Questions on finding integer/rational solutions of equations.

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3
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51 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 ...
3
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78 views

Solutions to $\prod_{i=1}^{k}p_i=\sum_{i=1}^{k}p_i^2$ with $p_i$ distincts primes

Does $\prod_{i=1}^{k}p_i=\sum_{i=1}^{k}p_i^2$ with $p_i$ distincts primes and $k\geq2$ have a solution ? Here is what I already know : There is no solutions if $k\equiv0\bmod2$ or if ...
3
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65 views

About pythagorean triples

In the circle of diameter $AB$ it is well known each point $C$ determines a right triangle $\Delta ABC$ and so it is with every point $D$ on the circle of diameter $AC$ determining a right triangle ...
3
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63 views

Pythagorean rectilinear polygons

Polygons all of whose edges meet at right angles are called rectilinear polygons. I am interested in rectilinear polygons with integer distance between each pair of vertices. Such rectilinear polygons ...
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160 views

Some Diophantine problems for equal sums with high powers

Given rationals $R = a,b,c,d,e,f$. Define, $$F_n = a^n+b^n+c^n-(d^n+e^n+f^n)$$ If $F_\color{red}1=0$, is there a rational solution to $7F_3x^4+7F_5x^2+F_7 = 0$? Then for $k=1,2,8$, ...
3
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73 views

Number of integral solutions to a polynomial

Given a polynomial of $n$th order, represented by $$f(x)=a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-2}x^{2}+a_{n-1}x+a_{n}=0$$ Is it possible to find the number of integral solutions/roots to ...
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144 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 + ...
3
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57 views

Solution of a equation in natural number nvolving reciprocal of prime

Let $p$ be a prime and $n$ a natural number . Solve in $\mathbb{N}$ the equation $$\sum_{k=1}^{n}\frac{1}{x^k_k}=\frac{1}{p}$$
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113 views

How to solve $x^4+y^4=n$?

How to solve Diophantus equation $$x^4+y^4=n $$ where $x,y$ and $n$ are positive integers. We know that Theorem: A natural numbern $n$ can be represented as a sum of two squares if and only if ...
3
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113 views

how many natural numbers on a sphere

how many natural solutions are there to the following equation: $$ \sum_{i=0}^k x_i^2 = n$$ where $n,k \in\ \Bbb{N}$ i well like to get a answer for every n and k, but could do with just $k=2,3$.
3
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350 views

Diophantine with Gaussian Integer

I'm trying to find the set of solutions to a specific diophantine equation over $\mathbb{Z}[i]$. The equation is the following: $$ z_1^2 + z_2^2 + z_1*z_2 + 39 = 0$$ with $ z_1$ (resp $z_2$) such ...
3
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401 views

$XX^t=A$, $X=?$. Where $X \in \{0,1\}^{n \times m}$

The problem: $XX^t=A$, $\quad$ ($X_{ij}\in{0,1}$, $\quad$ $\sum_{j=1}^m x_{ij}=2$), $\quad$ $X=?$ Details: $n,m \in N$ $A \in \{0,1,2\}^{n \times n}$ $X \in \{0,1\}^{n \times m}$ $A$ is a ...
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281 views

Counting Solutions of Diophantine Inequalities

I understand that Diophantine Analysis is an enormous field! Without first determining the solution set, suppose I'd like to calculate the number of non-negative integer solutions $(x,y,z)$ of ...
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214 views

Upper bound for the quality of an $abc$-triple

A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the quality or power of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log ...
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17 views

If $a\not\equiv 0\mod{p}$ then there are $p-1$ solutions (ordered pairs) to $x^2-y^2\equiv a\mod{p}$

Let $p$ be an odd prime, and let $a\in\mathbb{Z}_p$ such that $a\not\equiv 0$. I need to show that there are $p-1$ ordered pairs $(x,y)$ such that $x^2-y^2\equiv a \mod{p}$. As I see it, the ...
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39 views

Diophantine System Solution

Could you please help with finding of general solution of diophantine system for rational a, b, c, d $(a^2+b^2)(c^2+d^2)=A^2$ $(a^2-b^2)(c^2-d^2)=B^2$ for some rational A and B. This is related ...
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71 views

Quartic Diophantine equation $ 2 x^4 - 2 x^2 = 3 (y^2 - 1)$

About the quartic Diophantine equation: $$ 2 x^4 - 2 x^2 = 3 (y^2 - 1)$$ On oeis.org/A180445 it says that all positive solutions $(x,y)$ are: $$(1,1)\ \ (2,3)\ \ (3,7) \ \ (6,29)\ \ (91,6761)$$ ...
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54 views

On the integer solutions to $u^2+163v^2=w^3$ and others

It seems the solution of, $$u^2+dv^2 = w^3\tag1$$ involves the class number $h(d)$. Assume $\gcd(u,v)=1$. Q: For which $\color{red}{prime}\; d$ is the complete solution of $(1)$ in the integers ...
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58 views

Without solving it, is there an elementary way to show that $X^3+Y^3=Z^3$ has a finite number of primitive [and non-trivial] integer solutions?

