Questions on finding integer/rational solutions of polynomial equations.

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6
votes
2answers
1k views

Are there any $n$ for which $ n^4+n^3+n^2+n+1$ is a perfect square?

Are there any positive $n$ for which $ n^4+n^3+n^2+n+1$ is a perfect square? I tried to simplify \begin{align*} n^4+n^3+n^2+n+1 &= n^2(n^2+1)+n(n^2+1)+1\\ &= (n^2+n)(n^2+1)+1 \\ &= ...
3
votes
1answer
248 views

Heronian triangles

How to prove that all Heronian triangles can be found using formulas described here? I understand that the described substitution will give Heronian triangle, but how to prove that using the ...
2
votes
0answers
415 views

A Quadratic diophantine equation

How to prove or disprove this statement: For all $c<z<0<s$, there exists $0<k\leq i$, $0\leq j<s+i$, such that all conditions hold simultaneously: ...
4
votes
3answers
611 views

Heronian triangle Generator

I'm trouble shooting my code I wrote to generate all Heronian Triangles (triangle with integer sides and integer area). I'm using the following algorithm $$a=n(m^{2}+k^{2})$$ $$b=m(n^{2}+k^{2})$$ ...
1
vote
2answers
542 views

Solution of Diophantine equation for polynomials

I'm trying to solve this polynomial Diophantine equation for $R$ and $S$: $ AR + BS = G $ where $A(x) = a_0x^{n_a} + a_1x^{n_a-1} + a_2x^{n_a-2} + ... + a_{n_a}$ $B(x) = b_0x^{n_b} + b_1x^{n_b-1} + ...
14
votes
2answers
345 views

Solutions of $q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$ s.t. $q \geq 3$

Is it true that for every rational $q \geq 3$ , the following equation has a solution over $\mathbb N$ ? $$q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$$
2
votes
1answer
244 views

Number of integer solutions of $\frac{1}{x} + \frac{1}{y} = \frac{1}{1000}$

What is the number of integer solutions of: $$\frac{1}{x} + \frac{1}{y} = \frac{1}{1000}$$ How to solve these type of problems if am comfortable of solving $x+y=z$. But how to do if multiplicative ...
4
votes
1answer
156 views

A simple-looking diophantine equation

Consider the diophantine equation $Q(x,y,z)=0$, where $x$, $y$ and $z$ are nonnegative integer unknowns and $$ Q(x,y,z)=x^3 + (-2y + 2)x^2 + ((z - 6)y + (2z + 1))x + ((2z - 4)y + 3z) $$ Since the ...
5
votes
3answers
306 views

Number of positive integers that cannot be expressed as $ax+by$

Prove that there are exactly $$\displaystyle{\frac{(a-1)(b-1)}{2}}$$ positive integers that cannot be expressed in the form $$ax\hspace{2pt}+\hspace{2pt}by$$ where $x$ and $y$ are non-negative ...
3
votes
1answer
127 views

Is there no univariate integer polynomial that takes on the same positive values as the multivariate polynomial $x^2+y^2$?

Is there no univariate integer polynomial that takes on the same positive values as the multivariate polynomial $x^2+y^2$? The values are numbers such that each prime factor of the form $4k+3$ occurs ...
0
votes
1answer
437 views

The diophantine equation $a+b+c+d+e = abcde$

Now you may think that I am annoying, but if I am not asking this question, then it seems not so complete and I can't grasp the whole idea... refer to this question: Positive rationals satisfying ...
4
votes
1answer
437 views

Positive rationals satisfying $a+b+c+d=abcd$?

Victor has posted a couple of problems involving finding real and rational solutions of $a+b+c=abc$. Two techniques have been given: using triangles, and using scaling. Neither seems to work for the ...
4
votes
1answer
498 views

How to find all rational numbers satisfy this equation?

Find all rational number $a,b,c$ satisfy: $$a+b+c=abc$$ I try to change this in different forms like $(ab-1)c = a+b$, $(ac-1)b = a+c$, $(cb-1)a = b+c$ etc but it won't help...
5
votes
1answer
108 views

Does $4x^2+1=5^y$ have a solution in integers with $y>1$?

Consider the following equation : $4x^2+1=5^y~$ with $y>1$ Has this equation solutions in integers ? I wrote small Maple program in order to find solutions but couldn't find anyone . ...
2
votes
0answers
143 views

Does $~4^y+1=4xy^2+x~$ have infinitely many solutions in integers?

Consider the equation : $~4^y+1=4xy^2+x$ I have found that this equation has integer solutions for following values of $~y$ : $y\in \{1,2,193,10068,29570,..\}$ Question : Are there finitely or ...
29
votes
2answers
898 views

Does $2x^2-1=y^5$ have a solution in integers, with $y>1$?

