Questions on finding integer/rational solutions of polynomial equations.

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0
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5answers
64 views

Finding $7$ inverse modulo $11$

I'm trying to find the inverse of $7$ modulo $11$. From what I understand, the steps are: \begin{align} &11 = 1(7) + 3 \\ &7 = 2(3) + 1 \\ \end{align} From here, you work backwards ...
0
votes
3answers
69 views

Solving the congruence $7x \equiv 41 \mod{13}$

I have to solve the following linear congruence: $$7x \equiv 41 \mod{13}$$ The question where I got this from comes in two parts. The first is that it asks to find the set of the inverses of $7 ...
0
votes
4answers
68 views

solve and explain the Diophantine equation [closed]

Solve Diophantine equation and find the value of $x$ and $y$. For the value of $x$ and $y$ we solve through Diophantine equation. $$199x -98y = -5 $$
1
vote
1answer
65 views

Linear diophantine equation word problem

I have the following word problem: A small clothing manufacturer produces two styles of sweaters: cardigan and pullover. She sells cardigans for $\$31$ each and pullovers for $\$28$ each. If her ...
0
votes
1answer
29 views

Find the general solution to diophantine equation $-221x + 187y - 493 = 0$

I have to find the general solution to $$-221x + 187y - 493 = 0$$ The main issue, I'm figuring out if I have found the general solution or not. Below, are my steps: The $\gcd{(-221,187)} = 17$ and ...
-1
votes
2answers
48 views

Reminder of equation

I have a simple(maybe too simple) question - how to find the reminder of equation? For example: $(85^{74}+17^{95})^{15} \equiv \ ? \ (mod\ 13)$ I know that it is something simple, but I couldn't ...
0
votes
1answer
11 views

Solve the comparison

I have difficulties with these type of problems: Solve the comparison: $\displaystyle67x + 17 \equiv 0\pmod{28}.$ I'm sure it is something very simple but I'm stuck on it more than $2$ hours :( . ...
2
votes
1answer
92 views

Solutions to $x^p+y^q = z^r$

Is there any $(p,q,r)$ with $\gcd(p,q,r) = 1$ and $\frac{1}{p}+\frac{1}{q}+\frac{1}{r} < 1$ for which we know that the only integer solutions (not necessarily primitive) to the equation ...
1
vote
1answer
59 views

Pythagorean triples,consecutive terms of an arithmetic progression

I am looking at the exercise: Find all the positive Pythagorean triples that are consecutive terms of an arithmetic progression. $$$$ So,according to the solution that I saw in my notes,we want to ...
1
vote
1answer
55 views

Random Algebra Problem

Prove that if a, b, c, x, y, z, and $\alpha$ are natural numbers. For every given set of x, y, z, the number $\alpha$ obtained from the following equation: $$\frac{a^2}{x^2} + \frac{b^2}{y^2} + ...
2
votes
2answers
185 views

number of solutions of x*y<N [closed]

How many solutions (unique pairs (x,y) ) exist for equation $xy < N$ ? constraints : $x >1 , y>1 , N<=50000$ I tried following method , but it fails for say N=24 , in which i calculate ...
0
votes
1answer
28 views

The trivial solutions of the diophantine equation $x^2+y^2=z^2$

The trivial solutions of the diophantine equation $x^2+y^2=z^2$ are the following: $$x=0, y=n, z=\pm n, n \in \mathbb{Z}$$ $$x= \xi , y=0, z= \pm \xi, \xi \in \mathbb{Z}$$ $$$$ My question is, why is ...
0
votes
1answer
44 views

What is the best way of solving for $x,y\in\mathbb N$ given the conditions $\begin{cases}x\mid y+a\\y\mid x + b\end{cases}$?

What is the best way of solving for $x,y\in\mathbb N$ given the conditions $\begin{cases}x\mid y+a\\y\mid x + b\end{cases}$? The letters $a,b\in\mathbb N$ denote constant known numbers. The ...
1
vote
3answers
313 views

Solutions to $ax^2 + by^2 = cz^2$

The integer solutions to the equation $x^2 + y^2 = z^2$ are very well studied. I'm wondering if there's any literature about the integer solutions to the equation $ax^2 + by^2 = cz^2$ where a,b,c are ...
0
votes
0answers
25 views

prove that $ax+by=b+c$ has solution in $\mathbb Z$ if and only if $ax+by=c$ has solution in $\mathbb Z$

Let $a,b,c,d\in \mathbb Z$ prove that: a)$ax+by=b+c$ has solution in $\mathbb Z$ if and only if $ax+by=c$ has solution in $\mathbb Z$ b)$ax+by=c$ has solution in $\mathbb Z$ if and only if ...
1
vote
2answers
72 views

Equation in rational numbers?

Is it true that this equation $6=\frac{x^2}{y^2+1}$ has no solutions in rational numbers? If so, why? It is quite evident that it has no solutions in integers (because $y^2+1$ never divides $3$).
11
votes
1answer
243 views

Prove that if $n(a^2+b^2+c^2)=abc$ then $2\mid n$

Is it true that if $n\in\mathbb N$ and the diophantine equation $$n(a^2+b^2+c^2)=abc,\\(a,b)=(b,c)=(c,a)=1\tag1$$ has positive integer solutions $a,b,c$, then $2\mid n$? I can prove that ...
1
vote
1answer
26 views

How many n-tuple to genereate zero from some random several variable equation that use constant power and a variable as the base?

