Questions on finding integer/rational solutions of polynomial equations.

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3
votes
1answer
30 views

Solve in positive integers: $5x^2+6x^3=z^3$

Solve in positive integers: $5x^2+6x^3=z^3$. $x^2(6x+5)=z^3$ If $(x,5)=5$, let $x=5k$. So $k^2(6k+1)=\left(\frac{z}{5}\right)^3$, we're left with solving $6n^3+1=m^3$. If $(x,5)=1$, ...
0
votes
0answers
40 views

A nice set of squares. [duplicate]

Are there integers $a, b, c, d$ such that $$a^2+b^2=c^2$$ $$a^2-b^2 = d^2?$$ I have tried by showing that $a^2 = b^2 + d^2$ and thus $a^2+ b^2 = 2b^2+d^2 = c^2$ But how do I show that there are no ...
0
votes
1answer
30 views

Exponential diophantine equation

Need some help regarding the equation $$2^a-3^b=(2^c-1)\cdot d >0$$ where $a,b,c,d$ are integers; $a,b$ are fixed; and $c>2$. Can we show that $c,d$ exist? Thank you!
1
vote
2answers
47 views

Show that there are exactly 16 pairs of integers $(x,y)$ such that $11x+8y+17=xy$.

Original problem Show that there are exactly 16 pairs of integers $(x,y)$ such that $11x+8y+17=xy$. My work From case by case analysis I come to know that the equation will hold if and only if $x$ ...
3
votes
3answers
36 views

Split 16 Consecutive Integers into Two Subsets of 8 Integers

Show that any given set of sixteen consecutive integers {$x+1,x+2,\ldots,x+16$} can be divided into two eight element subsets with the properties that they have the same sum, the sums of the squares ...
6
votes
1answer
68 views

Only finitely many $a, b$ such that $2+3^n+5^{n^2}=2^a7^b$ for some $n$?

Let $(a,b)$ be a pair of positive integers such that $$2+3^n+5^{n^2}=2^a7^b$$ for some positive integer $n$. Is it true that there are only finitely many such pairs? I don't know the answer to such ...
0
votes
0answers
18 views

Solving the inverse loop

I was making playing around with the solutions of the equation of a circle in the origin $\{ (x^2)+(y^2) = (r^2) \mid\text{ solutions }y = \pm((r^2)-(x^2))^{1/2} \}$, when I tried to find the inverse ...
4
votes
2answers
72 views

$2^x+7^y=19^z$ has no solution in positive integers $x$, $y$, $z$

How do I show that the diophantine equation $2^x+7^y=19^z$ has no solution in positive integers $x$, $y$, $z$
1
vote
1answer
33 views

Trouble forming general solution for linear congruence

I was given $$ 6x+14y=4 \space \mod 5 $$ I took this approach: $$ 6x+14y-5z=4, \space \text{ for some } z $$ Let $$ w=\frac{6}{(6,14)}x+\frac{14}{(6,14)}y $$ Then, $$ (6,14)w+5z=4 \quad , \quad ...
11
votes
2answers
140 views

Diophantine equation $x^2 + xy + y^2 = \left({{x+y}\over{3}} + 1\right)^3$.

Solve in integers the equation$$x^2 + xy + y^2 = \left({{x+y}\over3} + 1\right)^3.$$
1
vote
1answer
72 views

Solving a Diophantine equation: $y^x=x^{2007}$, $x$ and $y$ integers.

I found this Diophantine equation and to solve it I used the definition of logarithm but the solution doesn't require the use of logarithmic rules. I solved it in this way: $$y^x=x^{2007}$$ ...
1
vote
1answer
33 views

Splitting Field of a real cubic

Let $f(X)$ be a cubic with 3 real roots, integer coefficients irreducible over $\mathbb{Q}$. Let $\alpha$ be one of these roots, and consider the number field $\mathbb{Q}(\alpha)$. Dirichlet's unit ...
2
votes
0answers
53 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) = ...
3
votes
1answer
28 views

Solvability of the Diophantine equation $x^{2} - y^{2} = 4z^{n}$?

It is known that for every integer $z$ there are integers $x, y$ such that $x^{2} - y^{2} = z^{3}.$ In fact, given an integer $z$, taking $x := z(z+1)/2$ and $y := z(z-1)/2$ suffices. But how is the ...
0
votes
1answer
38 views

The genus of a certain kind of cubic

I have a cubic curve that looks like $$ a_0 x^3 + a_1 x^2 y + a_2 xy^2 + a_3 y^3 = b $$ with $a_0, a_1, a_2, a_3$, and $b$ all integers, and $a_0$ and $b$ nonzero. I'm not sure but I think in my ...
3
votes
1answer
40 views

Is there any solution to this quadratic Diophantine 3 variables equation?

