Questions on finding integer/rational solutions of polynomial equations.

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3answers
115 views

Fastest way of finding solution to n*const1+ const2 = x^2

I am trying to solve the following equation: n*const1 + const2 = x^2 Where n, const1, const2 and x are integers > 0. Const1, const2 are known, n and x are ...
2
votes
1answer
781 views

Parametrization of a conic and rational solutions

How can we parametrize the conic $C$: $x^2+y^2 = 5$, by considering a variable line through $(2,1)$ and hence all rational solutions of $x^2 + y^2 = 5$? I'm thinking let $x = \sqrt{5}\cos t$, and $y ...
0
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3answers
230 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 ...
0
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3answers
307 views

Prove (or disprove) 2 equation of 6 variables

Given $$(s-x_{1})(s-y_{1})(s-z_{1}) = (s-x_{2})(s-y_{2})(s-z_{2})$$ Where $s=x_1+y_1+z_1$ and all variables are positive non-zero integers. I need to prove that such values of $x_{1}, x_{2}, ...
7
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2answers
1k views

Solve the Diophantine equation $ 3x^2 - 2y^2 =1 $

Solve $$ 3x^2 - 2y^2 =1 $$ in $ \mathbb{Z}$. How can we do it? ( All of answers gave me a great help. Thanks a lot kind stackexchangers.)
6
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2answers
571 views

Second degree Diophantine equations

I found a question whether there are general methods to solve second degree Diophantine equations. I was unable to find an answer so is this known? In particular, the original writer wants to know ...
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1answer
107 views

Diophantine Equation $x^n+y^n=z^n$

Problem Using simple mathematical operators (+,- ,> etc.) can it be shown that (assuming $ x<y$) Fermat’s theorem is always true when $$ n\ge x$$ Request I am sure this approach has been ...
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2answers
391 views

Finding positive integer solutions to $n = ax^2 +by^2 - cxy$

How can I find the positive integer solutions to $x$ and $y$, given that $n$, $a$, $b$ and $c$ are all positive integers, in an equation of the form: $$n = ax^2 + by^2 - cxy.$$ Specifically, I want ...
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3answers
551 views

Fermat's Last Theorem - A query

Problem Statement: In Fermat's Last Theorem $$x^n + y^n = z^n$$ $x,y,z$ are considered integers. But upon closer inspection it is seen that it is also true for any rational numbers $x,y,z$. And that ...
8
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2answers
284 views

On Pythagorean Triplets

The Problem: In the Pythagorean triplets (a,b,c) when a < b then b can't be a prime number. The Background: While searching the properties of Pythagorean triplets in web I saw quite a few listed, ...
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2answers
139 views

Trivial solution when solving in integers

Suppose we want to solve $4(x+y)^{2}-3xy-6(x+y)=0$ where $x$ and $y$ are both integers. Why we only get the trivial solution?
10
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1answer
293 views

Local solutions of a Diophantine equation

I am trying to prove that the equation $$3x^3 + 4y^3 +5z^3 \equiv 0 \pmod{p}$$ has a non-trivial solution for all primes $p$. I am sure that this is a standard exercise, and I have done the easy ...
10
votes
3answers
375 views

Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$

I am trying to find all solutions to (1) $y^3 = x^2 + x + 1$, where $x,y$ are integers $> 1$ I have attempted to do this using...I think they are called 'quadratic integers'. It would be ...
1
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1answer
120 views

Finding the all triples

How to find the all positive integer triples such that : $$ab+c=\gcd (a^2,b^2)+\gcd(a,bc)+\gcd(b,ac)+\gcd(c,ab)=239^2$$
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0answers
121 views

Is there always a telescopic series associated with a rational number?

Here is something I thought up while I was bored and my, erm, fish were busy: Given a rational number $p\in(0,1)$, are there always positive integers $n$, $c_m$, and $w_m$ such that ...
2
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2answers
210 views

How to find all solutions to equations like $3x+2y = 380$ using matrices/linear algebra?

I'm coming up blank on Wikipedia and other sources, though this seems elementary. I'd like to know what techniques or processes are used to find all (integer) solutions to an equation such as $3x+2y = ...
7
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1answer
121 views

Diophantine equation question concerning squares

How many squares there are of the form $d^2={b}^{2}-4ac$ if $a ,b ,c ,d$ are natural numbers between $1$ and $n$ such that $0\le{b}^{2}-4ac$? My first approach is for $d$, that must be an square of ...
5
votes
5answers
3k views

Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$

I'm working on a homework problem that is as follows: Suppose that $n$ is a positive even integer with $n/2$ odd. Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$. ...
2
votes
1answer
302 views

Pre-Wiles' results on Fermat's Last Theorem

IIRC, there was such a result as "there is no more than 1 non-trivial solution of $x^n+y^n=z^n$, if any", wasn't it? (IIRC, Siegel theorem implies that there are finitely many solutions for $n>3$; ...
3
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1answer
150 views

