Questions on finding integer/rational solutions of equations.

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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 ...
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0answers
49 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: ...
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1answer
33 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 ...
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4answers
105 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 ...
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4answers
70 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
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2answers
280 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.
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33 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 ...
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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 ...
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4answers
135 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? $a=-(k^2+2(p+s)k+p^2+ps+s^2)$ $b=2k^2+4(p+s)k+3p^2+3ps+2s^2$ ...
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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 ...
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0answers
39 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
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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
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1answer
52 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
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2answers
131 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
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4answers
427 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
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0answers
55 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
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3answers
67 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 ...
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1answer
45 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
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1answer
43 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.
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3answers
69 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 ...
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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
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1answer
77 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?
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1answer
39 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 ...
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1answer
52 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
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1answer
73 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 ...
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4answers
374 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) ...
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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 ...
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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
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1answer
47 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.
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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 ...
1
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1answer
74 views

What is quadratic equations in Algebra?

Yesterday someone asked a question in SE about indeterminate quadratic equations(of the form $x^2−ny^2=1$ which got me really interested in them and I thought I would try to learn something related to ...
3
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1answer
70 views

rational points on particular elliptic curve

I do have a few books that discuss elliptic curves, however... What are the rational points on $$ y^2 = 4 x^3 - 4 x = 4 x(x-1)(x+1)? $$ I think it ought to be $(-1,0), (0,0), (1,0).$ Maybe it's ...
4
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2answers
111 views

There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

According to Wikipedia, There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. I cannot find the proof in the citation provided. I am ...
2
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4answers
99 views

Solving $6ab+a+b-36pq-19p-13q=7$ where $a,b,p,q \in \mathbb{N}$, $a,b,p,q \neq 0$

Is there an efficient way to find solutions to the equation: $6ab+a+b-36pq-19p-13q=7$ where $a,b,p,q \in \mathbb{N}$ and $a,b,p,q \neq 0$ If the equation has no solutions, how could you prove that, ...
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1answer
103 views

$x^x + y^x=x^y + y^y$ positive integer solutions?

required is positive solutions for $x^x + y^x=x^y + y^y$? And negative integer solutions as well if possible?
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4answers
44 views

how do i prove $ab|n$ if $\gcd(a, b) = 1$ and $a|n $ and $b|n$?

Suppose that, for integers $a, b,$ and $n,$ $$\gcd(a, b) = 1\text{ and }a|n\text{ and }b|n.$$ How do I prove that $ab|n$ using linear Diophantine equations? Can I extend the above result to the ...
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1answer
162 views

Solve Diophantine Equations with powers

Find all integer solutions x; y to the following Diophantine equations: a)$$x^2=y^3$$ b)$$x^2=y^4-77$$ Can anyone please help me get started? I don't have any idea ( Im guessing it will involve ...
3
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1answer
92 views

solving $x^3-2y^3=1$ using cubic number field

I am trying to solve the diophantine equation $x^3-2y^3=1$ using $\mathbb{Q}(\sqrt[3]{2}).$ I've read this link: Solve $x^3 +1 = 2y^3$ The following is what i have tried: Finding all integer ...
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1answer
80 views

Problem with Diophantine equation

Let $a,b \in \mathbb N$ be coprime. Prove that for all $n\in \mathbb N$ such that $n>ab$ there are $r,s\in \mathbb N$ such that $n=ra+sb$. I'm really stuck on this problem. I know that since ...
7
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4answers
98 views

Diophantine equation: $(x-y)^2=x+y$

I have to solve the following equation: $(x-y)^2=x+y$, where $x$ and $y$ are non-negative integers. This equation has an infinite number of solutions, but how to prove that there exists a positive ...
3
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1answer
100 views

Solving $x^p + y^p = p^z$ in positive integers $x,y,z$ and a prime $p$

The question is from Zeitz's ''The Art and Craft of Problem Solving:" Find all positive integer solutions $x,y,z,p$, with $p$ a prime, of the equation $x^p + y^p = p^z$. One thing I noticed is ...
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0answers
101 views

15th power Diophantine equation

I'd appreciate some help (a hint) for the following. If $x,y>1$ are so that $2x^2-1=y^{15}$ then $x$ is a multiple of $5$. Don't know if this helps but the equation can be rewritten as ...
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1answer
59 views

$px^2-2y^2=1$,for what p,the pell equation has the result

$p$ is a odd prime ,If $px^2-2y^2=1$ is solvable,we can get Jacobi symbol $(\frac{-2}{p})=1$ ,so $p=8k+1,8k+3$ but when $k=12,p=97$, the pell equation $97x^2-2y^2=1$ is unsolvable.I think it's ...
8
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0answers
148 views

Solve the Diophantine equation $a^2(2^a-a^3)+1=7^b$.

The problem is to find all positive integers $a$ and $b$ such that $a^2(2^a-a^3)+1=7^b$. I found a=10, and my intuition tells me there are no more solutions. I've also shown that $a=42k+10$ for some ...
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1answer
49 views

Co-primality of coefficients of coprime integers

Given that $a,b$ are co-prime, we have infinitely many solutions for $x,y$ to the equation $$ax+by=c.$$ Furthermore, solutions have the form: $x=ca^{-1}+tb,y=cb^{-1}-ta$. Given that $c$ ...
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2answers
72 views

Can 11 be represented by $a^2 - 3b^2$ where a and b are integers?

I know the answer is no, just wan't to know how. From a similar question on the site I got that $a^2 - 3b^2$ should always equal a square modulo 3 which 11 is not. But I don't understand how to get to ...
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1answer
52 views

Diophantine Equation with 15th power

So I'm working on the Diophantine equation $2x^2-1=y^{15}$ (1) with x,y>1 In particular I want to show that x must be a multiple of 5. I have found that it suffices to show that for $y=1 mod 10$ ...
4
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1answer
72 views

Integer solutions of $ x^3+y^3+z^3=(x+y+z)^3 $

Consider the equation $$ x^3+y^3+z^3=(x+y+z)^3 $$ for triples of integers $(x, y, z) $. I noticed that this has infinitely many solutions: $ x, y $ arbitrary and $ z=-y $. Are there more solutions? ...
0
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2answers
76 views

Proving that if $a^2+b^2=c^2$ for $a,b,c \in \Bbb Z^+$, then either $a$ or $b$ is even. [closed]

Prove that if $a, b, c \in \Bbb Z^+$, and $a^2+b^2=c^2$, then either $a$ or $b$ is even. It seems like a proof by contradiction can be used here. I have my own proof below but it may need some ...
2
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2answers
128 views

How to solve equations like $x^2+y^2=2004^{2005}$?

I've found this kind of equation but I think I haven't enough mathematical tools to solve it. What would you do? $$x^2+y^2=2004^{2005}$$ Another kind: $$x^2+y^2=2005^{2004}$$