Questions on finding integer/rational solutions of polynomial equations.

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2
votes
0answers
169 views

An argument for “Brocard's problem has finite solution”

Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard%27s_problem. According to Brocard's problem ...
3
votes
5answers
109 views

Can every perfect square exist as the sum or difference of two perfect squares?

I believe this is trivial and I'm over-complicating it. But can every squared integer be expressed as the sum of two squared integers OR the difference of two squared integers? And is there a proof ...
0
votes
1answer
35 views

Solutions $3 p\sin x - (p+\sin x)(p^2-p \sin x +\ sin ^{2} x) =1$

$3 p \sin x - (p+\sin x)(p^2-p \sin x + \sin ^{2} x) =1$ has a solution for $x$. Then number of integral solutions of $p$ are ?
3
votes
2answers
70 views

Faster Sage Code for Diophantine Equation? [closed]

I'm having trouble with the computation time. Does anyone have any ideas for faster code? ...
2
votes
3answers
135 views

Integer solutions to $x^{x-2}=y^{x-1}$

Find all $x,y \in \mathbb{Z}^+ $ such that $$x^{x-2}=y^{x-1}.$$ I can only find the following solutions: $x=1,2$. Are there any other solutions?
9
votes
1answer
84 views

Solving a Diophantine equation: $p^n+144=m^2$

I found this Diophantine equation: $$p^n+144=m^2$$ where $m$ and $n$ are integers and $p$ is a prime number. I solved it but I want to know if there exist other proofs through the use of rules of ...
1
vote
1answer
45 views

How many non-negative integral solutions?

How many non-negative integral solutions does this equation have? $$17x_{17}+16x_{16}+ \ldots +2x_{2}+x_1=18^2$$ I add some conditions that bring more limitations: $$\sum_{i=1}^{17}x_{i}=20 \quad 0 ...
0
votes
0answers
29 views

Diophantine eqution with a parameter

My question is about the problem when is the number $$\frac{m^3 + n^3}{n^2+m^2+m+n+c}$$ a natural number. Here $c\in \mathbb{N}$ is a constant and $m, n \in \mathbb{N}$ are the variables. This ...
3
votes
1answer
39 views

Bounding $x^2+6x$ between consecutive cubes when solving $y^3=x^2+6x$

I am familiar with the method of bounding a polynomial between consecutive squares to prove it is not a square. For example, this method can prove $y^2=x^2+x+1$ has no solutions since ...
0
votes
2answers
57 views

The diophantine equation $z^2=a^2+bx^2+cy^2$

Is there a way to obtain (enumerate) the integer solutions $(x,y,z)$ of the following quadratic Diophantine equation $z^2=a^2+bx^2+cy^2$ where $a$ is an integer and $b, c$ are positive integers? I ...
2
votes
2answers
46 views

A simple question:

let $a,b,c,d$ be all positive integers such that $a-bc \neq 0$,and $\gcd(a,b)=1$. Under what conditions, $(a-bc)$|$(a-b^d)$? In other words, does it exist any integer $k \neq 1 $ such ...
2
votes
4answers
344 views

Sum of square patterns

Can anyone give the name of this pattern $$136^2+137^2+138^2+139^2+140^2+141^2+142^2+143^2+144^2 =\\ 145^2+146^2+147^2+148^2+149^2+150^2+151^2+152^2$$
1
vote
3answers
51 views

How can i solve this diophantine equation:$x^2-(6p-4q)x+3pq=0$?

I found this diophantine equation $$x^2-(6p-4q)x+3pq=0$$ (p and q both prime numbers) and i posted my answer but i want to know if there are other methods to find the solutions of this equation. What ...
1
vote
1answer
26 views

Number of coins using Diophantine equation

I'd like to solve this question using Diophantine equations: We have an unknown number of coins. If you make 77 strings of them, you are 50 coins short; but if you make 78 strings, it is exact. ...
4
votes
2answers
55 views

Solving the diophantine equation $p^2+n-3=6^n+n^6$

What are the pairs ($p,n$) of non-negative integers where $p$ is a prime number, such that $$p^2+n-3=6^n+n^6$$ How can I solve this diophantine equation?
9
votes
2answers
294 views

diophantine equation $x^3+x^2-16=2^y$

Solve in integers: $x^3+x^2-16=2^y$. my attempt: of course $y\ge 0$, then $2^y\ge 1$, so $x\ge 1$. for $y=0,1,2,3$ there is no good $x$. so $y\ge 4$ and we have equation $x^2(x+1)=16(2^z+1)$, ...
0
votes
0answers
28 views

Two variables, one equation

I have the equation: $(x + y - 1)(xy - 1) = 2015$. I solved this breaking up 2015 into its prime factors (5, 13, and 31). After a bunch of guess and check, I got that $x = 2$, and $y = -32$. Is ...
0
votes
1answer
27 views

