Questions on finding integer/rational solutions of equations.

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9
votes
4answers
108 views

How many answers to $|3^x-2^y|=5$?

How many answers are there to the equation $|3^x-2^y|=5$ such that $x$ and $y$ are positive integers? Are there infinite? I've found $(2,2)$, $(3,5)$, and $(1,3)$. It seems to explode with larger ...
0
votes
1answer
57 views

Solve $x^2 = 2^x$. [duplicate]

One can see that the solutions are $x=2, 4$ and $x=-0.77$(approximately) seen from the graph. I am posting this to find if there is a way to solve this and find solutions like polynomial equations. ...
1
vote
3answers
108 views

Find all pairs of nonzero integers $(a,b)$ such that $(a^2+b)(a+b^2)=(a-b)^3$

Find all pairs of nonzero integers $(a,b)$ such that $(a^2+b)(a+b^2)=(a-b)^3$ My effort Rearranging the equation I have \begin{array} \space (a^2+b)(a+b^2)-(a-b)^3 &=0 \\ ...
19
votes
2answers
2k views

Find a thousand natural numbers such that their sum equals their product

The question is to find a thousand natural numbers such that their sum equals their product. Here's my approach : I worked on this question for lesser cases : \begin{align*} &2 \times 2 = 2 + ...
1
vote
1answer
51 views

Proof that $ax+by+cz=0$ has infinitely many solutions. [closed]

For all non-zero integers $x,y,z$ clearly there exist infinitely many non-zero integers $a,b,c$ such that $$ax+by+cz=0$$ How can I prove this simple statement?
1
vote
1answer
105 views

Integer solutions for $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$?

Find all triples $(x, y, z)$ where $x, y, z$ are coprime integers such that $$\dfrac{1}{x^2}+\dfrac{1}{y^2}=\dfrac{1}{z^2}$$ I did the following: ...
0
votes
1answer
30 views

Solving $\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1$ in integers

Find all pairs $(x,y)$ of integers such that $$\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.$$ Since $x,y$ are symmetric, we can assume $x\geq y$, so the right-hand side is $x-y+1$. If $x=y$, the equation ...
0
votes
0answers
16 views

Solving equation with sin and exp

I'm trying to solve the following equation but getting nowhere. $\pi*\sin(\pi*x/b)/(2*b)+m*\exp(n-m*x) = 0$ With b,m,n constants. I remember from a long time ago that there are some types of ...
0
votes
0answers
61 views

Integer solutions to $y^2 = \frac{x^5-1}{x-1}$

$$y^2 = \frac{x^5-1}{x-1}$$ has integer solutions. How many pairs $(x,y)$ are there? My Work If $\sqrt{x^4+x^3+x^2+x+1}$ is an integer then there is a solution. But what to do now. Note: This ...
2
votes
2answers
59 views

A triple of pythagorean triples with an extra property

I'm trying to prove the non-existance of three positive integers $x,y,z$ with $x\geq z$ such that\begin{align} (x-z)^2+y^2 &\text{ is a perfect square,}\\ x^2+y^2 &\text{ is a perfect ...
1
vote
2answers
32 views

Find all $n$ such that $m = an$ or $m =\dfrac{n}{a}$

$a$ is the 1st digit (from the left) of a $3$-digit number $n$. We get the number $m$ by removing a from $n$ and putting it on the right of the unit-digit. For example, the number $123$ becomes $231$. ...
1
vote
1answer
12 views

When does:$(p^y+1 )\bmod (p^x+1)=0 $ if $(y,x)=1$ and $p $ is a prime number?

