Questions on finding integer/rational solutions of polynomial equations.

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3
votes
1answer
234 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
322 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).
10
votes
1answer
199 views

Diophantine equation $x^y-y^x=11$

How can one find all integer solutions to $x^y-y^x=k$, for a given k? Example case $x^y-y^x=11$
5
votes
1answer
110 views

Numbers of the form $a^m-b^n$

Can all positive integers $k$, be written as a difference of two perfect powers $k=a^m-b^n$, with $m,n>1$ and $a,b$ positive integers? A number is imperfect if it can not, which numbers are ...
17
votes
1answer
496 views

Finding solutions to equation of the form $1+x+x^{2} + \cdots + x^{m} = y^{n}$

Exercise $12$ in Section $1.6$ of Nathanson's : Methods in Number Theory book has the following question. When is the sum of a geometric progression equal to a power? Equivalently, what are ...
3
votes
1answer
112 views

Solutions to $x^2-2=y^p$ for $p\geq3$

Do any integral solutions exist for $x^2-2=y^p$ for $p\geq3$?
15
votes
2answers
380 views

Solution in integers to $2^n+n=3^m$

How to find all positive integers $m,n$ such that $2^n+n=3^m$ ? We have by inspection $(m,n)=(0,0)$ and $(1,1)$ And there are no more for m and n both less then $100$.
1
vote
1answer
114 views

Showing this equation has no small integer solutions?

Let $n > 2$ be an integer. I've got a pair of equations $$x^2 = (n^2 - 2)y^2 \pm 1,$$ one of which has a solution when $y=n$ (with +1); I'm trying to say that, when $1 \leq y < n$, neither ...
8
votes
1answer
303 views

Solutions of $p!q! = r!$

The title says it all, more or less. Obviously, there are infinitely many "trivial" integral solutions of the form $p=n, q=(n!-1), r= n!$. How many non-trivial solutions are there? I came across this ...
2
votes
3answers
273 views

Are there any integer solutions to $x^2 - (n^2 - 2)y^2 = -1$?

I was just wondering if there are any integer solutions to the Diophantine equation: $x^2 - (n^2 - 2)y^2 = -1 \ \ $ for $n > 2$ I don't think there are any but can't prove why.
7
votes
2answers
405 views

Prove that $x ^ 3-y ^ 2 = 2$ only has one solution $(3,5)$

Fermat claimed that $x ^ 3-y ^ 2 = 2$ only has one solution $(3,5)$, but did not write a proof. Who can provide a proof that a high school student can accept? Thank you for your help An answer ...
2
votes
2answers
93 views

Unique solutions for $ab = n ^ 2$

How many unique solutions are there to the equation $ab = n^2$ , where $n$ is a constant, $a,b \geq 1$ and $a,b,n$ are integers. Is there any way of counting the number of solutions?
-1
votes
2answers
321 views

Aristotelian syllogisms in modern mathematics?

Somewhere (?) in the writings of Gian-Carlo Rota, I recall a statement that old-fashioned Aristotelean syllogisms are not used in modern mathematics. I know of one gaudy counterexample, and wondered ...
3
votes
2answers
213 views

Count the number of integer solution to $\sum_{i=i}^{n}{f_ig_i} \geq 5 $

How to count the number of integer solutions to $$\sum_{i=i}^{n}{f_ig_i} \geq 5$$ such that $\displaystyle \sum_{i=1}^{n}{f_i}=6$ , $\displaystyle \sum_{i=1}^{n}{g_i}=5$ , $\displaystyle 0 \leq f_i ...
8
votes
1answer
305 views

Find all positive integers $L$, $M$, $N$ such that $L^2 + M^2 = \sqrt{ N^2 +21}$

Sorry, this is very much 'can you do my homework' but I have a little competition at work that requires me to solve (and prove) the following. Find all positive integers $L$, $M$, $N$ such that ...
7
votes
2answers
2k views

(Diophantine?) Equations With Multiple Variables

A normal linear Diophantine equation would be some: $ax + by = c$ where $a$ and $b$ are constants. Solution can be found using Euclid's Extended Algorithm. But what about for longer equations such ...
4
votes
1answer
376 views

Solving a Diophantine Equation using factorisation of ideals

I am stuck on the following question which is given as follows: Prove that the only integer solutions to the equation \begin{equation} x^2 + 13 = y^3 \end{equation} are $(70,17)$ and $(-70, 17)$. ...
6
votes
1answer
202 views

When is the sum of first $n$ numbers equal to the sum of the next $k$ numbers?