In my solution to this MSE problem, I noted that $2x^2-1=y^5$ is unlikely to have solutions in integers with $y>1$. Recently, I've tried to find a proof, without success. Following Thomas ...
2
votes
5answers
136 views

How can I make the following 2 fractions integers?

Let $m,n$ be integers. I want to find the possible values of $m,n$ such that $4(m+n)\over (2m+n)^2+3n^2$ and $4n\over (2m+n)^2+3n^2$ are both integers too. Would someone please help? Of course letting ...
3
votes
2answers
210 views

Generalization of Pythagorean triples

Is it known whether for any natural number $n$, I can find (infinitely many?) nontrivial integer tuples $$(x_0,\ldots,x_n)$$ such that $$x_0^n + \cdots + x_{n-1}^n = x_n^n?$$ Obviously this is true ...
2
votes
1answer
923 views

All positive integers m,n such that $an+b=cm$?

Given positive integers a,b,c, how to find all positive integers m,n such that $an+b=cm$? Is there always infinitely many m,n for all a,b,c? If $(n_0, m_0)$ is the smallest solution, are all other ...
12
votes
1answer
315 views

$a^m+k=b^n$ Finite or infinite solutions?

Given positive integers k,a,b, is there a finite or infinite number of solutions in positive integers $m,n>1$, to $a^m+k=b^n$? Pillai's conjecture states that each positive integer occurs only ...
4
votes
1answer
204 views

Diophantine equation $x^3+z x^2-z y^2=0$

I'm not familiar with diophantine equations. At most my approaches doesn't give results. I need to solve the following equation $$ x^3+zx^2-zy^2=0 $$ where $x,y,z\in\mathbb{Z}$
11
votes
2answers
343 views

Finding all the numbers that fit $x! + y! = z!$

I have the formula $x! + y! = z!$ and I'm looking for positive integers that make it true. Upon inspection it seems that x = y = 1 and z = 2 is the only solution. The problem is how to show it. ...
3
votes
1answer
302 views

find all positive integers satisfying $2x^2 - y^{14} = 1$

The following problem was posted to usenet forum de.rec.denksport two weeks ago and no progress was made. Find all positive integers $x$,$y$ satisfying the equation $$2x^2 - y^{14} = 1$$ $(1,1)$ ...
0
votes
1answer
72 views

Linear Diophantine Set Proof

Let's say I have set S and T being the set of all integer solutions to $ax+by=c$ and $ax+by=nc$ respectively, and set S* might be the same as set T. S* = $\{ (n x_0 + n y_0) | (x_0, y_0) \in S\}$ ...
12
votes
0answers
617 views

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers ...
1
vote
3answers
131 views

Number of distinct graphs with y-intercepts that are integers between $-10$ and $10$

I wanted to make a test bank of graphs of linear equations for my algebra classes. I want the $y$-intercept of each graph to be an integer no less than $-10$ and no greater than $10$. Generally, you ...
2
votes
2answers
127 views

Finding relatively prime integers with specified remainders $\pmod{m}$.

Let $m$ be a positive integer. Let $a,b$ be integers with $0 \leq a,b < m$, $a,b$ not both zero, $\gcd(a,b,m)=1$. Do there necessarily exist integers $x,y$ such that $x \equiv a \pmod{m}$ $y ...
4
votes
2answers
486 views

Solutions of some Diophantine equations

Respected Mathematicians, The Diophantine equation $$2^x + 5^y = z^2$$ has solutions $$x = 3, y = 0, z = 3$$ and $$x = 2, y = 1, z = 3$$ I got these solutions by trial and error method. To be ...
1
vote
1answer
108 views

Diophantine equations of the form $xy = n$

I am looking for a general way to estimate the number of possible solutions to solve the diophantine equation of the form $xy = n$, where $n$ is a positive integer. Note that $n$ can be an unusually ...
2
votes
2answers
503 views

Sum of squares diophantine equation

How can I count all integer solutions to the equation $$x^2+dy^2=n$$ given $n$ and $d$, where $n$ is composite (or prime)? By counting I mean any algorithm faster than brute force enumeration.
1
vote
2answers
218 views

Finding the positive integer solutions $(x,y)$ of the equation $x^2+3=y(x+2)$

Finding the positive integer solutions $(x,y)$ of the equation $x^2+3=y(x+2)$ Source: Art of Problem Solving Vol. 2 Any help would be appreciated.
12
votes
2answers
592 views

diophantine equation in positive integers

would you please help me solve this? solve this equation in positive integers: $x^2+y^2+z^2=3xyz$ I could prove that it's solutions are infinite, for if $(x,y,z)$ is a solution, with $x\le y \le ...
4
votes
2answers
282 views

How many ordered triple $ (p,a,b) $ is possible such that $p^a=b^4+4$?

