Questions on finding integer/rational solutions of equations.

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4
votes
2answers
93 views

Show that $x^2 + y^2 + z^2 = x^3 + y^3 + z^3$ has infinitely many integer solutions.

Show that $x^2 + y^2 + z^2 = x^3 + y^3 + z^3$ has infinitely many integer solutions. I am not able to find an idea on how to proceed with the above questions. I have found only the obvious ...
0
votes
1answer
39 views

Coprime - Irreducibility - Natural numbers

In reference to this question, is anyone could deduce that if $x^2+2=y^3$ and $x,y \in \mathbb{N}$, then $x=5$ and $y=3$. I already prove that the only natural number $x$ for which $x+\sqrt{-2}$ is a ...
2
votes
1answer
61 views

Non-linear Diophantine equation on integer quadruples

Find all integer quadruples $\{a,b,c,d\}$ such that $$ad = b + c$$ $$bc = a^2 - d$$ Working $\bmod 8$ (very messy) gives $d = 3 - 8k \quad \forall k \in \mathbb{N}$. Numerical searching has so ...
-1
votes
2answers
67 views

Solve equation over $\mathbb{N}\setminus\{0\}$ [closed]

I wonder whether there are any solutions besides considering $c=2^{5k+1}$ for this equation: $a^5+b^5=c^{2016}$, where $a,b,c\in\mathbb{N}\setminus\{0\}$
2
votes
1answer
88 views

Quadratic Diophantine Equation $x^2 + 2y^2 = 2013$ [closed]

Find integer values of $x$ and $y$ (if any) such that $x^2 + 2y^2 = 2013$.
3
votes
1answer
55 views

Parametrization of $a^2+b^2+c^2=d^2+e^2+f^2$

Is there an existing parametrization of the equation above that is similar to Brahmagupta's identity for $a^2+b^2=c^2+d^2$? I need either a reference to look it up or a hint to solve it. Thanks.
8
votes
1answer
476 views

Why there isn't any solution in positive integers for $z^3 = 3(x^3 +y^3+2xyz)$?

Consider the following Diophantine equation $$z^3 = 3(x^3 +y^3+2xyz)$$ Is there any elementary proof for the non solubility in positive integers for this Diophantine equation, where $x, y$ and $z$ ...
2
votes
1answer
51 views

Find all nonnegative integer solutions to $x^3 + 8x^2 − 6x + 8 = y^3$.

Find all nonnegative integer solutions to $x^3 + 8x^2 − 6x + 8 = y^3$. The only solution I have found is $x=0$. I have tried proving it by congruences and have had no success. I don't know how ...
1
vote
1answer
67 views

Help Project Euler Problem 269

I am stuck on prob 269 Project Euler. I've just tried brute force method to attempt this problem the example provided by PE For example, $P_{5703}(x)$ = $5x^3 + 7x^2 + 3$. We can see that: ...
1
vote
1answer
72 views

Transforming Diophantine quadratic equation to Pell's equation

I have been discussing the fastest and most efficient ways of solving QDEs in a separate question record (Alternative method to solve quadratic Diophantine equations). However, as suggested by ...
0
votes
0answers
23 views

Looking for info on representation of a diophantine equation as system of equations over finite field/boolean algebra

Suppose that $x$ is a positive integer. Fix some prime $p$. Then there exists some non-negative integer, $L$, and $\{x_0, x_1, . . . , x_L\} \subseteq \{0,1,...,p-1\}$ such that, $$x = ...
0
votes
1answer
47 views

Alternative method to solve quadratic Diophantine equations

For most types of quadratic Diophantine equations there exists an algorithm which makes it possible to find a solution (or solutions) over integers (good reference is here: ...
3
votes
1answer
62 views

Solve the equation $x^3 + 117y^3 = 5$ over the integers.

Solve the equation $x^3 + 117y^3 = 5$ over the integers. I have tried solving this. It is clear that one of $x$ or $y$ must be negative. $117$ seemed a strange number. So I found out that $117 = ...
2
votes
1answer
59 views

Why chord and tangent method does not give further points on Fermat's curve?