Is it possible to find all positive integer triplets $(x,y,z)$ satisfying the parametric equation : $$x^2 + 2ax + y^2 + 2by = z^2 + 2cz$$ Here $a, b, c$ are fixed positive integers.
41
votes
3answers
592 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
0answers
31 views

How to prove the cubic formula without root extraction

I'm trying to prove the cubic formula, in the following form: Given a field $F$ and $x,p,q\in F$, define $m=\frac p3$ and $n=\frac q2$, and suppose also that $\gamma,\tau$ are given such that ...
3
votes
2answers
97 views

Diophantine Equations : Solving $a^2$ $+$ $ b^2$ $=$ $2c^2$

I was working through some number theory problems , when I came across the following question : Find all solutions of $a^2$ $+$ $b^2$ $=$ $2c^2$ My Solution (Partial) : We can rewrite the ...
0
votes
3answers
38 views

Diophantine Equations : Solve $a^2 + b^2 = 4c + 3$

I was working my way through some number theory problems , when I came across the following question : Find all solutions to the equation $a^2 + b^2 = 4c + 3$ My Solution (partial) : If ...
2
votes
3answers
66 views

$a^2 = 2b^3 = 3c^5$ Find the smallest value of $abc$.

We have following equation: $a^2 = 2b^3 = 3c^5$ Where $a, b, c$ are natural numbers. Find the smallest possible value of product $abc$.
2
votes
2answers
57 views

Find the probability of solutions of an equation.

Let $x+y+z=20$. What is the probability that all the solutions are distinct? (No two variables have the same value). Assuming that the solutions are only positive integers or zero. I have tried- ...
0
votes
2answers
36 views

Strategy to find the most money to use.

As a reward for a week of good behavior, Tommy was given 7 dollars to spend at the canteen. By the time Tommy got to the canteen, there were only chocolate bars, meat pies and pizza pieces left. The ...
-2
votes
1answer
41 views

Trouble with two equations with 4 unknowns [closed]

I was wondering if I could receive assistance for the following system: $$\begin{cases}(x/a)^{3.2}+(y/b)^{3.2}=1\\ a/b = 174.1/86\end{cases}$$ I'm looking for integer solutions or how to find them ...
2
votes
2answers
43 views

How to find the number of values for $x$ and $y$?

I have come across numerous questions where I am asked for example, if $x$ and $y$ are non-negative integers and $3x + 4y = 96$, how many pairs of $(x,y)$ are there? Usually, I just use trial and ...
1
vote
2answers
22 views

Proving expressibility of integers as the difference of two squares.

I'm given the task: Prove that a positive integer is expressible as the difference of two squares of integers if and only if it is not of the form $4n+2, n\in\mathbb{Z}$ I was given a hint that I ...
1
vote
0answers
29 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 $\gamma >0$ ...
1
vote
2answers
53 views

Intersection of integer sets

This is probably a trivial question for mathematicians but I am not seeing how to approach the following problem: Imagine two sets defined by: ...
1
vote
1answer
82 views

Gorgeous diophantine equation [closed]

How to find all integer solutions of the following equation? $$y^7=14 \cdot 3^{100}x^6 + 70 \cdot 3^{300} x^4 + 42 \cdot 3^{500} x^2 + 2 \cdot 3^{700}$$
10
votes
1answer
290 views

How prove this systems-equation has least two postive integers solution

Show that: for any $k\ge 100,(k\in N^{+})$, there exsit $p\in N^{+}$, such $$\begin{cases} a+b+c=k\\ abc=p\\ a>b>c \end{cases}$$ has at least two postive integers solution $(a,b,c)$ ...
6
votes
2answers
83 views

Pythagorean Triples : Is every positive integer $\gt$ $2$ part of at least one Pythagorean triple?

I was doing some basic number theory problems from Rosen and came across this problem: Show that every positive integer $\gt$ $2$ is part of at least one ...
0
votes
1answer
40 views

How to solve a equation with special conditions?

I have this equation: $z = 11n + 13m$ Conditions: $z < 2015$ $z$, $n$ and $m$ must be natural numbers ($>0$). How many options are possible for $z$?
5
votes
1answer
96 views

Pythagorean Triples : Show that exactly one of $x$, $y$, and $z$ is divisible by $5$

I was doing some basic number theory problems from Rosen and came across this problem: Show that if $(x, y,z)$ is a primitive Pythagorean triple, then exactly one of $x$, $y$, and $z$ is divisible ...
6
votes
3answers
142 views

Finding integer solutions of $m^2-n^5 = m - n$

How to list all integer solutions of $m^2-n^5 = m - n$ Here $m$ and $n$ are some positive integers. Also, I want to know the name of this type equations (if name exist). Regards Rosy
1
vote
1answer
29 views

If equation has integer solution it has solution for every prime p.

