Questions on finding integer/rational solutions of equations.

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2
votes
1answer
116 views

Points on the elliptic curve for Ramanujan-type cubic identities

Given the rational Diophantine equation, $$t^3 - t^2 - \tfrac{1}{3}(n^2 + n)t - \tfrac{1}{27}n^3=w^3\tag1$$ Two points are, $$t_0 = 0\tag2$$ $$t_2 = \frac{-(1 + 2 n) (1 + 11 n + 42 n^2 + 14 n^3 + 13 ...
6
votes
4answers
423 views

General solution of Pell's equation

If we know the minimal solution or any specific solution of Pell's equation $x^2-ny^2=1$ , is there is any general formula to write all solution of Pell's equation?
0
votes
3answers
69 views

Solve $56x+63y=1$

Solve $56x+63y=1$ What I did: $$\gcd(56,63):$$ $$\qquad63=56\cdot 1+7\\56=7\cdot 8$$ $$\Longrightarrow \gcd=7$$ Since $7\nmid 1$ there is no solution, but I think that Wolfram says something ...
8
votes
4answers
561 views

Are there ways to solve equations with multiple variables?

I am not at a high level in math, so I have a simple question a simple Google search cannot answer, and the other Stack Exchange questions does not either. I thought about this question after reading ...
2
votes
2answers
107 views

Is $(2,5)$ the only solution?

Find all pairs $(m,n)\in{\mathbb{N^2}}$ such that $$(m^2-1)^3-n^2=2$$ Is $(2,5)$ the only solution?
1
vote
1answer
86 views

diophantine equation: $ (1-ab^3)(a^3b-1)=c^2$ [closed]

I wonder if the Diophantine equation $(1 -ab ^ 3 ) (a ^ 3b -1) = c^2$ admits rational solutions we must choose the numbers $a$ and $b$
6
votes
2answers
142 views

Why can't we use the law of cosines to prove Fermat's Last Theorem?

In investigating approaches to Fermat's Last Theorem I came across the following and I can't figure out where I am going wrong. Any input would be greatly appreciated. We want to show that $a^n + b^...
7
votes
0answers
119 views

Generalizing Ramanujan's cube roots of cubic roots identities

(This extends this post.) Define the function, $$\sqrt[3]{G(t)} = \sqrt[3]{t+x_1}+\sqrt[3]{t+x_2}+\sqrt[3]{t+x_3}\tag1$$ where the $x_i$ are roots of the cubic, $$x^3+ax^2+bx+c=0\tag2$$ While $G(t)...
2
votes
0answers
47 views

How to find all the integral solutions of $x^2-by^2=z^k$ where $k$ is an odd integer >2 and $b>0$?

Consider the bivariate quadratic polynomial of the form: $$ x^2-by^2=z^k$$ where $k$ is an odd integer>2 and $b>0$. It's well-known that Euler's method: $$x^2-by^2=(p^2-bq^2)^k $$ provides a class ...
0
votes
0answers
20 views

Rational Solutions to equations like $a^2+3b^2=k^3$

I'm working on the number field $\mathbb{Q}(\sqrt{-3})$, and I want to find elements $\alpha \in \mathbb{Q}(\sqrt{-3})$ such that the polynomial $X^3-\frac{\overline{\alpha}}{\alpha^2}$ be irreducible....
6
votes
5answers
320 views

Solve $4 \times2^x+3^x=5^x$ without any sort of calculator

Is there a way i can solve the following equation only by using high school mathematics? $$4 \times2^x+3^x=5^x$$ I tried writing $5$ as $2+3$ but didn't get any result. After that i tried to divide ...
3
votes
2answers
95 views

The number of positive integer solutions to the equation $x_1+x_2+…+x_n=n^2.$

I'm working on this problem. To solve it I need this lemma: Let $n\ge2, n\in \mathbb N$. Let $X$ be the number of solutions in positive integers to the equation $x_1+x_2+...+x_n=n^2$. Let $Y$ be ...
1
vote
2answers
63 views

For which integers $q \ge p\ge 1$ with $q^2-2p^2=2$ is $2p^2+1 \pm pq$ an integer square?

