Questions on finding integer/rational solutions of equations.

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3
votes
0answers
45 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)^...
12
votes
1answer
444 views

If $p$, $q$ are naturals, solve $p^3-q^5=(p+q)^2$.

In If $p,q$ are prime, solve $p^3-q^5=(p+q)^2$., the author asks to solve the equation $p^3-q^5=(p+q)^2$ for primes $p$ and $q$. A proof is given that $p=7, q=3$ is the only solution. In this "...
8
votes
0answers
156 views
+50

Solving (n+1)(n+2)…(n+k)−k = x^2

Let $n$ and $k$ be positive integers. Need to find all pairs of $(n,k)$ such that $$(n+1)(n+2) \cdots (n+k)−k = x^2,$$ where $x^2$ is a perfect square.
1
vote
0answers
133 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 ...
2
votes
1answer
55 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 ...
0
votes
0answers
37 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 ...
4
votes
0answers
54 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, \...
9
votes
0answers
343 views

Power Diophantine equation involving primes: $(p+q)^q-p^q-q^q+1=n^{p-q}$

Suppose $p$ and $q$ are prime numbers, and $n>1$ is a positive integer. Find all solutions to the following Diophantine equation:$$(p+q)^q-p^q-q^q+1=n^{p-q}$$ What I have tried: Obviously $p>q$...
2
votes
0answers
49 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 ...
1
vote
0answers
113 views

Strategies for solving rational Diophantine equations

Are there any strategies for solving Diophantine equations where the solutions can be any rational number, not just an integer, besides substituting $x=p/q$ and $y=r/s$, with $p,q,r,s$ integers with $\...
8
votes
0answers
91 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{\...
17
votes
0answers
225 views
+50

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
46 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?
1
vote
3answers
115 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??
2
votes
2answers
52 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
36 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
93 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
4answers
153 views

Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.

Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?
0
votes
0answers
30 views

What is the best time complexity for this case?

I only want to know if the following system has any integer solution or not. Actually, I do not need to know the solution(s), and only need to know the answer of question "Does the system have any ...
5
votes
1answer
146 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?
7
votes
3answers
143 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
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) ...
3
votes
1answer
66 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?
0
votes
0answers
22 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 \...
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=...
0
votes
1answer
31 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!!
2
votes
6answers
114 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 ...
2
votes
3answers
259 views

Find all integer solutions to $7595x + 1023y=124$

Find all integer solutions to $7595x + 1023y=124$ Using the Euclidean algorithm I have found the $\gcd(7595,1023)=31$ and found the Bezout identity $31=52\cdot1023-7\cdot7595$ but I'm not really sure ...
1
vote
2answers
39 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 ...
14
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 ...
-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
2answers
133 views

Solving Diophantine equation $1/x^2+1/y^2=1/z^2$

How can we find positive integers solutions $(x,y,z)$, where $\gcd(x,y,z)=1$ for the equation: $$1/x^2+1/y^2=1/z^2$$ Can we conclude that $x$ and $y$ are not coprimes for it to have solution?
3
votes
3answers
91 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
76 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 ...
0
votes
0answers
7 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
30 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-...
1
vote
0answers
79 views

Existence of positive solution of a diophantine

Is there any way to find if there are any positive solutions of a diophantine equation of the form $ax$ $+$ $by$ $=$ $c$ It is not necessary to find such $x$ and $y$. I just wan't to determine ...
1
vote
2answers
77 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
2answers
52 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
vote
3answers
32 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 ...
4
votes
0answers
123 views

Diophantine equation with binomial coefficient

Suppose that $p$ is a prime number and $p \le q \le p^2$ is an integer. How many solutions are there to the following equation? $$\binom{p^2}{q}-\binom{q}{p}=1$$ This question was proposed ...
5
votes
0answers
30 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 ...
4
votes
4answers
80 views

Showing that there are infinitely many integer solutions for the hyperbolic formula $|a^2 - 26 b^2| = 1$

I want to show that the formula $$ | a^2 - 26\cdot b^2| = 1$$ has infinitely many solutions $(a, b) \in \mathbb{Z}^2$. First I tried to solve the formula for one of the two variables, to get ...
-1
votes
1answer
55 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 {...
3
votes
3answers
104 views

Solutions to $\lfloor x\rfloor\lfloor y\rfloor=x+y$

Find all solutions to $$\lfloor x\rfloor\lfloor y\rfloor=x+y$$ and show that the non-Integral solutions lie on two unique lines. Also determine the equations of these 2 lines. I divided the problem ...
3
votes
0answers
206 views

Sum of the cubes of a Pythagorean triple equal a cube.

Apart from (3, 4, 5, 6) are there any more primitive solutions to $x^3+y^3+z^3=w^3$ where $x^2+y^2=z^2$ ? I’ve noted that if gcd(x ,y ,z) = k, then k divides w, so non-primitive Pythagorean triples ...
0
votes
1answer
44 views

Solve the following diophantic equations

I can't seem to find the solution to two problems in my textbook. They ask us to solve the diophantic equations: 1) $xy²-2y²-x-6=0$ $4x²-4xy+y²-9=0$ I tried several things but these two just ...
4
votes
2answers
117 views

Nature and number of solutions to $xy=x+y$

Find all solutions to $$xy=x+y$$ Initially the given condition was $x,y\in \Bbb{Z}$. $$$$In this case, I just guessed that the solutions were $(0,0)$ and $(2,2)$. As far as I can see, these are the ...
4
votes
1answer
143 views

Three Colour Analogue of Boolean Pythagorean Triples Problem

Having read about the Boolean Pythagorean Triples Problem (see here and this question), it occurred to me that a related problem would require the integers to be coloured in three rather than two ...