How to prove that if the equation in the form: $a_0 x_0^2 + a_1 x_1^2 + \dots + a_nx_n^2 = 0$ where $a_0, a_1, \dots , a_n \in \mathbb{Z}$, has an integer solution, then it has solution in ...
2
votes
0answers
43 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 = ...
1
vote
1answer
72 views

Diophantine equation $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{3}{5}$

I'm trying to solve the diophantine equation $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{3}{5}$ for all $x,y,z \in \mathbb{Z}$. I did this. Since that the diophantine equation is symmetric, we can ...
1
vote
0answers
41 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 ...
1
vote
3answers
76 views

Solve in non-negative integers: $m^2+n^2=1997 (m-n)$

Solve in $\mathbb{N}$:$$m^2+n^2=1997(m-n)$$ I try with quadratic equation or with factorising, but I have no idea what to do after that.
0
votes
0answers
31 views

Argument for finite solution of power Diophantione Equation.

Assume the equation $4x^3=y^2+3$ has infinite positive integer solution. If $x,y$ has general solution then it is clear that for any $x$(rational, integer), there is a $y$. It can be said there is a ...
0
votes
1answer
91 views

Integer solution to the equation

Does there exists an integer solution (for every integer $m\geq 1$) for the following equation? $$x_1x_2...x_n+(2y+1)z+y=4m+3$$ where, $1\leq x_1\leq x_2\leq...\leq x_n\leq l$,$0\leq y \leq ...
0
votes
0answers
14 views

minimising multivariate quadratic function over integer variables

I have a quadratic function $x_1^2+x_2^2-(u_1x_1+u_2x_2)^2$ which I need to minimise over integer $x_1$ and $x_2$; also, the coefficients $u_1,u_2<1$. In other word, assuming coefficients ...
1
vote
1answer
43 views

How do I solve these questions using Diophantine equations?

I have been told that it is easier to solve the below 2 questions using Diophantine equations instead of simply trial and error. 1) Find the smallest positive integer which, when divided by 6, ...
4
votes
0answers
83 views

Solve $(x+1)^n-x^n=p^m$ in positive integers

Solve in positive integers: $$(x+1)^n-x^n=p^m$$ $p$ is prime, $n\ge 2$. Seemingly Zsigmondy's Theorem and LTE won't work here. Though you can tell (as suggested by user barto), using ...
0
votes
0answers
14 views

Diophantine equation involving three variavles

I wonder if it is possible to enumerated all the integer solutions of the Diophantine equation $ax+bxy^2=xz^2+4z$, with $a,b\in\mathbb{N}$. Of course we may solve it for $x$ and obtain rational ...
1
vote
2answers
59 views

Diophantine equation resembling FLT

I was wondering if the equation $x^p+y^p=2z^p$ has been studied. For small cases it is seen that the only solutions are trivial: $x=y=z$. There are probably methods to solve this for regular ...
4
votes
1answer
103 views

Sums of three cubes in arithmetic progression equal to a cube $x^3+(x+y)^3+(x+2y)^3 = z^3$

Using exhaustive search, small positive and primitive integer solutions to, $$x^3+(x+y)^3+(x+2y)^3 = 3 x^3 + 9 x^2 y + 15 x y^2 + 9 y^3= z^3\tag1$$ are, $$x,y = 3,1,\quad x+y =2^2$$ $$x,y = ...
2
votes
5answers
62 views

Possible solutions of a diophantine equation: $p^2+pq+275p+10q=2008$

What are couples of prime integers that verify this diophantine equation: $$p^2+pq+275p+10q=2008?$$ I tried to solve this equation trough the rules of modular-arithmetic. I rewrite the equation as: ...
0
votes
1answer
30 views

Solving a Diophantine Equation of the form $N(N-1) = 2X(X-1)$ for $N, X > 0$

When working on a problem on Project Euler I came up with a formula I need to solve: $N(N-1) = 2X(X-1)$ for $N > 10^{12}, X > 0$ with $N$ and $X$ being integer numbers. After some ...
1
vote
1answer
82 views

Diophantine equation $a^3 + b^3 + c^3 = 2$

I have a pretty difficult math question that I have no idea even how to begin. Here it goes: Find the nonzero integers $a$, $b$, $c$ such that $a^3 + b^3 + c^3 = 2$? I would assume that at ...