The title says it all… I’m looking to prove (in an elementary way, if possible) the following question: Conjecture: If $q$ and $p$ are positive integers such that $q^2-2p^2=2$ and $2p^2+1 \pm pq$ is ...
3
votes
3answers
109 views

Number of positive unequal integer solutions of $x+y+z+w=20$

What is the number of positive different integer solutions of $x+y+z+w=20$, where $x,y,z,w$ are all different and positive? It would be nice if coding is not used. I am given the answer $552$.
2
votes
0answers
58 views

Diophantine Equation: $4x^r = 3y^2 + 1$

If $r \ge 3$ is an integer, show that $4x^r = 3y^2 + 1$ does not have positive integer solutions $(x, y)$ except for $(1, 1)$. (I am not sure whether this is an open problem; in any case, it is a ...
6
votes
2answers
128 views

The number of positive integer solutions to the equation $x_1+2x_2+…+nx_n=n^2.$

Let $n \ge 2, n \in \mathbb N$. $A_n$ denotes the number of positive integer solutions to the equation $$x_1+2x_2+...+nx_n=n^2.$$ Prove inequality $$\frac{n^n(n-1)^{n-1}}{2^{n-1}\left(n!\right)^...
1
vote
0answers
50 views

A divisibility conjecture related to the Ramanujan-Nagell equation

The Ramanujan-Nagell equation is $$ x^2+7=2^n, $$ where it has been proven (using non-elementary methods) that the complete solution is $n \in \{3, 4, 5, 7, 15\}$. I've found an elementary way to ...
6
votes
1answer
145 views

Solve in integers the equation $\sqrt{x^3-3xy^2+2y^3}=\sqrt[3]{13x+8}$

Solve in integers the equation $$\sqrt{x^3-3xy^2+2y^3}=\sqrt[3]{13x+8}$$ My work so far: I used www.wolframalpha.com. Then $x=9,y=8 -$ solution. My attempt: 1) Let $\sqrt{x^3-3xy^2+2y^3}=a, \...
2
votes
1answer
58 views

Integer Solutions to an Ellipse

I'm trying to find positive integer solutions to the ellipse $$x^2 - xy + y^2 - k^2 = 0$$ where $k$ is a constant. Specifically, I already have two solutions for a given $k$, and I'm trying to find a ...
2
votes
0answers
55 views

Finding solutions in modulo

If I know $x$ modulo m and n, then under what conditions on m,n and p will i necessarily know $x$ modulo p? My initial guess is only in trivial cases, i.e. p is a multiple of m or n, but i cant seem ...
2
votes
3answers
68 views

Check if a positive solution exist of a linear equation with two variables?

Let's say there's an equation $$a x + b y = c$$ where $a,b,c > 0$ are given. I want to know if positive solutions $x, y >0$ exist for this equation.
1
vote
2answers
44 views

Existence of positive integer solution of a equation

I'm trying to find if the following equation has positive integer solutions $$x + (x+y) + (x+2y) + (x+3y) + \cdots + (x+(n-1)y) = z$$ where $z$ and $n$ are given. I can't progress further. -> $xn +...
1
vote
1answer
42 views

Diophantine relations using an equation with polynomials of degree at most 4

I'm completely stuck at exercise 5.8.5 of Mathematical Logic, Chiswell & Hodges: Here are the mentioned definition and theorem: I'm stuck because I failed to use the hint given in the ...
1
vote
3answers
94 views

Number of integer solutions (ordered and unordered)

$$\frac1 a + \frac 1 b +\frac 1 c = \frac 34$$ Find number of triplets of $a\ , b\ , c\in \mathbb{Z}^+$ Should it not be infinite since it can be $\frac 34$ or $\frac38$ or $\frac9{12}$ etc. ...
2
votes
3answers
135 views

Find all natural roots of $\sqrt{x}+\sqrt{y}=\sqrt{1376}$ given that $x\leq y$

Find all natural roots of $\sqrt{x}+\sqrt{y}=\sqrt{1376}$ given that $x\leq y$ I'm confused of this equation because $1376$ is not a square!! So maybe it has no natural root! Am I right??
1
vote
1answer
39 views

Integer solutions to $210y^2=(x)(x+1)(2x+1)$

I'm looking to find integer solutions for large positive $y$ values (say over 1000) to the following equation: $210y^2=(x)(x+1)(2x+1)$ What I know so far: Integer solutions include (0,0) and (7,2) ...
1
vote
0answers
38 views

Given some arbitrary roots of a polynomial p(x,y,z,…) with integer coefficients, is it possible to tell if p has a root in the Gaussian integers?