(0,1) and (1,0) are two rational points on x^3 + y^3= 1. But why doesn't chord and tangent method yield any further points on the curve?
4
votes
4answers
104 views

Does the Pell-like equation $X^2-dY^2=k$ have a simple recursion like $X^2-dY^2=1$?

If $d \ne 0$ is a non-square integer, and $(u,v)$ is an integer solution to the Pell equation $$ X^2 - dY^2 = 1, \tag{$\star$} $$ then each solution $(x_i,y_i)$ can be recursively calculated using ...
1
vote
1answer
39 views

An extension to Pell's equation

During my number theory seminary, I found this interesting problem and I didn't know how to solve it. Given Pell's equation $$x^2-3y^2=1,$$ where $x,y \in \Bbb N,$ show that there are infinitely many ...
-1
votes
2answers
76 views

Find all positive integers n such that $2^2 + 2^5+ 2^n$ is a perfect square. [closed]

Find all positive integers n such that $2^2 + 2^5 + 2^n$ is a perfect square. Explain your answer.
30
votes
0answers
462 views

On Ramanujan's curious equality for $\sqrt{2\,(1-3^{-2})(1-7^{-2})(1-11^{-2})\cdots} $

In Ramanujan's Notebooks, Vol IV, p.20, there is the rather curious, $$\sqrt{2\,\Big(1-\frac{1}{3^2}\Big) \Big(1-\frac{1}{7^2}\Big)\Big(1-\frac{1}{11^2}\Big)\Big(1-\frac{1}{19^2}\Big)} = ...
2
votes
0answers
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: $$(1,1)\ \ (2,3)\ \ (3,7) \ \ (6,29)\ \ (91,6761)$$ ...
3
votes
1answer
80 views

Solve the Diophantine Equation $x^2 + 1 = 2y^4$ over $\mathbb{Z}$.

Solve the Diophantine Equation $x^2 + 1 = 2y^4$ over $\mathbb{Z}$. I have found few elementary solutions like $(1,1)$. I have tried it with variable replacements. After solving it a bit it ...
2
votes
0answers
55 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 ...
2
votes
0answers
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 ...
1
vote
3answers
50 views

Integer solutions for $n$ for $|{\sqrt{n} - \sqrt{2011}}| < 1$

$$|{\sqrt{n} - \sqrt{2011}}| < 1$$ What is the number of positive integer $n$ values, which satisfy the above inequality. My effort: $ ({\sqrt{n} - \sqrt{2011}})^2 < 1 \\n + 2011 ...
7
votes
1answer
82 views

$a,b,c \in \mathbf{Z}$ such that $a^7+b^7+c^7=45$

Do there exist integers $a,b,c$ such that $a^7+b^7+c^7=45$? [I have an ugly argument for a negative answer, is it possible to give a "manual" solution?]
1
vote
3answers
132 views

Three questions about the form $X^2 \pm 3Y^2 = Z^3$ and a related lemma

In Ribenboim’s Fermat’s Last Theorem for Amateurs, he gives the following lemma [Lemma 4.7, pp. 30–31]. Lemma. Let $E$ be the set of all triples $(u, v, s)$ such that $s$ is odd, $\gcd(u,v) = 1$ and ...
2
votes
2answers
55 views

The Diophantine Equation: $x^3-3=k(x-3)$

I wish to know how to resolve the diophantine equation: $x^3-3=k(x-3)$ ? The problem is: Find all integers $x\ne3$ such that $x-3\mid x^3-3$. - From 250 Problem's in Elementary Number Theory, by ...
2
votes
1answer
44 views

Solving Diophantine Equations? What are the common techniques?

What does it mean to solve a Diophantine Equation? Some questions are like Solve $x^2+1=2y^4$ over the integers. What does it actually mean to solve a diophantine equation? Do we have to find a ...
1
vote
1answer
85 views

Find the largest integer $n$ such that $n^2$ is the difference of two consecutive cubes and $2n +79$ is a perfect square.