I'm interesting to look the solution of this equation :$$(p^y+1 )\bmod (p^x+1)=0 $$ at a least to see an example of the two coprime $y, x$ for which $(p^y+1 )\bmod (p^x+1)=0 $ but i don't succed , ...
1
vote
2answers
63 views

$X^4 - 4Y^4 = -Z^2$ has no solutions in non zero integers

I am trying to prove that $X^4 - 4Y^4 = -Z^2$ has no solutions in non zero integers. I know there are similar questions on MS, but that minus signs before the $Z$ gives me a hard time. For the ...
2
votes
1answer
77 views

Solve in integers $b^{11}-1=a^{2016}+a^{2015}+\dots+1$

Find all integers $(a,b)$ satisfying $$b^{11}-1=a^{2016}+a^{2015}+\dots+1.$$ Obviously, we can get the factorisation $(b-1)(b^{10}+\dots+1)=a^{2016}+a^{2015}+\dots+1$, but I'm not sure how to ...
0
votes
3answers
32 views

Solve $x^2+2xy-782y=0$ diophantine equation

I'm trying to solve $x^2+2xy-782y=0$ diophantine equation. With these steps: a) $(*4)$; $4x^2+8xy-3128y=0$ b) $(+/-4y^2)$; $4x^2+8xy+4y^2-3128y-(4y^2)=0$ c) Reducing; $(2x+2y)^2+(-3128y)-(4y^2) = ...
2
votes
2answers
42 views

A fundamental tool used in the study of Diophantine equations…

Notations: $K$ is a perfect field, $\overline K$ an algebraic closure and $V\subseteq \mathbb P^n(\overline K)$ is a projective variety on $\overline K$. If $V$ is defined over $K$, in symbols ...
5
votes
3answers
99 views

Solutions to $x+y+z-2 = (x-y)(y-z)(z-x)$

Show that the equation $$x+y+z-2 = (x-y)(y-z)(z-x)$$ has infinite solutions $(x,y,z)$ with $x, y,z$ distinct integers. In my attempt to solve the problem only found solutions form $x=y, z=2-2x$. ...
3
votes
5answers
110 views

Prove $a/b+b/a$ for $a$ and $b$ natural is only natural for $a=b$ [closed]

Is it possible to prove that for any natural $a,b$ the value of $a/b+b/a$ will not be natural with exception $a=b$?
5
votes
1answer
54 views

Proving minimum number of chairs is $567$

Hints only please! I am trying to figure this out somehow. A row can have as many girls, and a column can have as many boys. Proof by contradiction seems like a good technique, but I am not sure ...
0
votes
1answer
16 views

Finding other solutions to diophantine equations

I understand how to find the first solution to these equations but can't grasp how the other solutions are found. E.g. $102x\equiv 12 \pmod{174}$ So I can find the $gcd(174,102)=6$ (showing that ...
0
votes
0answers
43 views

Solving for 2 unknowns in a perfect square

Basically was looking at a sequence of perfect squares for a given constant integer A, where in the first instant we can easily and trivially generate a sequence of perfect squares using the ...
0
votes
2answers
38 views

Integral Solutions in Diophantine Equation

How do you solve this problem: Describe the integral solutions to the equation $317a + 241b = 9.$ I know the answer is $(a, b) = (35 + 241k, −46 − 317k)$ for integers k but I don't know how ...
1
vote
1answer
40 views

Find integers $x$ and $y$ such that $\frac{27^{x+y}}{9^{xy}}=27$ and $\frac{4^{2xy}}{8^{x+y}}=512$ .

Find all the integers $x$ and $y$ such that : $$\frac{27^{x+y}}{9^{xy}}=27$$ and :$$\frac{4^{2xy}}{8^{x+y}}=512$$ I'm in Algebra two and I feel like there are certain types of math I haven't ...
1
vote
0answers
42 views

Generate All Triangular Square Numbers Recursively?

We define a triangular number as follows: $$\sum_{n=1}^{n} x_{i}$$ As in $T_3$ = $3+2+1$, or $6$. Generating these triangular numbers is rather simple and done by the equation: $$T_n ={(n^2 + ...
0
votes
0answers
14 views

Find all numbers $x, y\in\mathbb{Z}$ satisfying certrain equations

I'm looking for hints on solving such equations as $(x+2)^4=y^3+x^4$ or $x^2+y^2=1997(x-y)$. They cannot be solved using typical techniques. Again, I'm looking for hints.
0
votes
0answers
11 views

Reference Request For Hermite normal form of non full row rank matrix

Could someone recommend me some references which discuss the problem of the reduction of a matrix which is not full row rank into its Hermite normal form?
1
vote
1answer
43 views

Is every integral point of an arithmetic scheme contained in an affine open set?