I'm trying to find if p(x,y,z,...)=0 has a Gaussian integer root (more specifically, I want to find if p has a Gaussian integer root where the imaginary components are even, but if that can't be done, ...
7
votes
3answers
144 views

How many integer solutions are there of the equation $|x_{1}|+|x_{2}|+\cdots +|x_{k}|=n$?

How many solutions are there to the equation $$|x_{1}|+|x_{2}|+\cdots +|x_{k}|=n$$ for $n,k\in \mathbb N$ and $\forall\ 1\leq i\leq k,\ x_{i}\in \mathbb Z$? Any ideas? I don't know how to ...
1
vote
0answers
137 views

solve $x^y-y^x=xy^2-19,$ $x,y\in\mathbb{Z}$

I have been struggling to solve this exercise but with no result: $$x^y-y^x=xy^2-19,$$ $x,y\in{\mathbb Z}$ I have started to think it has no solutions at all. I have no idea how to solve it so I was ...
0
votes
0answers
29 views

Solutions to Diophantine Moving Window Inequations

I am interested in finding the number of non-negative integer solutions, $N(m,h,u)$, to this system of inequations $$ \left\{ \matrix{ 0 \le x_{\,0} + x_{\,1} + \cdots + x_{\,m} \le u \hfill \...
3
votes
1answer
67 views

Find all natural roots of: $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=d$ given that: $a<b<c$

Find all natural roots of: $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=d$ given that: $a<b<c$ Rearranging the equation gives: $$ab+bc+ac=abcd$$ What can we do with this?
2
votes
2answers
57 views

Diophantine equation, 3 variables

How do I solve the following equation, where $x,y,z$ have to be positive integers? $$ \frac{x^2}{y} + \frac{y^2}{z}+ \frac{z^2}{x}= \frac{y^2}{x} + \frac{z^2}{y} + \frac{x^2}{z} $$ Given that $$xyz=...
5
votes
1answer
150 views

Find the integer $x$ such $x^6+x^5+x^4+x^3+x^2+x+1=y^3$

Find the equation integer solution $$\color{red}{y^3=x^6+x^5+x^4+x^3+x^2+x+1}$$ It is obvious $x=0,y=1$ or $x=-1,y=1$ are solutions. How to find all solutions?
0
votes
1answer
32 views

Solve $z^3=kx+ny$ , ($k\neq{n},k,n\in \mathbb{N}$)

Solve $z^3=kx+ny$ , ($k\neq{n},k,n\in \mathbb{N}$) for positive integer unknowns $x,y,z$ I have really no idea for this!!
8
votes
0answers
95 views

Generalizing the growth of sums of two squares

Consider the set $S$ of numbers which are the sum of two (integer) squares, and define $S(n)$ as the number of members of $S$ in $\{1,2,\ldots,n\}.$ It is well-known that $$ S(n) \sim \frac{Kn}{\sqrt{\...
2
votes
7answers
122 views

Find all positive integer roots of : $5xy=19x+96y$

Find all positive integer roots of : $5xy=19x+96y$ I tried using decomposition technique but no success...,it seems suitable factorization of this equation is IMPOSSIBLE!! Handy calculations show ...
16
votes
1answer
1k views

Failure of an elementary 'proof' of Fermat's Last Theorem?