Find the integer solution of $ a^b = 2^{2 c + 1} + 2^c + 1 $

Find the possible number of integer solution for this equation, such that $ b>1$ $$ a^b = 2^{2 c + 1} + 2^c + 1 $$ From $1$ to $1000$, $ {a = 2, b = 2, c=0} $ and $ {a = 23, b = 2, c=4} $ ...
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3answers
176 views

Solutions of exponential diophantine equation

How would I go about finding the solutions to the exponential diophantine equation $18n+10=2^k$ ?
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1answer
110 views

Still another diophantine equation

Can any of you guys provide a hint for thew following exercise? Exercise. There is no $3$-tuple $(x,y,z) \in \mathbb{Z}^{3}$ such that $x^{10}+y^{10} = z^{10}+23$. Thanks a lot for your insightful ...
3
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3answers
311 views

Show that the curve $y^2 = x^3 + 2x^2$ has a double point, and find all rational points

Show that the curve $y^2 = x^3 + 2x^2$ has a double point. Find all rational points on this curve. By implicit differentiation of $x$, $-3x^2 - 4x$ vanishes iff $x = -4/3$ and $0$. By implicit ...
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2answers
1k views

prove that $x^2 + y^2 = z^4$ has infinitely many solutions with $(x,y,z)=1$

Prove that $x^2 + y^2 = z^4$ has infinitely many solutions with $(x,y,z)=1$. Do I use the terms $x= r^2 - s^2$, $y = 2rs$, and $z = r^2 + s^2$ to prove this problem? Thanks for any help.
3
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2answers
159 views

show that $x^2+y^2=z^5+z$ Has infinitely many relatively prime integral solutions

How to show that this equation: $$x^2+y^2=z^5+z$$ Has infinitely many relatively prime integral solutions
8
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5answers
1k views

How to solve this Pell's equation $x^{2} - 991y^{2} = 1 $

How to solve the following Pell's equation? $$x^{2} - 991y^{2} = 1 $$ where $(x, y)$ are naturals. The answer is $$x = 379,516,400,906,811,930,638,014,896,080$$ $$y = ...
4
votes
1answer
101 views

Integers that are a sum of two $k$th powers in $n$ different ways

Do there exist infinitely many $k$ such that for all $n$ we can find a sequence $x_i$ of distinct natural numbers such that $x_1^k+x_2^k=x_3^k+x_4^k=\cdots=x_{2n-1}^k+x_{2n}^k$ ?
4
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2answers
235 views

$n = a^2 + b^2 = c^2 + d^2$. What are the properties of a, b, c and d?

If n is a positive integer that can be represented as the sum of two odd squares in two different ways: $$ n = a^2 + b^2 = c^2 + d^2 $$ where $a$, $b$, $c$ and $d$ are discrete odd positive integers, ...
4
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0answers
168 views

How many natural numbers $x, y$ are possible if $(x - y)^2 = \frac{4xy}{(x + y - 1)}$.

How many natural numbers $x$, $y$ are possible if $(x - y)^2 = \frac{4xy}{x + y - 1}$. Does this system has infinite solutions which can be generalized for some integer $k \geq 2?$ $(x - y)^2(x + y) ...
1
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2answers
404 views

$(x^n-x^m)a=(ax^m-4)y^2$ in positive integers

How do I find all positive integers $(a,x,y,n,m)$ that satisfy $ a(x^{n}-x^{m}) = (ax^{m}-4) y^{2} $ and $ m\equiv n\pmod{2} $, with $ax$ odd?
4
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1answer
140 views

Pythagorian quadruples

From my work on hyperelliptic equations I found how to get infinitely many solutions of the equation $a^4+b^4+c^2=d^4$. I call these solutions harmonic: $$\begin{array}{rcccccl} 1^4 &+& 2^4 ...
39
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2answers
2k views
+200

Decomposing polynomials with integer coefficients

Can every quadratic with integer coefficients be written as a sum of two polynomials with integer roots? (Any constant $k \in \mathbb{Z}$, including $0$, is also allowed as a term for simplicity's ...
3
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2answers
342 views

Squares of the form $x^2+y^2+xy$

How can I find all $(a,b,c) \in \mathbb{Z}^3$ such that $a^2+b^2+ab$, $a^2+c^2+ac$ and $b^2+c^2+bc$ are squares ? Thanks !
0
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3answers
95 views

method for finding integral points outside of a polynomial?

consider some function like $y = x^2 + 3x$ and then some family of related polynomial functions (like: $y = x^2 + 3x$, $y = 2x^2 + 4x$, $y = 3x^2 + 5x$, $y = 4x^2 + 6x$, etc.) what method or ...
4
votes
4answers
667 views