Find the largest integer $n$ such that $n^2$ is the difference of two consecutive cubes and $2n +79$ is a perfect square. This is an AIME problem. I have been trying and have been going round in ...
1
vote
0answers
34 views

Integer solutions of $\sum_{j=1}^m\frac{n-m}{j}+\sum_{j=m+1}^n\frac{n}{j}=k$

This is in reference to an answer I gave to this question. I am curious to know if my intuition is correct. Given $n,m\in\mathbb{N}$, with $n>m$, is it possible for $\exp(2\pi ...
0
votes
1answer
40 views

Number of integer solutions $(x, y)$ of $x(x+6) = y^2 + k$ for different integer values of $k$

Let $n$ be the number of pairs $(x, y)$ of integer solutions to the following equation:$$x(x+6) = y^2 + k$$ Can there be an integer $m$, $k$ can be given an integer value so that $n=m$ ?
1
vote
1answer
26 views

Prime solutions to an exponential diophantine equation

Suppose that $a,n,t$ are positive integers greater than $1$ and $q$ is a prime number. I write $a_{n} = \binom {a^{n}} {a}$. In general I was curious about $a_{n}-k =q^{t}$ in particular the case ...
3
votes
1answer
75 views

Trying to prove $c^3a^2+(9c^2-b^2)a+(27c-10b)=0$ has no positive integer solutions

I'm trying to prove (or, I suppose, disprove) the following claim, in either version. Conjecture (Strong Version): There are no positive integers $a,b,c$ such that $$c^3a^2+(9c^2-b^2)a+(27c-10b)=0.$$ ...
0
votes
0answers
20 views

Let $x,y,z$ be 3 coprime integers where $u|x, v|y, w|z$, is $x^3+y^3+z^3\equiv 0 \pmod{(uvw)^3}?$

Let $u,v,w \neq \pm1$ be 3 non-zero integers respective factors of 3 relatively prime integers $x,y,z$. Is the following equivalence possible: $$x^3+y^3+z^3\equiv 0 \pmod{(uvw)^3}?$$ It is obvious if ...
2
votes
2answers
37 views

How many solutions in integers to the following equation

What is the number of positive integer solutions $(a, b)$ to $2016 + a^2 = b^2$? We have, $2016 = (b-a)(b+a) = 2^5 \cdot 3^2 \cdot 7$ $b - a = 2^{t_1} 3^{t_2} 7^{t_4}$ and $a - b = 2^{t_5} ...
0
votes
0answers
19 views

Regarding hypothetical counterexamples to Mihailescu's Theorem (Catalan's Conjecture)

Mihailescu's Theorem (Catalan's Conjecture) states that the Diophantine equation $$X^p = Y^q + 1$$ has only one positive integer solution, namely $3^2=2^3+1$. Given a hypothetical [positive integer] ...
0
votes
1answer
46 views

Is there any algorithm to find all the solutions of the following special linear Diophantine system?

Consider the following system. 1) $a_{11}x_1 + a_{21}x_2 + \cdots + a_{m1}x_m=d_1$ 2) $a_{12}x_1 + a_{22}x_2 + \cdots + a_{m2}x_m=d_2$ $\vdots$ n) $a_{1n}x_1 + a_{2n}x_2 + \cdots + a_{mn}x_m=d_n$ ...
1
vote
2answers
112 views

Seeking general methods to attack $ax^4+bx^3y+cx^2y^2+dxy^3+ey^4=z^2$ in integers/rationals

In my current number theory work, I'm running into many equations of the form in the title. Q1: Am I correct in calling this a quartic Thue equation? Q2: Are there general methods to attacking this ...
0
votes
1answer
32 views

Diophantine Factorial Equation [closed]

Prove that there exist pairwise distinct positive integers $a_0, a_1, a_2, \ldots, a_{1000}$ such that $a_0! = a_1!a_2! \cdots a_{1000}!.$
1
vote
2answers
72 views

Parametric solution of the Diophantine equation $\frac{3}{n}=\sum\frac{1}{a}$

Assmue $n>3$ is a odd number,Prove that there exists distinct odd numbers $a,b,c$ such $$\dfrac{3}{n}=\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\tag{1}$$ I'm reading a lot about the Erdös-Straus ...
0
votes
2answers
45 views

Representations of some primes as $3x^2-4y^2$?

