Questions tagged [exponential-diophantine-equations]

Diophantine equation where the variable is in the exponent (ex.: find all sols for $3^x-5^y=7$)

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Solutions to $4^\alpha+1=q^\beta$ [duplicate]

Problem. Solve the equation $$ 4^\alpha+1=q^\beta $$ for prime number $q$ and positive integers $\alpha,\beta$. Background. This question arose in the context of the following question: Find all ...
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Find solution set of Diophantine equation: $\frac{a^b-1}{a-1}=\frac{c^d-1}{c-1}$

After reading up on Catalan's Conjecture, a related equation piqued my interest: Let $a>c>1$ and $b,d>2$, where $a,b,c,d \in \mathbb{Z}$. I am looking for integer solutions to $$ \frac{a^b-1}{...
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2 votes
2 answers
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Collatz Conjecture and non-trivial cycles

Consider $$n \rightarrow ... \rightarrow an+b$$ to be a sequence of natural numbers to which we apply $3n+1$ or $\frac{n}{2}$ operations. This sequence is called a cycle, if and only if $an+b=2n$. (...
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Solve this equation $12^x-5^y=19$ positive integers

Find all $x,y$ be positive integers,such $$12^x-5^y=19$$ I found $(x,y)=(2,3)$ is solution,maybe have other,so I consider case $x,y>3$ and $\pmod 9$,since $$12^x\equiv 0\pmod 9,x\ge 2$$ then $5^y\...
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Solving the exponential Diophantine equation $2^{m+1} = zk + 1$ where $m,k,z \in Z$ for a given very large $z$

I am working on a problem where I have ended with an exponential Diophantine equation of the form $$2^{m+1} = zk + 1$$ where $m,k,z \in Z$ for a given very large $z$ (i.e., factoring $z$ is ...
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Query on $3^x \pm 2^{x-a}$ with relation to prime and semiprime

While doing some research(more closer to some playing) with the formula $3^x\pm2^{x-a}$ for $x\in \mathbb{N}$ and $ \{a\mid a \in \mathbb{Z}_{\geq 0},\hspace{1mm} a\le (x-1)\}$ I've become to observe ...
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On relation between prime numbers and exponential Diophantine equation $c\cdot a^x\pm b=z$, feat. $71999999\cdots$

While dealing with some integers which are the elements of the following set $$\{p\mid p\in\mathbb{P}, p=72\times(10^n)-1\}$$ I've could observed that when $n\in\{6,7,8,9\}$, they are all primes.(Such ...
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  • 331
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Solving $5^x - 2^x = 117$ using modular arithmetic

I'm trying to solve $5^x - 2^x = 117$. The solution is very easy to find by inspection, however I'm having difficulty trying to find reasoning specifically using modular arithmetic where the solution ...
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Prove that $4^m + 11^n = x^2$ with $(m,n,x) \in \mathbb{N}^3$ does not admit solutions.

My problem is the following :\ Show that $4^m + 11^n = x^2$ with $(m,n,x) \in \mathbb{N}^3$ has no solution. My first idea is to separate the problem into two cases: when $n$ is even, and when $n$ ...
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Integral solutions to $20^m-10m^2+1=19^n$.

Find postive integral solutions of the equation $20^m-10m^2+1=19^n$. My solution, Using modulo $10$, I got the information that $n$ is even. Using modulo $19$, I got the information that $m$ should ...
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4 votes
5 answers
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Exponential diophantine equation $2^x+7^y=9^z$.

The challenge is to solve this equation $2^{x}+7^{y}=9^{z}$ in positive integers. The obvious solution is $x=y=z=1$. Using brute force, I found $3$ possible solutions: \begin{eqnarray*} (x_1,y_1,z_1)&...
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How to solve this exponential/diophantine equation?

The equation is: $$ 2^x3^{x-1}=y\cdot3^{x-1}\cdot2+z\cdot2^{x-1}$$ for natural numbers. I’ve tried to divide this expression or try various substitutions, but nothing is working.
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Solve the Diophantine equation $ 12^x + y^4 = 56^z $ where $x, y, z$ are non-negative integers.