3 Variables, One Equation

What triples (x, y, z) will satisfy the following equation?: $x^2$ + $y^2$ + $z^2$ = $7(x+y+z)$ I tried factoring the left side as $(x+y+z)^2 - 2xyz$, and I wasn't sure how to continue from there. ...
0
votes
3answers
41 views

Linear Diophantine Equation iff statement [closed]

Let $a,b,c$ be integers. For every integer $x_0$, there exist an integer $y_0$ such that $ax_0+by_0=c$. Determine conditions on $a,b,c$ such that the statement is true iff these conditions hold.
1
vote
0answers
77 views

find all integer solutions of $y^2=x^3-2$ [duplicate]

I’m blind about integer solutions of a polynomial. I have no number theory background, but I’m curious about how to figure out all integer solutions of a polynomial, for example this question. It is ...
5
votes
0answers
88 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
0
votes
1answer
43 views

Linear diophantine equation $97y-299x=10$

Here is my equation: $$97y-299x=10$$ I tried to solve like this: $$-299 =-3\cdot97-8$$ $$97=-12\cdot-8+1$$ $$-8=-8\cdot1+0$$ I'm not sure if I am correct or can I ignore the negative signs?
3
votes
5answers
70 views

If $a,b,c$ are integers such that $4a^3+2b^3+c^3=6abc$, is $a=b=c=0$?

If $a,b,c$ are integers such that $4a^3+2b^3+c^3=6abc$ , then is it true that $a=b=c=0$ ? I was thinking of infinite descent but can't actually proceed , please help. Thanks in advance
1
vote
3answers
128 views

Solve $x^p + y^p = p^z$ when $p$ is prime

Find the solutions in positive integers of $x^p + y^p = p^z$, where $p$ is a prime number. Particular case $p=2$: For $z=0$ there are no solutions. For $z=1$ the only solution is $x=y=1$. For ...
3
votes
2answers
57 views

The diophantine equation $x^2+y^2=3z^2$

I tried to solve this question but without success: Find all the integer solutions of the equation: $x^2+y^2=3z^2$ I know that if the sum of two squares is divided by $3$ then the two numbers ...
3
votes
1answer
67 views

Partition problem for consecutive $k$th powers with equal sums (another family)

This is the partition problem as applied to a special set, namely the first $n$ $k$th powers. Assume the notation, $$[a_1,a_2,\dots,a_n]^k = a_1^k+a_2^k+\dots+a_n^k$$ I. Family 1 The following ...
4
votes
1answer
52 views

For a given positive integer $n>1$ , how to find all positive integers $s,t$ such that $n^s-(n-1)^t=1$ ?

For a given positive integer $n>1$ , how to find all positive integers $s,t$ such that $n^s-(n-1)^t=1$ ? $s=t=1$ is clearly a solution . One more thing is clear that for any such $s,t$ we must have ...
0
votes
0answers
41 views

A bivariate quadratic diophantine equation

Given $a,b,c>0$, is there a procedure to solve $(x,y)\in\Bbb Z:ax^2+by^2=c$ in $O(\log^d c)$ arithmetic operations (either randomized or deterministic) with $d>0$ being fixed? Is there a ...
1
vote
0answers
30 views

Diophantine linear Equation Gaussian Integers

We know that $ax+by=c$ with $gcd(a,b)=1$ could be solved over $\Bbb Z$. Supposing if $a,b,c\in\Bbb Z[i]$, is there an analogous framework to find $x,y\in\Bbb Z[i]$ (at least of minimum norms)?
0
votes
1answer
101 views

On a theorem of Kronecker! [closed]

Let $\alpha$ be an irrational number and $\beta$ be an arbitrary real number, Prove that there are infinitely many pair of integers $(x,y)$ with $x\in\mathbb{N}$ such that: ...
7
votes
1answer
171 views

Solve $x^3=y^2-y+1$ in positive integers.

I recently started doing number theory and have finished with all the basic, intermediate and some of the advanced stuff with ease. However, I encountered this question and have been stuck for about a ...
5
votes
1answer
54 views

An integer sequence with integer $k$ norms

Find the maximum value of $n$(if exists) such that there exists a sequence $a_1,a_2,\ldots,a_n$ of positive integers such that for every $2\leq k \leq n$ $$\sqrt[k]{a_1^k+a_2^k+\cdots+a_k^k}$$ is ...
3
votes
0answers
48 views

A question on the Pell equation $x^2-pqy^2 = -1$, with prime $p,q$.

We know that a necessary but not sufficient condition such that, $$x^2-dy^2 = -1\tag1$$ is solvable is that $d$ is not divisible by a prime of form $4m+3$. It is not sufficient because the prime ...
0
votes
1answer
40 views

Question about the chakravala method on solving Pell's equation

I am currently reading on this old way of Pell's equation: http://en.wikipedia.org/wiki/Chakravala_method Looking at the section where they consider $N = 61$, it is not clear to me if the solution ...
5
votes
2answers
264 views

Diophantine system of two equations with four variables

Find all integer solutions for the system: $$\left\{\begin{array}{rcl}xy + vw &=& 5 \\ xv - yw &=& 6\end{array}\right.$$ It's supposed to be solvable by 9-graders...
0
votes
1answer
61 views

How do you solve the Problem below?