Can someone explain to me why this does not constitute a proof of Fermat's Last Theorem, please? Basically, using something I've read online, it appears you can write an equation for $(a, b, c)$ to ...
18
votes
0answers
247 views

Finding primes so that $x^p+y^p=z^p$ is unsolvable in the p-adic units

On my number theory exam yesterday, we had the following interesting problem related to Fermat's last theorem: Suppose $p>2$ is a prime. Show that $x^p+y^p=z^p$ has a solution in $\mathbb{Z}_p^{\...
-2
votes
1answer
30 views

find the number of tuples of positive integers [closed]

find the number of tuples (a,b,c,d) of positive integers \begin{array}{l} {a^3} = {b^2}\\ {c^3} = {d^2}\\ c - a = 64 \end{array} answer should be one of 0 , 1 , 2 , 4
3
votes
3answers
93 views

Number of solutions of: $3x+y=5702$

Find the number of ordered pairs $(x,y)$ satisfying $3x+y=5702$ in natural numbers restricted by: $x+y\le2003$ I don't know any method for counting number of solutions of such equations...
1
vote
4answers
83 views

If $x$ and $y$ are non-negative integers for which $(xy-7)^2=x^2+y^2$. Find the sum of all possible values of $x$.

I am not able to reach to the answer. I have used discriminant as $x$ and $y$ are both integers but it didn't give any hint to reach to answer. I am not able to understand how should I deal with these ...
1
vote
2answers
44 views

Determine if quadratic diophantine equation in two variables will generate perfect squares

I have come across two equations with variables $x,y$ \begin{align*} (x+ay)^2+ 4 x y\\ (x-y)^2-4 c x y \end{align*} where $a,c\in \mathbb{Z}_+$ are some constants. I would like to determine the ...
2
votes
1answer
47 views

Solution of equation of the form $n = 1234a + 56b + 7c$

I have $n = 1234a + 56b + 7c$. Is there a way to check if a triplet $(a,b,c)$ exists, such that all three are non-negative?
0
votes
0answers
8 views

What is the easiest way to solve diophantine equation with three unknowns?

Suppose we have a diophantine equation of the form: $$ ax + by + cz = d $$ What is the best (simplest, easiest) way to find the solution(s)? Should I apply extended Euclidean algorithm?
1
vote
1answer
32 views

How to find all positive integer solutions of a Diophantine equation?

Here is the equation $$ 6a+9b+20c=16 $$ To solve this, i follow the below steps : $\gcd(6,9)(2a+3b)+20c = 16$ let, $w = 2a+3b$ So, $3w+20c =16$ then, specific solution of $w = 112+20n$, $c = -16-...
2
votes
3answers
93 views

System of diophantine equations $x^2+3y=u^2$, $y^2+3x=v^2$

Solve the following system of Diophantine equations(the unknowns are positive integers): $$ \left\{ \begin{array}{c} x^2+3y=u^2 \\ y^2+3x=v^2 \end{array} \right. $$ I worked as follows: ...
5
votes
0answers
32 views

How many generators needed for Pell-equation-related group

Let $d$ be a positive integer which is not a perfect square. We have the norm multiplicative group homomorphism, $N:{\mathbb Q}[\sqrt{d}] \to {\mathbb Q}$ defined by $N(x+y\sqrt{d})=x^2-dy^2$. It ...
1
vote
3answers
34 views

Diophantine equations using Euclidean algorithm

I solved two systems of Diophantine equations using the Euclidean algorithm and I can't figure out where I went wrong because the solutions I test aren't working but I have rechecked my work several ...
5
votes
2answers
53 views

Pell equation in ${\mathbb Q}(x)$

Is it known whether the equation $A^2-(x^2+3)B^2=1$ has a solution $A,B\in{\mathbb Q}(x)$ with $B\neq 0$ ? My thoughts : I think that there is no solution, as the fundamental solution of $A^2-(x^2+3)...
-1
votes
1answer
59 views

Given $N$ find the number of natural numbers less than $N$ that may be written in the form $\frac{(k)(k+1)}{2}$

Given $N$, find the number of natural numbers less than $N$ that may be written in the form $$\frac{k(k+1)}{2},$$ where $k\in \Bbb N$. I know that the answer to this problem is approximately $\sqrt {...