I am looking for (elementary) proofs (idea of the proofs is also OK) or references to proofs of the followings: $$ p\equiv11\pmod{12}\longrightarrow p=3x^2-4y^2 $$ Any help appreciated.
0
votes
2answers
36 views

Find all natural solutions of $14x+21y=91$

Find all natural solutions of $14x+21y=91$ where $x,y\in \mathbb N$ My attempt: Divided by $7$: $$2x+3y=13$$ And $\gcd(2,3)=1\mid13$ I found a private solution $x_0=\color{blue}5,\quad ...
0
votes
1answer
49 views

is A an even number?

Let $a,b,c,d$ be positive integers such that $(3a+5b)(7b+11c)(13c+17d)(19d+23a)=2001^{2001}$ hence, prove that $a$ is even. I tried to approach this problem reducting it modulo 6. From which we ...
3
votes
0answers
122 views

the system of diophantine equations: $x+y=a^3$; $xy=\dfrac{a^6-b^3}{3}$ has only trivial solutions.

Without using Fermat's Last Theorem, how can one prove that the following system of diophantine equations has only trivial solutions: $$x+y=a^3$$ $$xy=\dfrac{a^6-b^3}{3}$$ We suppose of course that ...
0
votes
1answer
32 views

Representations of some primes as $x^2-2y^2$?

I am looking for (elementary) proofs (idea of the proofs is also OK) or references to proofs of the followings: $$ p\equiv\pm1(\mod8)\longrightarrow p=x^2-2y^2 $$ Any help appreciated.
0
votes
1answer
47 views

Hyperbolic Diophantine Equations: Application of Euclidean Algorithm?

I'm trying to determine whether or not I can find the integer solutions to $(x+a)$$(x+b)$ $=$ $x(x-1)$ + $x(a-b)$ (with a known $x$ value you choose, i.e. $707$). Plugging in my example value on ...
1
vote
0answers
39 views

Problem about Diophantine's equation and congruences

This problem is from Niven's, 5.4.8. Let $\,f(\mathbf{x})=f(x_1,x_2,x_3) = x_1^4+x_2^4+x_3^4-x_1^2x_2^2-x_2^2x_3^2-x_3^2x_1^2-x_1x_2x_3(x_1+x_2+x_3).$ Show that $f(\mathbf{x}) \equiv1$ ...
3
votes
1answer
76 views

How to solve $p^n+12^2=m^2$

Find all triples $(m,n,p) \in \mathbb{N}^3$, with $p$ prime, which satisfy $$p^n+12^2=m^2$$
0
votes
0answers
44 views

A farmer bought some chickens and cows from a local rancher…

A farmer bought some chickens and cows from a local rancher. If a chicken costs 2 dollars and a cow costs 5 dollars, how much of each can he purchase if the total cost is 38 dollars and he purchases ...
4
votes
2answers
89 views

Solve in positive integers the equation $a^3+b^3=9ab$

Solve in positive integers the equation: $$a^3+b^3=9ab$$ I try to: $$\dfrac{a^2}{b}+\dfrac{b^2}{a}=9\Longrightarrow a^2<9b,b^2<9a$$ Of course, I can't solve it. Can anyone help?
0
votes
1answer
115 views

Solutions to simultaneous Diophantine equations $2y^2-3x^2=-1$ and $z^2-2y^2= -1$

I am looking for integer solutions for the following set of equations: $2y^2-3x^2=-1$ $z^2-2y^2= -1$ I know that there are the solutions (1,1,1) and (-1,-1,-1) for this set of ...