I need help to solve the following Diophantine equation:$$ 12^x + y^4 = 56^z $$ where $x,y,z$ are non-negative integers. My attempt: We can see immediately that $x=y=z=0$ is a solution. By taking mod $...
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How to solve Diophantine Equations $2^{4m}-1=(10k-5)^2$ and other similar ones?

My motivation is my try for solving the following great question and then I stuck: Find all positive integer solutions of $(a,b,n)$ s.t. $(2^a-1)(2^b-1)=n^2$ so any variable in this question $\in \{1,...
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5 votes
2 answers
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Does the exponential Diophantine equation $2^nn^{-n}s^s(n-s)^{n-s}=1/9,9$ have a solution?

Assuming $0^0=1$, how to prove/disprove the existence of integers $n,s$ such that $0\le s\le n\ge 1$ and $$\left({2\,s\over n}\right)^s\left({2\,(n-s)\over n}\right)^{n-s}\in\left\{{1\over 9},\,9\...
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What does it take to find integer solutions to this exponential division equation?

Consider this equation where $a$ and $b$ are positive integers. $$k = \frac{2^a - 1}{2^{a+b} - 3^b}$$ This equation has the trivial solution $k=1, a=1, b=1$. How would I find more solutions, or show ...
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3 answers
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How to prove there exists at least one positive integer solution of $k$ and $n$ in $2^{k} = 3^{n-1}(m + 1) - 1$, where m is any even integer?

Here's the related question Is it possible to prove there exists at least one positive integer solution for $k$ and $n$ in $2^{k} = 3^{n-1}(m + 1) - 1$, where m is any even positive integer? So we ...
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Understanding the Lambert W function

I am asking this question because I wish to get a feel for how it would be to invent the Lambert W function. I would like to really understand it. We have the equation: $$xe^x=y \tag{1}$$ and we want ...
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The solutions of the diophantine equation $z^x=1+(z-1)y^2$

I'm trying to determine the positive integer solutions of the diophantine equation $z^x=1+(z-1)y^2$. This equation can be written also as $$y^2=\frac{z^x-1}{z-1}=\sum_{i=0}^{x-1}z^i$$ Trivial ...
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The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$,$y$ are integers

The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$,$y$ are integers I tried using hit and trial and got only one solution $x=3$ and $y=1$ which is the answer. How can we prove that it is the ...
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9 votes
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Collatz Conjecture: For a cycle where the maximum odd integer is $x_{max}$, does it follow that $x_{max} < 3^n$

I am working on understanding the upper limit in the case where a non-trivial cycle exists for the Collatz Conjecture. Is the following reasoning valid for establishing that the maximum odd integer in ...
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1 answer
117 views

Establishing a lower bound with integers $m,n$ for $2^m - 3^n$ when $2^m > 3^n$

When I look at the encyclopedia of integers, it is clear that this grows even though it doesn't grow continuously. It seems to me that it should be possible to establish a lower bound based on ...
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For integers $m>2, n>1, c> 0$, with $2^m - 3^n=c$, does it follow for integers $x>0,y>0$ that $2^{m+x} - 3^{n+y} \ne c$

I am attempting to generalize an answer given by Will Jagy here. I may have made a mistake since the conclusion is stronger than I typically have seen and my argument is pretty much the same as Will's....
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7 votes
2 answers
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Solve over the positive integers: $7^x+18=19^y.$

Solve over the positive integers: $$7^x+18=19^y.$$ Progress:- I first took $\mod 7,$ so we get $5^y\equiv 4 \mod 7$ since $5$ is a primitive root of $7$ and $5^2\equiv 4\mod 7.$ So we get $y\equiv 2\...
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Solution of the Diophantine equation

What are the possible triples (x,y,z) in positive integers such >that, $$(x+y)^{2}+3x+y+1=z^{2}$$ I have used the inequality approach and many others but wasn't able to find an answer.
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Can you prove mu-recursive functions are Diophantine without bounded universal quantifiers?