Define algebraic number tuple as the aphabetical order sequence of variables that use in the equation. How many algebraic number n-tuple (x,y,z,...) are able to genereate zero to input into a several ...
1
vote
2answers
75 views

Proof that $y^2=x^3+x$ has a unique integer solution

Prove that the equation $y^2=x^3+x$ has only one integer solution, namely $x=y=0$.
0
votes
2answers
80 views

Solve and explain diophantine equation

A Diophantine equation ax+by = c always has a solution whenever a and b are relatively prime. Find x ,y such that $$93x-81y=3 $$
0
votes
0answers
42 views

The solutions to $x^2+y^2=5$ in $\mathbb{Q}$. [duplicate]

Consider the following equation: $$x^2+y^2=5.\tag{1}$$ What are the solutions to this equation if $x,y\in\mathbb{Q}$, where $\mathbb{Q}$ is the set of all rational numbers? My attempt: Because ...
1
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0answers
51 views

Diophantine Equation $2x^2+25=y^3$

I'm trying to find integer solutions to: $2x^2+25=y^3$. Here's what I've managed to do so far: y is odd. y and x are co-prime. In $\mathbb{Q}(\sqrt{2},i)$ we can write: ...
1
vote
1answer
34 views

Is every unit vector in $\mathbb{Q}^n$ the first column of a rational orthogonal matrix?

Equivalently, does every unit vector in $\mathbb{Q}^n$ belong to some orthonormal basis for $\mathbb{Q}^n$? This is clearly true for $\mathbb{Q}^2$, and for $\mathbb{Q}^3$ it seems to be true for ...
1
vote
4answers
106 views

The solutions to $x^2+5=y^2$.

Consider the equation $$x^2+5=y^2.\tag{1}$$ If $x,y\in\mathbb{Z}$, what are solutions to (1)? If $x,y\in\mathbb{Q}$, what are solutions to (1)? Note: $\mathbb{Z}$ is the set of all integers and ...
-1
votes
4answers
73 views

solve this problem with diophantine equation

A man arrives in a bank to cash a cheque. for some stated amount. The teller on the counter makes a mistake and interchanges dollars and cents. I donated 5 cents to a charity box at the bank. Later, I ...
4
votes
2answers
288 views

solve for all integer solutions to the diophantine equation

$$1/x+1/y=1/14$$ Find all integer solutions for x and y. I can solve linear diophantine equations without a problem normally but this has me stumped.
0
votes
0answers
35 views

Evaluation of certain trigonometric sums

In trying to approximate the number of solutions to the equation $3^n - 2 = k^2$ for positive integers $n, k$, I tried to use the circle method. In doing so, I had to bound the trigonometric sum for ...
1
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2answers
99 views

Integer solutions of $800000007 = x^2+y^2+z^2$

Prove that the equation, $800000007 = x^2+y^2+z^2$ has no solutions in integers.(That is $8$ followed by $7$ zeroes, with a $7$ at the end). I tried checking modulo $3$, $5$, $7$, and $10$, but ...
1
vote
4answers
138 views

One Diophantine equation

I wonder now that the following Diophantine equation: $2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$ have only this formula describing his decision? $a=-(k^2+2(p+s)k+p^2+ps+s^2)$ $b=2k^2+4(p+s)k+3p^2+3ps+2s^2$ ...
1
vote
1answer
35 views

On representing the general solution for the diophantine equation $a_1x_1+\dotsb+a_nx_n=c$

On representing the general solution with the special solutions for the diophantine equation $$a_1x_1+a_2x_2+\dotsb+a_nx_n=c$$ here $a_1 ,a_2, \dotsb,a_n,c\in\Bbb Z,(a_1 ,a_2, \dotsb,a_n)=1$. Can ...
0
votes
0answers
42 views

Divisors of Pell Equation Solutions

Let $d > 0$ be square-free. Let $\epsilon = x_0 + y_0 \sqrt{d}$ be the minimal solution to the Pell's equation $x ^ 2 - d y ^ 2 = 1$. Let $x + y \sqrt{d} = \epsilon ^ l, l \geq 1$ be a solution. ...
2
votes
3answers
103 views

Show that $0 = 2a^3-5ab^2+25b^3$ has no other integer solutions than $a = b = 0$.