Considering the cubic case of Fermat’s Last Theorem, I make the following claim: Proposition: The Diophantine equation $$ X^3 + Y^3 = Z^3 \tag{$\star$} $$ has a finite number of primitive [and ...
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21 views

Reformulating diophantine inequalities

Assume we are given inequalities $x \not\equiv a_i\text { (mod }b_i)$ for $i=1,\ldots,n$ where $1 \leq a_i \leq b_i, x \in \mathbb{Z}$. Can we somehow reformulate the problem as $x \not\equiv ...
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16 views

Largest Erdős–Diophantine graphs

A Diophantine graph is a set of vertices in the plane with integer coordinates, all at integer distances from eachother. An Erdős–Diophantine graph is a maximal Diophantine graph, so that it cannot ...
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135 views

Prove that $(a-b)^n\mid (a^n-b^n) \iff n=1$ under given conditions

Suppose that $a,b,(a-b)$ are pairwise co-prime (i.e. $a\perp b\perp (a-b)\perp a$), and that $\frac{a}{2}<b<a$, where $a$ and $b$ are both positive integers greater than $2$. Let $n$ be odd. ...
2
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82 views

An interesting equation in natural numbers

Let $n$ be a fixed natural number. How to solve the following equation in natural numbers: $$ \frac{1}{x_1} + \frac{2}{x_2} + \cdots + \frac{n}{x_n} = 1 $$ (I can find many soltions but I am looking ...
2
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57 views

Help solving the quadratic equation $ax^2-4bx+4bc-\frac{d^2}{a}=0$

I have been struggling to solve this quadratic equation in the variable $x$ with integral coefficients: $$ax^2-4bx+4bc-\frac{d^2}{a}=0$$ $a\neq 0$ of course.How do I ensure that $x$ is an integer? ...
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73 views

Which positive integers satisfies $a^{b^2} = b^a$

How one can find all integers satisfying $a\geq 1,b\geq 1,a^{b^2} = b^a$? I think that the solutions are $ (a,b)=(1, 1), (16, 2),(27, 3)$.
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46 views

Show that $x^4+py^4+p^2z^4=p^3w^4$ has no solutions, where $p$ is any prime.

I am trying to show that the equation: \begin{equation}x^4+py^4+p^2z^4=p^3w^4\end{equation} has no solutions. Assuming there is a nonzero solution $(x_0,y_0,z_0,w_0)$, with $w_0$ minimal, then it ...
2
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23 views

Constellations of three powers

How can I prove that for all $i, j \in \mathbb{N}$ there are only a finite number of solutions to $x^a + i + j = y^b + j = z^c$ with $a,b,c,x,y,z \in \mathbb{N}$ and $a,b,c \ge 2$? This is a weaker ...
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77 views

How to prove every term of this sequence is not a natural number

Sorry for the repost and for my "bad" English. I made a lot of errors in the previous one, so here's my actual question: Let's take a look at this sequence: (1) $[a_1,a_2,a_3,a_4,...,a_x]$ where ...
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23 views

$L$-existential and $L$-diophantine

Could you explain to me the last sentence: "Whenever we want to stress dependence on the language, we will use the self-explanatory terms and $L$-existential and $L$-diophantine" ? What does ...
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49 views

Finding solutions to a symmetric divisibility condition $x\mid p(y),\;y\mid p(x)$

In general, are there strategies for finding all integers $x$ and $y$ such that $x \mid p(y)$ and $y \mid p(x)$ for some polynomial $p$ with integer coefficients? For example, could we find all ...
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43 views

Diophantine eqution with odd prime

HOW to find all possible set of solutions of an equation type $y^p \pm 2 = x^2$, where $p$ is any odd prime High regards to one and all
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79 views

Finding all solutions: $a^2 + b^2 = c^2 + d^2$

I want to find all solutions to the problem of two squares equaling two other squares. $$a^2 + b^2 = c^2 + d^2 \qquad b \le N$$Clearly, without loss of generality, I can assume that $$gcd(a,b,c,d) = ...
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68 views

