Questions on finding integer/rational solutions of polynomial equations.

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Solving a general Diophantine Equation

For "normal" equations in one variable we have several techniques for solving equations, such as $\sin(5x) = 5\pi\cos(5x)$ or $\ln(x + 2) = 4$. However, imagine we have the following equations: ...
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0answers
44 views

Looking for solutions to $xy^2 = (1 + z)^2 (5 + 8z)$ in integers

I've been reading about Weierstrass equations and shifted Weierstrass equations and Mordell curve and elliptic curves, but so far I haven't been able to transform my equation to any of this type. ...
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0answers
60 views

How to solve the diophantine equation $x^y = y^x $? [duplicate]

I know this might be an obvious question as we all know that the answers (beside $(x,y)=(1,1)$) are $(x,y)=(2,4)$ but the problem is, how is this exactly solved? Tags might be inaccurate so feel free ...
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2answers
90 views

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

Given the equation: $$x^2+y^2+z^2=kxyz$$ with: $(k,x,y,z)\in\mathbb{N}$, the only solution for $k=2$ is: $x=0,y=0,z=0$. For what values of $k$ the equations has solutions in which $x,y,z$ are ...
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2answers
271 views

Is $7^{8}+8^{9}+9^{7}+1$ a prime? (no computer usage allowed)

Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer. I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.
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0answers
29 views

Polynomials that represent a function

Let $D(x,n_1,\dots,n_k) \in \mathbb{Z}[x,n_1,\dots,n_k]$ be a polynomial. Every such polynomial represents a semi-decidable property of natural numbers by $$P(x) :\equiv (\exists n_1,\dots,n_k)\ ...
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0answers
37 views

Matrix Multiplication Integer Solution

Given a matrix multiplication and a vector addition. (A,b has rational entries) $$Ax+b$$ how do i get an $x$ for that $Ax+b$ is integer or show that there is not such a solution? $x$ has no ...
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2answers
35 views

Finding Solutions to a Diophantine Equation with Factorials

How many ordered pairs of positive integers $(a, b)$ are there such that $a!+\dfrac{b!}{a!}$ is a perfect square? Is the number of solutions finite? Source: Ran into it on Facebook. I have plugged ...
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1answer
91 views

An operation with respect to which the set of prime numbers is closed

Like every (semi-)decidable set of natural numbers the set $P$ of prime numbers is diophantine, i.e. there are two polynomials $p(x)$, $q$ with natural coefficients and exponents – the first of ...
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2answers
33 views

Rational Number of a given fraction

Find all rational numbers $\frac pq$ such that $\frac pq=\frac {p^2 +30}{q^2 +30}$. How can I go about it. If I substitute p and q by real values $\frac pq$ gets innumerable rational numbers
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1answer
75 views

Has anyone solved this general Diophantine Equation?

I know that Pythagorean triples have been parameterized, I also know that Andrew Wiles has proved that there are no distinct integer solutions for $ a^n + b^n = c^n$, when $ n \ge 3 $. However we may ...
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0answers
21 views

Diophantine equations (Mordell theorem)

I have a really serious problem with this exercise, I don`t know how I can resolve it. Could you help me? I study in Spanish, so if you don't understand my translation, please ask me... We have the ...
2
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1answer
98 views

Solutions to a diophantine equation

I tried to find integer solutions to the following diophantine equation $$x^3 - 3y^3 + 5z^3 - 3xy^2 + 3x^2y + 9xz^2 + 7x^2z + 3yz^2 - 3y^2z + xyz = 0$$ but was unable to do so. I suspect that there ...
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0answers
30 views

Differential Diophantine Equations?

So this is both a question on its own as well as a request for where I can find information on a given topic. Consider Differential Equations in two variables of the form: $$P(Z,Z', Z'' ... Z^{[n]}, ...
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0answers
48 views

Diophantine eqn, general solution?

Here's the equation: $$ 4 \left( x^2+y^2-z^2 \right)=\left( 2k+1 \right) \left( x+y-z \right) $$ Is there a nontrivial solution for this in integers? If not, why not? If there is, can a general ...
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3answers
90 views

How to solve $b^2-a^2=d^2-c^2$

I'm looking for how to solve the equation $b^2-a^2=d^2-c^2$ where $a,b,c,d$ are naturals and $d>c>b>a>0$ , an algorithm would be appreciated Regards
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2answers
37 views

Quarters weigh 6 grams while dimes weigh 2 grams.