I am trying to solve the following problem: I have the equation $0 = 2a^3-5ab^2+25b^3$, where $a,b \in \mathbb Z$. Obviously, $a = b = 0$ is a solution of this equation. But how can I show that there ...
1
vote
1answer
59 views

Solving equation involving the ceiling function

How can I solve the equation $$\lceil \log_{b}{1024} \rceil = n$$ where $n \in \mathbb{N}$ in terms of $b$? I have seen equations of a similar form (Solving an equation with floor function before), ...
5
votes
2answers
135 views

Integers can be expressed as $a^3+b^3+c^3-3abc$

$$S=\{a^3+b^3+c^3-3abc|a,b,c\in\Bbb Z\}$$ Can we decide $S$? that is, we want to find all integers of the form $a^3+b^3+c^3-3abc$. obviously, if $m,n\in S$, then $mn\in S$, so we only need to ...
13
votes
4answers
434 views

prove Diophantine equation has no solution $\prod_{i=1}^{2014}(x+i)=\prod_{i=1}^{4028}(y+i)$

show that this equation $$(x+1)(x+2)(x+3)\cdots(x+2014)=(y+1)(y+2)(y+3)\cdots(y+4028)$$ have no positive integer solution. This problem is china TST (2014),I remember a famous result? maybe ...
2
votes
0answers
56 views

An algorithm for solving linear diophantine equations?

I am entering an interesting team based math contest called the purple comet, and quite a lot of questions on this contest involve Diophantine equations. For this contest, you are given a computer, ...
3
votes
3answers
68 views

Solving for $3^x - 1 = 2^y$

Besides $x=2, y=3$, are there any other solutions? I know that if there is another solution: $y$ is odd since $2^y \equiv -1 \pmod 3$ $x$ is even since $3^x - 1 \equiv 0 \pmod 8$ $3 | y$ since $-1 ...
1
vote
1answer
46 views

Gap between smooth integers tends to infinity (Stoermer-type result)?

Consider the following claim : (*) Let $P$ be a finite set of primes, let $S$ be the set of natural numbers all of whose divisors are in $P$, and let $s_n$ denote the $n$-th element of $S$. Then ...
3
votes
1answer
46 views

Finding all possible values

we have to find all possible prime values $(p,q,r)$ such that $ pq = r + 1 $ $ 2(p^2+q^2) = r^2 + 1 $ I do not know how to start looking for an answer.
1
vote
3answers
71 views

Can we find positive integers $a$ and $k \geq 2$ with $2^n - 1 = a^k$?

I would appreciate if somebody could help me with the following problem: For a given positive integer $n$, can we find positive integers $a$ and $k$ ($k\geq 2$) such that $2^n-1=a^k$? The ...
0
votes
1answer
96 views

Prove that $2^x \cdot 3^y - 5^z \cdot 7^w = 1$ has no solutions

Prove that $2^x \cdot 3^y - 5^z \cdot 7^w = 1$ has no solutions in $\mathbb{Z}^+$, if $y\ge 3$.
3
votes
1answer
78 views

What's the set p,if $37x^2-113y^2=p$ is solvable,with p a prime

if $37x^2-113y^2=p$ is solvable.with p a odd prime. What's the set of all $p$? Does it have a formula?
1
vote
1answer
41 views

Real valued function associated with the Diophantine equation $a^2(2^a-a^3)+1=7^b$

The parent question that maybe still remains to be answered at this moment is:Solve the Diophantine equation $a^2(2^a-a^3)+1=7^b$ . As far as the parent question is concerned, when generalizing to ...
1
vote
1answer
55 views

Solution of a simple linear diophantine equation

I'm having a slight problem with a simple equation of the sort $a_1+a_2+a_3...=n$. Where $n,a_1, a_2, a_3... \in N$. I do know how to find the number of solutions to these equations when they are of ...
3
votes
1answer
81 views

Cubic diophantine equation

How can I solve the equation $x^3+x-1=y^2$ in positive integers? I know this equation defines an elliptic curve but this seems to be a non-elementary way to solve the question. Is there a more ...
8
votes
4answers
377 views

Equation with an infinite number of solutions

I have the following equation: $x^3+y^3=6xy$. I have two questions: 1. Does it have an infinite number of rational solutions? 2. Which are the solutions over the integers?($ x=3 $ and $ y=3 $ is one) ...
1
vote
1answer
112 views

$\left(x+{2\over x^2+x}\right)\left(y+{2\over y^2+y}\right)$ product is equal to positive integers, general solution

Given $\left(x+{2\over x^2+x}\right)\left(y+{2\over y^2+y}\right)$ this product is equal to positive integers. $x,y$ are both positive. Conditions for general solution is required. List a few ...
0
votes
2answers
52 views

does this equation has an answer?

This is the equation:$$x=\log(a+bx)$$, where $a$ and $b$ satisfies the conditions that let the equation makes sense. Does it have an answer that can be expressed explicitely? Thanks a lot.
0
votes
1answer
49 views

$(x+\frac{1}{x})(y+\frac{1}{y})$ is equal to positive integer, solutions. [closed]

$(x+\frac{1}{x})(y+\frac{1}{y})$ is equal to positive integer. General proof/(conditions?) for positive the solution.
4
votes
3answers
81 views

Four integers that satisfy $a+b+c+d\; =\; -3$ and $a^{3}+b^{3}+c^{3}+d^{3}\; =\; 3$

Find a set of 4 integers that satisfy $$a+b+c+d\; =\; -3$$ and $$a^{3}+b^{3}+c^{3}+d^{3}\; =\; 3$$ I am really not sure how to proceed. I tried letting $d = -c$ to see if that would yield a ...