All integer solutions for $x^4-y^4=15$

I'm trying to find all the integer solutions for $x^4-y^4=15$. I know that the options are $x^2-y^2=5, x^2+y^2=3$, or $x^2-y^2=1, x^2+y^2=15$, or $x^2-y^2=15, x^2+y^2=1$, and the last one $x^2-y^2=3, ...
8
votes
1answer
346 views

On the Diophantine equation $a^2+b^2 = c^2+k$

Given the Diophantine equation, $$a^2+b^2=c^2+k$$ where k is a constant integer. Let $0 < a \le b$, and $\Delta_k(N)$ be the number of primitive solutions with $0 < c < N$ for some bound ...
2
votes
4answers
242 views

$|2^x-3^y|=1$ has only three natural pairs as solutions

Consider the equation $$|2^x-3^y|=1$$ in the unknowns $x \in \mathbb{N}$ and $y \in \mathbb{N}$. Is it possible to prove that the only solutions are $(1,1)$, $(2,1)$ and $(3,2)$?
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0answers
132 views

diophantine equation with squares over 3 variables

I am trying to find solutions for this diophantine equation $$x^2+y^2+x^2y^2=4z^2$$ I am looking for advice on a procedure to find all positive integer solutions for this equations.
4
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2answers
370 views

Finding solutions to $(4x^2+1)(4y^2+1) = (4z^2+1)$

Consider the following equation with integral, nonzero $x,y,z$ $$(4x^2+1)(4y^2+1) = (4z^2+1)$$ What are some general strategies to find solutions to this Diophantine? If it helps, this can also be ...
2
votes
4answers
236 views

Under what situations does $x+1$ divide $4n^2-x$?

I am looking at the equation $$\frac{4n^2-x}{x+1} = y$$ for even $x$ and $y$, both positive. Under what situations does $x+1$ divide $4n^2-x$?
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1answer
106 views

Minimal bivariate diophantine equation solution space

I am facing the following type of diophantine equations: $$ axy + bx + cy + d = 0 $$ Where $a$, $b$, $c$, $d$ are integers and solutions for $x$, $y$ in the integers are seeked. If $a=0$ one can ...
5
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2answers
303 views

Does this equation have infinitely many solutions?

I was considering some number theory problems which inspired me to write the following conjecture, which bears some resemblance to the Catalan problem, but is in fact different: Fix two distinct ...
11
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3answers
767 views

Proving a statement regarding a Diophantine equation

FINAL EDIT : Prove that if $p^z|n^2-1$ $$p^{x-z}(p^{z}-1)=\dfrac{ n^2-1}{p^z}-3$$ doesn't hold for any chosen values of $p,x,n$ and $z$. Here $p>3$ is an odd prime , $x=2y+z, \ ...
3
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0answers
313 views

Diophantine with Gaussian Integer

I'm trying to find the set of solutions to a specific diophantine equation over $\mathbb{Z}[i]$. The equation is the following: $$ z_1^2 + z_2^2 + z_1*z_2 + 39 = 0$$ with $ z_1$ (resp $z_2$) such ...
4
votes
2answers
284 views

A diophantine equation $x^3+y^3-xy^2=1$

What kind of methods there are to find integer solutions of $x^3+y^3-xy^2=1$? I tried some inequalities and congruences without success. I also found on Wikipedia that this might be a Thue equation ...
1
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2answers
73 views

Enumerating all $x$ such that $b^n$ divides $x^2-x$

Given $b$ and $n$, I need to efficiently enumerate all integers $x$ (and $k$) such that: $$x^2-x=kb^{n}$$ $$x∈⟦b^{n-1},b^n-1⟧$$ I simply don't know how to proceed. I tried the naive quadratic ...
3
votes
1answer
237 views

Simple exponential diophantine equations with huge solutions?

It seems like there's been an explosion of (exponential) diophantine equations with straightforward solutions lately and it would be great to have an example at hand of how such simple equations can ...
2
votes
2answers
323 views

All positive integers satisfying $n2^n=5^m+7$

How can one solve each of the equations below in positive integers? $$2^n=5mn+7$$ $$mn2^n=5^m+7$$ $$n2^n=5^m+7$$
9
votes
4answers
3k views

Diophantine equation $a^2+b^2=c^2+d^2$

I was reasonably certain I've seen this before, but I was wondering how to solve the Diophantine equation $$a^2+b^2=c^2+d^2$$ I tried a web search and found nothing on this one. I'm trying to avoid ...
0
votes
1answer
172 views

Count the number of integer solutions for $a \times b \geq k$?

count the number of integer solution for $a \times b \geq k$ given the conditions 1) $1 \leq a \leq p$ 2) $1 \leq b \leq q$ (k, p, and q are constant).