When is the sum $1+2+\cdots + n = (n+1) + (n+2) + \cdots +(n+k)$? The easiest solution $(n,k)$ is $(2,1)$. For example, $1+2 = 3$. Do any others exist? Roots of $(n+k)^2 + (n+k) = 2n^2 +2n$ give ...
2
votes
3answers
638 views

Prove that there are no integer solutions to $3m^2-1=n^2$.

How do I answer this? Prove that it is impossible to find any integer $n$ such that $n^2 \equiv 2 \pmod 4$ or $n^2 \equiv 3 \pmod 4$. Hence or otherwise, prove that there do not exist integers ...
2
votes
3answers
162 views

Solve $V_1+V_2+\cdots+V_k=A, V_1^2+V_2^2+\cdots+V_k^2=B$ in positive integers

There have been changes made to the second equation in the pair that will be worth looking at. All values for the solutions must be non-zero positive integers (natural numbers). Please note, all ...
10
votes
3answers
1k views

Solving $x^3-y^3=xy+61$ in integers

I'm trying to solve the following equation: $$x^3-y^3=xy+61$$ I got: $$(x-y)(x^2+xy+y^2)=xy+61$$ But I can't go any further. I am looking for a solution in integers. I need some hints to proceed, ...
0
votes
1answer
185 views

Total number of solutions of an equation

What is the total number of solutions of an equation of the form $x_1 + x_2 + \cdots + x_r = m$ such that $1 \le x_1 < x_2 < \cdots < x_r < N$ where $N$ is some natural number and $x_1, ...
1
vote
3answers
159 views

How to obtain all the solution for this diophantine equation?

How can I Find all solutions of the diophantine equation? : $$xy=\frac{3x+y}{2}.$$
1
vote
2answers
522 views

How to solve this Diophantine Equation. Step by Step

What are all the solutions to the diophantine equation: 6x-6y-xy=0.
3
votes
4answers
536 views

Methods for quartic diophantine equation

$$3x^2 + 2y^4 = z^4$$ How do I solve this?? I would like to use so-called "elementary number theory", not abstract algebra (e.g. $\mathbb{Z} ( \sqrt d)$) or elliptic curves. Note: I'm not asking ...
8
votes
1answer
315 views

diophantine equations $x^3-2y^3=1$

I'm not familiar with diophantine equations. At most my approaches doesn't give results. I need to solve the following equation $$x^3-2y^3=1$$ Where $x,y,z\in\mathbb{Z}$ I know $x=-1,y=-1.x=1,y=0,$ ...
2
votes
1answer
998 views

How to find integer solutions for indeterminate equations in $Ax + By = C$

I would like to find some positive integer solutions to an equation in the form $Ax + By = C.$ I have already seen some methods for doing this, such as the one outlined in this Math.SE post. What I ...
0
votes
2answers
304 views

Very interesting equation $x^y + y^z + z^x = x^z + y^x + z^y$

How to solve equation of the form $x^y + y^z + z^x = x^z + y^x + z^y$. I grouped like this: $(x^y-y^x) + (y^z -z^y) + (z^x - x^z) = 0$ one of the case is $x^y-y^x = 0; y^z - z^y = 0$ and $z^x - x^z = ...
-1
votes
3answers
1k views

No Integer Solutions and Congruences

Using congruences, I seek to prove two things: 1) $x^2 - 4y^2 = 3$ has no solutions in integers $x,y,z$. I think this can be done using modulo 4? How so? 2) $3x^3 - 7y^3 + 21 z^3 = 2$ has no ...
7
votes
2answers
2k views

Number of positive integral solutions of equation $\dfrac 1 x+ \dfrac 1 y= \dfrac 1 {n!}$

$$\dfrac 1 x+ \dfrac 1 y= \dfrac 1 {n!}$$ This is one of the popular equation to find out the number of solutions. From Google, here I found that for equation $\dfrac 1 x+ \dfrac 1 y= \dfrac 1 {n!}$, ...
0
votes
3answers
179 views

Numbers of the form $x^2+axy+by^2$

This book, which needs to be returned quite soon, has a problem I don't know where to start. How do I find a 4 parameter solution to the equation $x^2+axy+by^2=u^2+auv+bv^2$ The title of the ...
0
votes
2answers
86 views

conditional equation

Find all pairs $(x, y)$ of integers such that $x \ge 1$ and $y \ge 1$ and $x^{y^2} = y^x$. work done: if $d = \gcd (x, y)$. then $x = du$ and $y = dv \implies \gcd (u, v) = 1$ and the equation ...
4
votes
1answer
152 views

special equation $z^y$ - $y^z$ = $x^y$

I have prepared an equation by myself $z^y$ - $y^z$ = $x^y$. This equation has infinitely $(0, n, n)$ and $(n, 0, n)$ solutions for some positive integer $n$. Can this equation can be treated as ...
6
votes
3answers
342 views

