

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2
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0answers
20 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 and let $$A=\begin{pmatrix}a_{11}&\cdots&a_{1n}\\ \vdots&\ddots&\vdots\\ ...
2
votes
4answers
59 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
votes
2answers
78 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 ...
3
votes
0answers
61 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
0answers
13 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
votes
1answer
56 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 ...
6
votes
1answer
57 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?
4
votes
1answer
38 views

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
votes
0answers
80 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
56 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
votes
1answer
22 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
99 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
96 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
votes
2answers
84 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
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 ...
27
votes
5answers
403 views
+50

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
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 ...
0
votes
1answer
50 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
votes
3answers
52 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. ...
2
votes
2answers
194 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 ...
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 ...
0
votes
2answers
36 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 ...
1
vote
6answers
100 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
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?
1
vote
1answer
53 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
44 views

Find this this diophantine equation the number

Let $a,b$ be positive integer numbers. Find the number of pairs $(a,b)$ satisfying $$\dfrac{ab}{1998}=\sqrt{a^2+b^2}+a+b.$$
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 ...
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
votes
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
29 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 $$(m+1)^3-m^3=n^2\; ? $$
4
votes
4answers
443 views

Proof of inequality without calculus

So we are given the equation $3x+4y+xy=2012$ where $x$ and $y$ are positive integers. Prove that $x+y\geq83$. Using calculus optimisation methods, this can be proved. However, it requires a lot ...
2
votes
1answer
99 views

Quartic polynomial taking infinitely many square rational values?

I was wondering whether the value of $$P(x)=x^4-6x^3+9x^2-3x,$$ is a rational square for infinitely many rational values of $x$. Is there a general method to check this for a polynomial (in one ...
5
votes
2answers
124 views

Homogeneous diophantine equation $x^3+2y^3+6xyz=3z^3$

Is it known if there are infinitely (non-proportional) many integer solutions to $x^3+2y^3+6xyz=3z^3$ ? Motivation : if true, this would provide an alternative solution to that recent MSE question, ...
1
vote
1answer
49 views

How one can solves an equation of the form: $ap_{n}+bn=c$

My question is: How can one solve an equation of the form: $$ap_{n}+bn=c$$ where $p_{n}$ is the $n^{th}$ prime number, $a,b$ and $c$ are integers.
0
votes
0answers
48 views

NT problem (use mod) Solve this equation

This might not be solvable, but my NT teacher wanted us to give it a try. Solve $x_1^4 + x_2^4+\cdots+x_7^4 = 1\,000\,007$. What has been figured out so far: Mod 8 to get the equation = 7 (mod ...
3
votes
1answer
79 views

Why do the Diophantine equation $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=n$ gives an elliptic curve?

In a book "Which way did the bicycle go" was tought a problem of integer solutions of certain Diophantine equation. This is the idea, not an exact quotation: For which integers $n$ are there integers ...
-1
votes
0answers
72 views

Solving quadratic diophantine equation in two variables

Looking at the following quadratic diophantine equation: $r^2-(12r+ 100)x+36x^2$ = $y^2$ where r is a known positive integer and x and y unknown positive integers. How can we in general obtain ...
4
votes
0answers
82 views

Integer solutions to the equation $a^3+b^3+c^3=30$

The following problem was posed to me but I could not do much about it: Determine if there are any integer solutions to the equation $a^3+b^3+c^3=30$ I made a computer search that shows that ...
8
votes
3answers
186 views

Why does $x^2+47y^2 = z^5$ involve solvable quintics?

This is related to the post on $x^2+ny^2=z^k$. In response to my answer on, $$x^2+47y^2 = z^3\tag1$$ where $z$ is not of form $p^2+nq^2$, Will Jagy provided one for, $$x^2+47y^2 = z^5\tag2$$ as, ...
0
votes
2answers
59 views

Diophantine Equation Without Using Fermat's Last Theorem

I'm having trouble with a problem. The problem asks me to solve the equation $(x+1)^4-(x-1)^4=y^3$ in integers. I found out that the only integer solution is $(0,0)$. I found this answer by setting ...
1
vote
3answers
68 views

Number theory problem and Diophantine Equations

Suppose $m^3=n^4-4$ where $m,n \in \mathbb Z$. a) Show that $m$ cannot be even if $n$ is odd. b) Show that $m$ and $n$ cannot both be even. c) By considering the prime factors of ...
5
votes
3answers
106 views

Parametrization of $x^2+ay^2=z^k$, where $\gcd(x,y,z)=1$

$x,y,z$ be three coprime integers, $a \in \mathbb{Z}>0$ and $k$ an odd integer. How do I find all the non-trivial solutions of the diophantine equation? $$x^2+ay^2=z^k$$ Does the method which ...
0
votes
0answers
31 views

Simpler proof of the lemma: if $\gcd(a,b)=1$ then all odd factor of $a^2 + 3b^2$ has the same form?

Lemma: If $\gcd(a,b)=1$, then every odd factor of $a^2 + 3b^2$ has this same form. I was reviewing the proofs for this lemma online. Every proof is long and cumbersome. Is there a simpler method to ...
4
votes
1answer
74 views

Solution to Diophantine equation $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2} $

I have to prove the following, but I don't know how to start. The only solutions in positive integers of the equation $$ \frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2} \qquad \gcd(x,y,z)=1 $$ ...
0
votes
3answers
63 views

Solving this Cubic equation

$(x^2+y)(x+y^2)=(x+y)^3$ Can $x^2+y^2$ attain values $2$ and $13$? How to approach this question I tried solving this equation and couldn't solve after this: $$xy+1=3(x+y) $$
-1
votes
2answers
100 views

Solve $x,y\in \mathbb{Z}$ [closed]

Solve for $x,y\in \mathbb{Z}$ $$x^{6}=y^{2}+53$$ I tried but I couldn't complete
1
vote
1answer
52 views

All integer solutions to diophantine equation: $x^2+p y^2=z^2$?

I would like to find all integer solutions to the Diophantine equation $$ x^2+p y^2=z^2 $$ where $p\ge2$ is a given prime number. Also prove that my (probably parametric similar to Pythagorean triples ...