If we have a prime number $p$ and two natural numbers $a$ and $b$ such that $p^a=b^4+4$, then how many such ordered triplets $(p,a,b)$ exist? What should be the strategy to solve this one? The only I ...
-1
votes
4answers
1k views

Integer solutions to $x^2+y^2=N$?

This is not homework; but I am completely lost on how to go about handling this especially for large N
0
votes
2answers
203 views

The Diophantine equations $X^3=DY^3+A^3$

Does anyone know for which values of $D$ the equation $X^3=DY^3+A^3$ has solutions? All numbers non-zero naturals.
1
vote
1answer
641 views

Discriminant of derivative of cubic equation being a perfect square

Is it possible for the discriminant of the first derivative of a cubic polynomial (x+a)(x+b)(x+c), where a, b and c are distinct non-zero integers (i.e. Discriminant[d((x+a)(x+b)(x+c))/dx] in ...
2
votes
2answers
5k views

How many solutions are there to the equation $x + y + z + w = 17$?

How many solutions are there to the equation $x + y + z + w = 17$? I don't know if I'm doing this right, but I guessed that the solution would be $\binom{20}{3}$, which equals $1140$. Am I doing ...
4
votes
1answer
145 views

A quartic diophantine equation

Here is the statement: Let $a,b \in \mathbb{Z}$ positive integers such that $a^2=b^4+b^3+b^2+b+1.$ Prove $b=3.$ I've tried is the following: Let $\Sigma=b^4+b^3+b^2+b+1$. If $a\equiv 0\mod 3$, then ...
7
votes
4answers
703 views

Right triangle where the perimeter = area*k

I was doodling on some piece of paper a problem that sprung into my mind. After a few minutes of resultless tries, I advanced to try to solve the problem using computer based means. The problem ...
4
votes
5answers
2k views

Count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$

How to count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$ such that $x_1\ge 4,x_3 = 11,x_4\ge 7$ And how about $x_1\ge 4, x_3=11,x_4\ge 7,x_5\le 5$ In both cases, ...
5
votes
4answers
194 views

Use induction to prove a product of sums of squares is a sum of squares

For any natural number $n\ge 1$, given pairs $(a_1,b_1),(a_2,b_2),...,(a_n,b_n)$ of integer numbers, there exist integer number $c$ and $d$ such that $$\prod_{i=1}^{n}(a_i^2+b_i^2) = c^2+d^2$$ My ...
0
votes
3answers
184 views

Using induction to find a general pattern of $x,y$

Let $m$ be an even natural number. Find natural numbers $x$ and $y$ such that $$ m=(x+y)^2+3x+y$$ Try a few cases to find pattern and then use induction to prove that the pattern works. P.S I saw ...
5
votes
1answer
770 views

Integer coordinate set of points that is a member of sphere surface

I have a graphic application to develop which involve many spheres. I should determine then on run time. Supposing that I have a sphere of radius r, how can I determine the sub set of the sphere ...
3
votes
3answers
520 views

Finding all positive integer solutions to $(x!)(y!) = x!+y!+z!$

The equation is $(x!)(y!) = x!+y!+z! $ where $x,y,z$ are natural numbers. How to find out them all?
1
vote
0answers
213 views

Quadratic fields and solving Diophantine equations

I would like to learn to solve Diophantine equations and I think my next step would be quadratic fields or number fields. What are kind of methods there are to use those on solving equations? And what ...
1
vote
1answer
95 views

Proving $\frac{m-n}{(m+1)(n+1)}=\frac{1}{k}$ for every $k>1$

How can we show that for any integer $k>1$ there are positive integers $m$ and $n$ such that $$\frac{1}{k}=\frac{m-n}{(m+1)(n+1)}.$$ (Thanks to Arthur Fischer for the reformulation!)
4
votes
3answers
1k views

$\mathbb Z[\sqrt 3]$ contains infinitely many units

I'm asked to show that there are infinitely many units in the ring $\mathbb Z[\sqrt 3]$. But I don't really see a good approach to this one, so far. Some thoughts: The inverse of $a+\sqrt3 b$ ...
2
votes
4answers
259 views

Form of rational solutions to $a^2+b^2=1$?

Is there a way to determine the form of all rational solutions to the equation $a^2+b^2=1$?
2
votes
2answers
106 views

Find a couple of integers such that the third power of a given natural can be written as the difference of the squares of those integers

Given a natural number $n$, find inegers $a, b$ such that $n^3=a^2-b^2$. I've tried, but I'm a bit rusty. Please Help
2
votes
2answers
94 views

Integer points of a circumference which radius in $n^{3/2}$

The question is: with a fixed integer $n$, what are the points with integer coordinates $(a,b)$ so that $a^2 + b^2 = n^3$? The equation is symmetric in $a$ and $b$, so if $(x,y)$ is a solution, then ...