Whenever Pell's equation proof is solvable, it has infinitely many solutions

Prove that whenever the equation $x^2 - dy^2 = c$ is solvable, then it has infinitely many solutions. I consider that, if $u$ and $v$ satisfy $x^2 -dy^2 = c$ and then $r$ and $s$ satisfy $x^2 ...
2
votes
1answer
83 views

Is always $\small {rq-1 \over 2^B} +1 \le \min(q,r) $ with equality iff $\small q$ or $\small r$ is a divisor…

I had a simpler question before such that I could even answer it myself. For the next step I seem again to be too dense today. (Remark several days later: it's not only being dense... I still don't ...
5
votes
1answer
415 views

Integer solutions of $x^4 + 16x^2y^2 + y^4 = z^2$

I come across this question very long ago. I just got one solution by my computer search. If any one know the other solutions and resolvability, please let me know. $$x^4 + 16x^2y^2 + y^4 = z^2$$ has ...
2
votes
2answers
265 views

nature of diophantine solutions in general

I had a question in my mind from many years. we generally present the trivial solutions to Diophantine equations. Diophantine equations usually always have some sort of trivial solution (if you ...
0
votes
1answer
65 views

Is always $\small {rq-1 \over 2^B} \le q-1 $ with natural r,q,B and $\small r,q \in \{1, \ldots, (2^B-1)\} , odd$?

Consider the comparision in positive integers $$\small {rq-1 \over 2^B} \le q-1 $$ where $\small B $ is the given parameter and r and q are residues modulo $\small 2^B$ (which also implies, that the ...
12
votes
1answer
826 views

Finding pairs of triangular numbers whose sum and difference is triangular

The triangular numbers 15 and 21 have the property that both their sum and difference are triangular. There are another 4 pairs less than 1000. To complete this problem, I have done like this: To ...
0
votes
1answer
611 views

How to solve Linear Diophantine equations?

I have read about Linear Diophantine equations such as $ax+by=c$ are called diophantine equations and give an integer solution only if $\gcd(a,b)$ divides $c$. These equations are of great importance ...
1
vote
1answer
77 views

Greatest $n$ that can be written in the form of $ax+by=n$

In a diophantine equation $ax + by = n$ with $(a, b) = 1$, the greatest possible value of $n$ such that both $(x, y)$ are not positive is $ab − b − a$? This is given in my module (without any proof). ...
1
vote
0answers
331 views

Is this Diophantine equation proof correct?

It's probably a good thing I decided to try working on problems from the book. This section seems to be proving difficult. In any case, I'm asked to prove that $x^4+4y^4=z^2$ has no non-trivial ...
2
votes
2answers
225 views

rational triangles and cosines

I've recently started to try working on exercises from this book on Diophantine equations before I need to return it to the library. This one has me slightly stumped. It asks to show that the cosine ...
1
vote
1answer
2k views

Problem on Linear Diophantine Equation over 3 variables

How to solve $ax+by+cz=d$ over integers where $a,b,c,d$ are integers?
1
vote
2answers
137 views

Constructive proof need to know the solutions of the equations

Observe the following equations: $2x^2 + 1 = 3^n$ has two solutions $(1, 1) ~\text{and}~ (2, 2)$ $x^2 + 1 = 2 \cdot 5^n$ has two solutions $(3, 1) ~\text{and}~ (7, 2)$ $7x^2 + 11= 2 \cdot 3^n$ has ...
6
votes
1answer
4k views

Integer solutions (lattice points) to arbitrary circles

Wolfram Alpha will provide integer solutions to arbitrary circle equations. I'm trying to understand how it's able to calculate them, but despite a fair bit of digging I haven't found any discussion ...
2
votes
2answers
248 views

Triplets based equation

Let $p \ge 7$ be a prime number. Find the triples $(x, y, z)$ in $\mathbb{Z}$ such as $xyz$ is not equal to zero, $\gcd (x, y, z) = 1$ and $x^p + 2y^p = z^2$. I want triplets and proof/generalization. ...
3
votes
2answers
203 views

Solutions of $x^2 + 119 = 15 \cdot 2^n$ without trial and error

I seen this equation at math.stack exchange The equation $x^2 + 119 = 15 \cdot 2^n$ has only six solutions. Those are (1,3) (11, 4), (19, 5), (29, 6), (61, 8) and other one is I don't know. This ...