The exponent on Thue's theorem

I have been reading about Runge's theorem on diophantine approximation Theorem. Let $\xi$ be an algebraic real number of degree $d\geq 3$. For every $\epsilon >0$ there is a number $\gamma >0$ ...
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85 views

How to find whole number answers in systems of square root equations

Given the following 4 equations, can you find 4 whole number answers using whole number variable inputs? $x,y,z$ where $x>y>z$ $Eq 1 = (x^2-2xy+y^2-2xz+z^2)^{\frac{1}{2}} $ $Eq 2 = ...
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60 views

The Nagell-Ljunngren Equation

I have been trying to find the papers of Nagell and Ljunngren, which deal with the equation $$\frac{x^n - 1}{x - 1} = y^2$$ and solve it completely. Many papers cite these papers, but I haven't found ...
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351 views

An argument for “Brocard's problem has finite solution”

Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard%27s_problem. According to Brocard's problem ...
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62 views

Exponential diophantine: $2^x-7^y=z^2$

Find all integers $x,y,z$ such that $2^x-7^y=z^2$. For example: $2^3-7^1=1^2$ $2^5-7^1=5^2$ $2^7-7^1=11^2$ (But note that $\sqrt{2^9-7^1}\not\in \mathbb{Z}$.) The problem with this particular ...
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73 views

About Runge's method

I have been reading about some Diophantine equations (like Runge's theorem and Cassel's theorem) and in the text says that these theorems are solved using Runge's method, but it doesn't say what ...
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94 views

Transforming the cubic Pell-type equation for the tribonacci numbers

The Lucas and Fibonacci numbers solve the Pell equation, $$L_n^2-5F_n^2=4(-1)^n\tag1$$ The tribonacci numbers $z = T_n$ are positive integer solutions to the cubic Pell-type equation, $$27 x^3 - 36 ...
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0answers
85 views

The number of solution of a Diophantine equation

If we fixe $n\in \mathbb{N}$. I was wondring if there is an estimation of the number of the integer solutions of the equation : $$x_1^2+x_2^2+\cdots+x_n^2=n^3 $$ where $x_i>0$ for all ...
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87 views

How to solve this equation $x^5 +4^y=2013^z$ in positive integers?

I think to solve the equation in positive integers. It appears in a contest and I don't remember where. I obtain that $x$ must be an odd number and further $x=1 \, mod\, 4$. Any hint is appreciated.
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164 views

Primes as sum of squares.

If $p_{i}$ and $p_{j}$ are two primes of the form $4k+1$ , with $p_{j} > p_{i}$, show that if $p_{j} \neq$ sum of two squares $p_{i}$ is also not equal to sum of two squares. It is well ...
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0answers
79 views

An elliptic curve for the multigrade $\sum^8 a_n^k = \sum^8 b_n^k$ for $k=1,2,3,4,5,9$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ $$13^9+18^9+23^9-5^9-10^9-15^9 = 9^9+21^9+22^9-1^9-13^9-14^9$$ was found in 1967 by computer search by Lander et al. It stood ...
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79 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 ...
2
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69 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, ...
2
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0answers
81 views

Diophantine $7^a+2=3^b$

I want to find the solutions $(a,b)\in\mathbb{Z}^+\times\mathbb{Z}^+$ of $7^a+2=3^b$. One such solution is $(a,b)=(1,2)$. Looking modulo $4$, we have $(-1)^a+2\equiv(-1)^b$, so $a$ and $b$ are of ...
2
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0answers
136 views

Woking Heron's Formula In Reverse

I'm writing a program to generate randomized Heron's Formula word problems. I need to figure out how to work the problem in reverse so that the answer will come out to an integer. As an example, if I ...
2
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0answers
60 views

Quadruples of integers with $20^x + 14^{2y} = (x + 2y + z)^{zt}.$

Determine all quadruples $(x,y,z,t)$ of positive integers such that $$20^x + 14^{2y} = (x + 2y + z)^{zt}.$$ We can check that $20+14^2=216=(1+2+3)^3$. But how can we check if there are other ones?
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0answers
150 views

Is Legendre’s solution of the general quadratic equation the only one?

Legendre famously solved the general quadratic equation $$ ax^2+bxy+cy^2+dx+ey+f=0 $$ by noting that \begin{equation*} 4a(b^2-4ac)(ax^2+bxy+cy^2+dx+ey+f) = 0 \tag{$\star$} \end{equation*} along with ...
2
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0answers
119 views

How to solve $x^2+11=y^3$?

I've been trying to solve the diophantine $$x^2+11=y^3$$ recently but to no avail. I tried the "UFD trick", re-writing as $(x-i\sqrt{11})(x+i\sqrt{11})=y^3$, but it didn't give me all the solutions. I ...