Let $u,v,w\in \mathbb{Z}>0$ denote 3 relatively prime integers(Pairwise coprime). If $(mn)$ is irrational, can we find 2 non-zero coprime (non-square) integers $u,v$ such that: ...
1
vote
1answer
41 views

$a,b$ are integers , both greater than $1$ , such that $(a^n-1)(b^n-1)$ is a perfect square for every positive integer $n$ , then $a=b$ ?

If $a,b$ are integers , both greater than $1$ , such that $(a^n-1)(b^n-1)$ is a perfect square for every positive integer $n$ , then is it true that $a=b$ ?
3
votes
2answers
64 views

How to find all positive integers $m,n$ such that $3^m+4^n$ is a perfect square?

How to find all positive integers $m$, $n$ such that $3^m+4^n$ is a perfect square? I have found $m=n=2$ is a solution, but cannot find any other and cannot prove whether there is any other solution ...
1
vote
1answer
39 views

Diophantine Equations with Factor Exponents

I'm trying to prove that the following equations have no solutions to finish a problem. They're intuitively impossible but I'm looking for rigorous arguments (if they are actually possible, then prove ...
4
votes
3answers
49 views

How many pairs of positive integers $(n, m)$ are there such that $2n+3m=2015$?

I know that $m$ must be odd and $m\le671$. Also, $n\le1006$. I can't go any further, any help?
1
vote
1answer
45 views

is it possible to find $x$ where $y$ is equal to a whole number in a non iterative fashion

Given the equation $$\frac{635x+326}{637+x} = y$$ where $$x>0$$ Is it possible to find all positive values of $x$ (there is only one) where $x$ is positive and $y$ is a whole number. While I ...
3
votes
1answer
56 views

Help me finding $a+b+c$ in the given question

If $a,b,c$ are three positive integers such that $$abc+ab+bc+ca+a+b+c=1000$$ then what is the value of $a+b+c$?
1
vote
2answers
60 views

Sums of Consecutive Cubes (Trouble Interpreting Question)

Show that it is possible to divide the set of the first twelve cubes $\left(1^3,2^3,\ldots,12^3\right)$ into two sets of size six with equal sums. Any suggestions on what techniques should be used to ...
1
vote
1answer
34 views

Show that $x^2 − Dy^2 = 1$ has infinitely many integer solutions.

Let $D$ be a non-square positive integer. Suppose there are positive integers $a$ and $b$ such that $a^2 − Db^2 = 1$. Show that the Diophantine equation $x^2 − Dy^2 = 1$ has infinitely many integer ...
3
votes
3answers
95 views

Find rational points on $x^2 + y^2 = 3$ and on $x^2 + y^2 = 17$

$(a)$ Find all rational points on the circle $x^2 + y^2 = 3$, if there are any. If there is none, prove so. $(b)$ Find all rational points on the circle $x^2 + y^2 = 17$, if there are any. If there ...
0
votes
0answers
27 views

Exponential and regular Diophantines?

I am looking for a reference on connections between exponential and "regular" (polynomial) Diophantine equations. For example, I was wondering about the Catalan-Mihailescu problem and I thought of the ...
4
votes
1answer
43 views

How to solve $a\sqrt{x}\pm b\sqrt{y}=c\sqrt{z}$

Let $a,b,c,x,y,z \in \mathbb{Z}>1$ How do I prove if $x,y,z$ are square-free integers and: $$a\sqrt{x}\pm b\sqrt{y}=c\sqrt{z}$$ Then $\gcd(x,y,z)>1$? I know for some of you it may be ...
0
votes
1answer
104 views

Can you explain this identity's secret with this Equation $n-th$ powers.

For $k = 0,1,2,3,4,5,6,7,8$, we have the equality, $$(-5)^k + (-119)^k + (-101)^k + (-215)^k + (-197)^k + 43^k + 157^k + 31^k + 217^k + 169^k\\ =\\ (-47)^k + (-161)^k + (-35)^k + (-221)^k ...
4
votes
1answer
213 views

No of right angled triangles [closed]

How many right angled triangles are possible with the perpendicular side equal to 36 units. I took the side $x$ and $y$ and using Pythagoras theorem you have $(x+y)(x-y) = 1296$ and $1296$ has $25$ ...
2
votes
0answers
49 views

Exponential diophantine: $2^x-7^y=z^2$

Find all integers $x,y,z$ such that $2^x-7^y=z^2$. For example: $2^3-7^1=1^2$ $2^5-7^1=5^2$ $2^7-7^1=11^2$ (But note that $\sqrt{2^9-7^1}\not\in \mathbb{Z}$.) The problem with this particular ...