Most proofs I’ve come across for the unsolvability of Hilbert’s tenth problem show that every recursive function is Diophantine using the approach of mu-recursive functions, i.e. they show that the ...
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4 votes
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Find a positive integer $i$ such that $9i + 1$ divides $2 \times 10^i - 1$

I have written a Python program running over $i$, but up to billions there is no solution, so I guess there is no solution. Trying to prove that, I looked at multiplicative order, but I do not get a ...
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-2 votes
2 answers
222 views

Diophantine equation $2^x-3^y=2021$ [closed]

$$2^x-3^y=2021$$ where $x,y$ are non-negative integers. I only found $2^{11}-3^3=2021$.
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Exponential diophantine equation only need a single solution

Hello can someone please help me find a formula for at least one integer solution to the following without brute force $Y = ((g^x)-A)/p$ Where $g,p,A$ are constants, $P$ is a large prime, $Y$ and $X$ ...
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10 votes
1 answer
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Finding all natural $x$, $y$, $z$ satisfying $7^x+1=3^y+5^z$

The problem goes as follows: Find all possible pairs of $x,y,z \in \mathbb{N}$ which satisfy the equation $7^x+1=3^y+5^z$ My first instinct was to continue by modding, but I don't think I can get ...
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3 votes
0 answers
189 views

Integer solutions of $(2^{B_N}-1) \bmod \left( (-(3^N)) \bmod 2^{B_N}\right) =0$ other than $N\in \{1,2,3\} \{4,8,12\}$?

Re-viewing some older fiddlings on more elementary aspects of the existence of "1-cycles" in the Collatz-problem I came at the following problem. Let $N \gt 0$ denote the number of odd steps ...
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2 answers
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For which positive integer $n$, is $\frac{2^n-1}{3}$ a factor of $4m^2+1$ for some integer $m$? (How to prove that one answer group is the only one?)

Edit: Dietrich Burde kindly gave me a hint but I, in my inexperience, am not able to understand the logic. I've tried but failed. So, right now my problem is understanding him. So, if you can explain ...
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4 votes
2 answers
182 views

Find the integer solutions to $4^x - 9^y = 55$

I want to find the integer solutions of: $$ 4^x - 9^y = 55$$ For now, I see that $x = 3, y = 1$ is an integer solution to the equation. How can I rigorously prove there are no other solutions for $x, ...
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2 votes
1 answer
115 views

How many solutions for $37+\frac{3(x-1)x}2+3^{x+2}=y^2\,$?

How many solutions does the following exponential diophantine equation have in the positive integers? $$37+\frac{3(x-1)x}2+3^{x+2}=y^2$$ The solutions $\,(x,y)\,$ I have found are: $\;(1,8),\;(2,11),\;...
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1 vote
0 answers
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Integer solution of $2 m (2^{n+1} - 1) = n (3^{m+1} - 1)$

I was reading a lot of other questions here which are supposed to be similar, but none of the answers gave me a hint, how I can approach this. I wrote a Python program to solve it, but up to $2^n < ...
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0 votes
0 answers
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Divisibility of sum of three powers times two powers

I am trying to find some restrictions for what the exponents, $a_{i}\in\mathbb{Z}^{+}$, of two powers can be if the following equation yields an odd integer: $[3^{n}+3^{(n-1)}2^{(a_{2})}+3^{(n-2)}2^{(...
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0 votes
1 answer
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Minimal value of a diophantine expression

Given the expression $$x^a - y^b>0$$ what is the minimum positive value it can have given $x>y > 1$ and $a,b>1$. For example, if I have $4^a - 3^b$ I would conjecture that the smallest ...
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1 vote
0 answers
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Minimizing a diophantine equation