The schemes $$X = Proj \mathbb{Z}[s,t] = \mathbb{P}^1_{\mathbb{Z}}$$ and $$Y = Proj \mathbb{Z}[x,y,z]/(x^2 + y^2 - z^2)$$ both have isomorphic generic fibers as schemes over $\mathbb{Z}$, and there is ...
2
votes
1answer
60 views

On $4n+1 = x^2, 5n+1 = y^2$ and the Fibonacci numbers

While answering this question, I decided to look at the particular case, $$4n+1 = x^2\\5n+1 = y^2$$ to be solved simultaneously. The solutions for $n,x,y$ are A157459, A007805, and A049629, ...
5
votes
0answers
92 views

Find all $x,y,z \in \mathbb{N}$ that $x^xy^y=z^z$ [duplicate]

I was curious about the following diophantine equation $$x^xy^y=z^z$$ The equation seemed to have no positive integer solutions except for when $x$ or $y$ was $1$. Though I attempted to solve the ...
1
vote
2answers
65 views

Solve in positive integers

Solve the following equations in postivie integers: 1) $x^2+3y^2=z^2$ 2) $x^2+y^2=5z^2$ I have solved them using this as a reference, but I am interested in other solutions, more "elegant" ones. ...
3
votes
2answers
54 views

Is there a systematic way to solve in $\bf Z$: $x_1^2+x_2^3+…+x_{n}^{n+1}=z^{n+2}$ for all $n$?

Is there a systematic way to solve in $\bf Z$ $$x_1^2+x_2^3+...+x_{n}^{n+1}=z^{n+2}$$ For all $n$? It's evident that $\vec 0$ is a solution for all $n$. But finding more solutions becomes harder ...
0
votes
2answers
93 views

Find all triples of non-negative Integers $a,b,c$ such that $a!b!=a!+b!+c!$ [duplicate]

Exactly what it says in the Title; not much development from there :/
1
vote
1answer
19 views

Linear Diophantine equation solving

Find all the positive solutions in integers of $x+y+z=31$ $x+2y+3z=41$ For the first, I have subtract second equation from first , and by that I have found that $y=2k-10$ ,$z=k$,is it possible?
1
vote
1answer
48 views

Solution to $p^3-p+1=a^2$

What are the solutions to $p^3-p+1=a^2$ where $p$ is prime and $a$ is natural? I found the solutions: $p=3$ and $a=5$ $p=5$ and $a=11$ and one solution when $p$ is not a prime: $p=56$ and $a=419$, ...
8
votes
1answer
87 views

Diophantine equations for polynomials

I know that there has been work on diophanitine equations with solutions in poynomials ( rather than integers ) of the Fermat and Catalan type $x(t)^n+y(t)^n=z(t)^n$ ; $x(t)^m-y(t)^n=1$ and these have ...
1
vote
3answers
110 views

Solving $x^2 + y^2 = 1 + z^4$ with (x,y,z) = 1 and z < x < y

I have a computer programming problem where I need to find n many sets of integers that meet the condition $x^2 + y^2 = 1 + z^4$ with (x,y,z) = 1 and z < x < y I can do this relatively easily ...
3
votes
0answers
80 views

Symmetry and trivial solutions to Pell equations

Below is a representation of the solutions to the equation $x^2-Dy^2=1$ for $6(6-1)\leq D \leq 6(6+1)$: \begin{array}{c} & 30 & 31 & 32 & 33 & 34 & 35 & 36 & 37 & ...
2
votes
1answer
20 views

How to deal with an additive constant in a linear congruence equation?