Quarters weigh $6$ grams while dimes weigh $2$ grams. Tiffany has $\$5.35$ worth of quarters and dimes in her pocket weighing a total of $124$ grams. How many quarters does Tiffany have?
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3answers
132 views

Solving $a^2+3b^2=c^2$

I'm looking for how to solve the equation $a^2+3b^2=c^2$ where $a,b,c$ are integers and $b$ is even, I'm looking for the algorithm used to solve this kind of equations, not just the solution. Regards ...
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1answer
39 views

The sum of the cubes and the amount of combinations.

Quite simply turned out to solve this Diophantine equation, when he made the assumption that the solutions of these equations symmetric. So given this equation: ...
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0answers
141 views

$(b-a)^2-2ab$ is a perfect square.

I'm in need of some help if possible, about a formula, theorems, old works, ideas, or even an existing solution are welcome. The problem is that i have two distinct natural numbers as $b > a > ...
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1answer
55 views

Special kind of a linear Linear Diophantine equation

Could any one help me to point out some literature/ papers which solves a homogenous linear Diophantine equation (one equation) of the form $a_1 \times x_1+a_2 \times x_2 + a_3 \times x_3+....+a_n ...
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2answers
43 views

$A^7 \not\equiv A(\mod 13) \Rightarrow A^{78} + 1 \equiv 0 (\mod 169)$

Let variable $A$ is integer and $A^7 \not\equiv A(\mod 13)$. Prove that $A^{78} + 1 \equiv 0 (\mod 169)$ Could someone explain, how to solve this type of problems? Any help would be greatly ...
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2answers
43 views

Two diophantine equations with lots of unknowns

Is it possible (tractable) to determine if the following system of equations has any nontrivial solutions (ie, none of the unknowns are zero) in the domain of integers? $$A^2 + B^2=C^2 D^2$$ $$2 C^4 ...
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2answers
69 views

Need help with a diophantine expression

I'm faced with this problem. Under what conditions is this expression a positive odd integer: $$\frac{2^g(x^2+y^2-z^2)}{x+y-z}$$ where $g,x,y,z$ are nonnegative integers. x and z are odd, and y is ...
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2answers
342 views

Integer values of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$?

What are the possible integer values of $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$$ where $x$, $y$, and $z$ are positive integers? My suspicion is the the only integer values are $3$ and $5$, the former ...
3
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0answers
77 views

Does the special Pell equation $X^2-dY^2=Z^2$ have a simple general parameterization?

In Carmichael's Diophantine Analysis ($\S8$), he notes that the equation $$X^2-dY^2=Z^2 \qquad(\dagger)$$ has a two-parameter solution $$x=m^2+dn^2, \quad y=2mn, \quad z=m^2-dn^2. \qquad(\star)$$ He ...
3
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3answers
124 views

Rational points on a surface

I am trying to find rational points on this surface $$ \left( \left( 1-x \right) ^{2}+{y}^{2} \right) \left( \left( 1+x \right) ^{2}+{y}^{2} \right) ={z}^{2}$$ I am actually only interested in ...
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1answer
28 views

Base convertion and equations

I am studying for an exam in my course, and I will certainly have a question of the kind: In what base is the equation right, for example: 42-3=36 Another ...
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1answer
36 views

Solutions to diophantine equation $m^2-12mn-3m+2=0$

I am trying to find all solutions to the relativly simple diophantine equation $m^2-12mn-3m+2=0$. I suspect that the only solutions are $n=0$, $m=1$ and $n=0$, $m=2$, but I am currently unable to show ...
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3answers
65 views

How to show $n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$ has no nonzero integer solutions?

How do we prove that $$n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$$ has no nonzero integer solutions? I know two ways to prove this by taking a geometric interpretation but I don't want such a version. How ...
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2answers
36 views

Equation over Z

Solve the equation $xy+1=3x+y$ over $\mathbb{Z}^2$ Indeed, $$ xy+1=3x+y \Longleftrightarrow (x-1)(y-3)=2 $$ or $ \textrm{Div}(2)=\{k \in \mathbb{Z}/ k|2 \}=\{-1;1;-2;2\}$ Then $(x-1)/2 \implies ...
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3answers
63 views

Solving different types of Diophantine equation [closed]

In each of the following three equations I need help in finding all solutions in positive integers : i) $\dfrac 1x+\dfrac 2y-\dfrac3z=1 $ ii) $\dfrac 1{x^2}+\dfrac 2{y^2}+\dfrac 3{z^2}=\dfrac 23$ ...
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1answer
92 views

motivation for talking about torsion points on an elliptic curve

Let's consider elliptic curves over a fixed field $k$. I understand that viewing the set of $k$-rational as an abelian group is interesting and useful, but I am confused about why the torsion points ...
3
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1answer
57 views

For what positive integers $p$ and $q$: $(p+1)!+(q+1)!=(pq)^2$

I tried this problem using brute force and got the answers as $(3,4)$ and $(4,3)$,but is there a way to solve this question?
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0answers
54 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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3answers
62 views

How can I show the complete symmetric quadratic form has no zeros?