If we are given $x,y$ both integers where $x>y>1$ for the expression $$(x^{\frac{a}{b}} - y)\cdot y^{b}$$ Is there any way to find integer values for a,b in which this expression is minimal and ...
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  • 561
3 votes
1 answer
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Find all $n,x \in \Bbb N$ such that $3\cdot2^x+4=n^2$

Find all $n,x \in \Bbb N$ such that $3\cdot2^x+4=n^2$ Arranging the equation a bit one has that $$(n-2)(n+2)=3\cdot2^x.$$ Now considering the cases $n-2=3, n+2=2^x$ from $n-2=3 \Rightarrow n=5$. ...
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3 votes
0 answers
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When is $b^2 - \{b-1\}_2$ with odd $b$ a perfect power of $2$? (The bracket-notation explained below)

For the complete extraction of the factor $p$ and its powers from a natural number $m$ let's define the notation $$ \{n\}_p := { n \over p^{\nu_p(n)}} $$ While looking at the question of existences of ...
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1 vote
1 answer
116 views

Are there easier ways to solve this integer equation than brute-force

The following question searched for the solutions of $$ y^x=x^{50} $$ User JCAA was able to reduce this to finding $s$ and $q$ as the solutions of $$ \frac{s^q}{q} = \frac{50}{p} \quad\textrm{with}\ p\...
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1 vote
1 answer
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Are there any solutions to this nonlinear diophantine equation?

Are there any $k\gt2$ for which we have a solution of the nonlinear diophantine equation $x^{k-1}=\sum_{i=0}^{k-1}10^i$. This question arose when I tried to provide a simple solution to this question ...
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6 votes
2 answers
196 views

Power values of polynomial

$f(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n$ is a polynomial of degree $n$ with positive integer coefficients. Primary problem statement: Is the Exponential Diophantine Equation $f(f(a) + 1) = y^...
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  • 1,075
2 votes
3 answers
139 views

Diophantine equation power of 7 and 2

$$ 7^x = 2^y \cdot 3 + 1$$ Find all positive $(x,y) \in \mathbb{N}^2$ When I look at this equation $\mod 3$ or $\mod 7$ it does hold - but how can I continue from here? I know that $7^x -1$ is even so ...
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  • 779
5 votes
1 answer
200 views

Base-Exponent Invariants

A sum of powers is called a base-exponent invariant if its value does not change if each base and exponent are switched. The simplest example is $2^4$, which of course is equal to $4^2$. Another ...
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2 votes
1 answer
128 views

Find all non-negative integral solutions to the equation $2013^q+2014^w=2015^r$.

Let $\mathbb{N}$ denote the set of nonnegative integers. Find all solutions $(q, r, w) \in \mathbb{N}^3$ to the equation $$2013^q+2014^w=2015^r.$$ Can someone explain me this solution ? one can post ...
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  • 1,266
2 votes
2 answers
65 views

Integer exponent equation

Show that $$(2^a-1)(2^b-1)=2^{2^c}+1$$ doesn't have solution in positive integers $a$, $b$, and $c$. After expansion I got $$2^{a+b}-2^a-2^b=2^{2^c}\,.$$ Any hint will be appreciated.
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0 votes
1 answer
242 views

Linear diophantine equation of n variables.

I know how to solve a linear Diophantine equation of 2, 3 variables. But is there a way to solve directly a linear Diophantine equation of n variables. For example using matrix?
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2 votes
1 answer
132 views

Solving $n(4n+3)=2^m-1$ in positive integers

Find all positive integers $m$ and $n$ such that $$n(4n+3)=2^m-1\,.$$ This is an interesting equation which was sent to me by a friend (probably found online). I have been scratching my head about ...
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5 votes
2 answers
175 views

How is this set of solutions for exponential diophantine equations been found?(Generalized Collatz mx+1, 2-odd-step-cycle)

I'm doctoring many weeks on the problem of the existence and set of solutions of 2-step-cycles in the generalized Collatz-problem, written in the Syracuse-form: $$ Y_m(a): b = {ma+1 \over 2^A} \...
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