I am trying to solve the following equation: $10x+3 \equiv 2 \pmod{17}$. The problem I am having is that I don't know what to do with the number $3$. This is what I have done so far: $10x+3 = ...
23
votes
2answers
342 views

A diophantine equation with only “titanic” solutions

I made a note some time ago that I had read in some book that the equation $$313(x^3+y^3)=t^3$$ has positive integer solutions, but that these are so large that it would be absolutely hopeless to ...
11
votes
4answers
236 views

Find all natural numbers $x,y$ such that $3^x=2y^2+1$.

Find all natural numbers $x,y$ such that $$3^x=2y^2+1$$ solutions are $(1,1)$, $(2,2)$, $(5,11)$. I found that parity of both is same and If $x$ Is odd it is of the form $4k+1$.
6
votes
1answer
76 views

When $n!=m(m+1)(m+2)$: A Diophantine Equation

I believe that I saw this problem not long ago in a book: Solve the Diophantine Equation $k!=n(n+1)(n+2)$, where $k,n$ are positive integers. However, I am no longer able to find this question, and ...
-13
votes
3answers
321 views

Is there any flaw in the following proposed elementary " proof'' of FLT? [closed]

Earlier on today, I received an interesting mail from a certain exceptionally talented undergraduate student, claiming an elementary proof of FLT. He has also submitted his paper to a formal journal ...
3
votes
2answers
164 views

Pell equations upper bound

Consider the Pell equation $x^2-p_n y^2=1$, where $p_n$ is the $n$th prime. Is $n^{2 \sqrt{n}}$ a reasonable upper bound for the smallest integer solution for $y$? Above is a plot of $\log x$ ...
1
vote
1answer
54 views

On the genus of a curve and its set of rational points.

The genus $g$ of a nonsingular curve $C$ of degree $n$ is defined as $g = \frac{1}{2}(n-1)(n-2)$. Let $C(Q)$ denote the set of rational points on $C$. By Faltings, we know that $C (Q) < \infty$ for ...
0
votes
0answers
14 views

Finite solutions in $x^2+D=λk^n$

In http://www.math.tifr.res.in/~saradha/saradharev.pdf it is stated that in $$x^2+D=λk^n$$ The following result of Siegel shows that the number of solutions $(x, n)$ in each case is finite: If $f(x) ...
4
votes
1answer
78 views

Is it true that for every $k\in\mathbb N$ , there exist infinitely many $n \in \mathbb N$ such that $kn+1 , (k+1)n+1$ both are perfect squares ?

Is it true that for every $k\in\mathbb N$ , there exist infinitely many $n \in \mathbb N$ such that $kn+1 , (k+1)n+1$ both are perfect squares ? What I have tried is that I have to necessarily solve ...
1
vote
1answer
58 views

Find the diophantine equation $x^2(y^2-1)=z^2-1$ solution

How can I solve (find all the solutions) the nonlinear Diophantine equation Let $x,y,z$ be postive integers ,and $x,y,z\ge 2$,find this following equation solution $$x^2=\dfrac{z^2-1}{y^2-1}$$ I ...
1
vote
0answers
31 views

Solving a Diophantine Equation with 2 variables

This is my answer for the following question: Find all natural numbers $(a,b)$ for which $a^b-b^a=1$. When $a$ or $b$ equals $1$, $(a,b)=(2,1)$ is trivial. If $a,b>1$, I generalized the problem ...
2
votes
2answers
34 views

A diophantine related query

Supposing I give you a multivariate equation $$F\in\Bbb Z[x_1,\dots,x_n]$$ Following is undecidable: 'Is there an $(a_1,\dots,a_n)\in\Bbb N^n$ such that $F(a)=0$?' However is the following always ...
1
vote
1answer
18 views

Proof that $x^2 + D = AB^y$ has in every case of $D,A,B$ a finite amount of solutions $x,y$

Could somebody please find me a proof that $$x^2 + D = AB^y$$ has in every case of $D,A,B$ a finite amount of solutions $x,y$. I forgot how this is called and would greatly appreciate it if someone ...