The quadratic complete symmetric homogeneous polynomial in $n$ variables $t_1,\ldots,t_n$ is defined to be $$h_2(t_1,\ldots,t_n) := \sum_{1 \leq j \leq k \leq n} t_j t_k = \sum_{j=1}^n t_j^2 + ...
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1answer
77 views

How prove this diophantine equation $3^x-2^y=k$ have finitely many integral solutions

For any $\forall k\in N^{+}$ show that the diophantine equation $3^x-2^y=k$ have finitely many integral solutions. My try: if $k=2m$,then $$3^x=2^y+k=2^y+2m$$ It is clear there is no integer ...
0
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1answer
34 views

Solve equation. sum of negatie powers of two equal to one. Diaphantite.

Is the following correct? Let $\sum_{i=1}^n \frac{1}{2^{x_i}}=1$ where $x_i \in \mathbb{N}_0$ for $i \in \{1,\ldots,n\}$ than the only solutions is $$x_i=n-1, \quad \forall i \in \{1,\ldots,n\}.$$ ...
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0answers
20 views

Diophantine equations,is that what I have done right?

I have solved the following diophantine equations: $14x+35y=93$ $56x+72y=40$ That's what I have tried: $gcd(35,14)=7$ , but $7 \nmid 93,$ so the first diophantine equation has no solution. ...
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1answer
42 views

Why are these the solutions of the diophantine equation?

According to my notes: $$ \begin{align} & \text{ Let } a,b \text{ not both } 0. \\ & ax+by \text{ has a solution iff } (a,b) \mid c \\ & \text{ If } d:=(a,b) \mid c \text{ and } a=d \cdot ...
0
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1answer
69 views

Is there any easy way to find the positive integer solutions $(x,y,z)$ from this linear equation?

The equation is like this: $3^x -2^y = 19^z$ It seems that no way to find the solution except using trial and error. I got only one solution: $x=3, y=3, and z= 1$ by using trial and error. But, ...
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0answers
12 views

Polynomial Roots of Bivariates

I've got a few polynomials that I am trying to get some results for (shown below). They come from the characteristic equation of a matrix. I have two variables in the polynomials, $\eta$ (which is ...
4
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1answer
84 views

Integer solutions of $x^y-z^3=2$

Is it an open question to solve $x^y-z^3=2$ in integers (both positive, zero and negative)? If not, what kind of methods the solution requires?
3
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1answer
57 views

Is it true that $f(x,y)=\dfrac{x^2+y^2}{xy-t}$ has only finitely many distinct integer values with $x,y$ positive integers?

Prove or disprove that if $t$ is a positive integer, $$f(x,y)=\dfrac{x^2+y^2}{xy-t},$$ then $f(x,y)$ has only finitely many distinct integer values with $x,y$ positive integers. In other words, ...
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1answer
58 views

The Diophantine equation $x^2 + py + a = 0$.

Let $p$ be an odd prime. Show that the Diophantine equation $$x^2 + py + a = 0$$ with $\gcd(a, p) = 1$ has an integral solution if and only if $\left(\frac{-a}{p}\right) = 1$
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5answers
44 views

Diophantine Equation (Hint or Help)

Solve the equation below: $19x+29y=1000$ My try: $19x=1000-29y \rightarrow x=\frac {1000-29y}{19} , x \in N \rightarrow 1000-29y\equiv 0\pmod {19} \rightarrow 29y\equiv 1000\pmod{19} \rightarrow ...
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1answer
30 views

Positive integer solutions of $0=10x³-(2y+5)x²+(y-4)x+76$

To practise my mathematical skills, I often solve some problems in my free time. In this case, one should find every positive integer solution $(x,y)\inℤ^+\timesℤ^+$ of $0=10x³-(2y+5)x²+(y-4)x+76$. ...
2
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1answer
99 views

Quadratic Diophantine Equation $x^2 + axy + y^2 = z^2$

I have been reading about this quadratic Diophantine equation of the form $x^2 + axy + y^2 = z^2$ where x, y, z are integers to be solved and a is a given integer. All integral solutions are given ...
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1answer
74 views

Project Euler - task №390.

When this task was not clear what the equation to be solved. This equation? $x^2y^2+z^2y^2+x^2z^2=r^2$ in integers. It is not clear, because this equation is quite simple and I do not think that ...
2
votes
2answers
52 views

Solving $a^2-24 a=k^2$ across the integers

$$a^2-24 a=k^2$$ I was trying to solve another problem that essentially boiled down to finding all possible integer values of a such that $a^2-24 a$ was a perfect square, and I thus came up with the ...