Questions on finding integer/rational solutions of equations.

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2answers
43 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
95 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
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0answers
45 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
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1answer
76 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
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0answers
43 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 ...
5
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0answers
57 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
56 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
50 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
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3answers
55 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.
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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
38 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
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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. ...
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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??
1
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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) ...
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
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0answers
134 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
23 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
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?
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
147 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
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!!
8
votes
0answers
92 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
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 ...
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 ...
17
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0answers
235 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^{\...
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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
92 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...
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4answers
78 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
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2answers
41 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
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?
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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
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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
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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
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 ...
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 ...
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)...
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1answer
56 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 {...
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 ...
4
votes
0answers
124 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 ...
0
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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 ...
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 ...
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 ...
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?
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 ...
3
votes
1answer
38 views

Positive integer solutions to $1\cdot m!\cdot(m^2)!\cdots (m^p)!=2(n^p)!$

Let $m$ and $n$ be natural numbers and $p$ a positive integer such that $$1\cdot m!\cdot(m^2)!\cdot\ldots\cdot(m^p)!=2(n^p)!$$ One solution is $(m,n,p)=(2,1,1)$. Are there any others?
3
votes
0answers
209 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 ...
3
votes
1answer
67 views

Finding all pairs of integers that satisfy a bilinear Diophantine equation

The problem asks to "find all pairs of integers $(x,y)$ that satisfy the equation $xy - 2x + 7y = 49$. So far, I've got \begin{align} xy - 2x + 7y &= 49 \\ x\left(y - 2\right) + 7 &= 49 \\ ...
4
votes
2answers
195 views

Minimum of $|ax-by+c|$

Find the minimum of the function $$ f(x,y)=|ax-by+c|$$ where $a,b,c \in \mathbb N$ and $x,y \in \mathbb Z$. The questions here and here are similar but they are in cases where $x, y$ are ...
0
votes
0answers
28 views

Finding smallest positive value of a function.

Given four positive integers $A,B,C$ and $D$, we have to find the minimum absolute difference between $A+qC$ and $B+wD$ where $q$ and $w$ are non-negative integers. I know it has something to do with ...