Diophantine equation where the variable is in the exponent (ex.: find all sols for $3^x-5^y=7$)
Diophantine equation where the variable is in the exponent (ex.: find all sols for $3^x-5^y=7$)
Questions tagged [exponential-diophantine-equations]
147 questions
94
votes
6 answers
$x^y = y^x$ for integers $x$ and $y$
We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?
Paulo Argolo
- 4,370
25
votes
5 answers
Finding $(a, b, c)$ with $ab-c$, $bc-a$, and $ca-b$ being powers of $2$
This is a 2015 IMO problem. It seems difficult to solve.
Find all triples of positive integers $(a, b, c)$ such that each of the numbers $ab-c$, $bc-a$, and $ca-b$ is a power of $2$.
Four such triples are…
user253631
19
votes
2 answers
$x^y y^x$ in more than one way
I've submitted to OEIS the sequence consisting of numbers that can be written in more than one way as
$x^y y^x$ for integers $x,y$ with $1 < x \le y$. The first $6$ examples are
$$ \eqalign{ 4^{16} \cdot 16^4 &= 8^8 \cdot 8^8 = 281474976710656 \cr
…
Robert Israel
- 470,583
17
votes
6 answers
Finding solutions to the diophantine equation $7^a=3^b+100$
Find the positive integer solutions of the diophantine equation $$7^a-3^b=100.$$
So far, I only found this group $7^3-3^5=100$.
math110
- 94,932
- 17
- 148
- 519
15
votes
1 answer
Positive integers satisfying $a^b = cd$ and $b^a = c+d$
Yesterday, at 23:18, I thought it was a remarkable moment of the day. The digits on the watch were providing a quadruplet of positive integers that satisfy the following system of equations: $$\begin{align} a^b &= cd \\ b^a &= c+d \end{align}$$
I…
Fikilis
- 1,013
15
votes
8 answers
Elementary solution of exponential Diophantine equation $2^x - 3^y = 7$.
The title says it all. I would like to have a solution, preferably one which is as elementary as possible, of the exponential Diophantine equation
$$
2^x - 3^y = 7
$$
where $x,y$ are non-negative integers. Note that some small solutions are…
R.P.
- 1,546
12
votes
0 answers
Integer solutions to $2^n - n = m^2$?
Is $(n,m) = (7,11)$ the only solution in positive integers to the equation
$2^n - n = m^2\text{?}\tag*{}$
It's not hard to verify by direct calculation that there are no other solutions for $n < 10000$ but that's no way to establish a general…
Ted Hopp
- 707
11
votes
1 answer
Finding all natural $x$, $y$, $z$ satisfying $7^x+1=3^y+5^z$
The problem goes as follows:
Find all possible pairs of $x,y,z \in \mathbb{N}$ which satisfy the equation $7^x+1=3^y+5^z$
My first instinct was to continue by modding, but I don't think I can get anything out of it. The obvious solutions seem to…
Peter Allen
- 895
10
votes
3 answers
Does $23^n+1=2x^2$ have only two [positive integer] solutions?
I’m working on a problem, and believe I have reduced it to the solution of the Diophantine equation
$$23^n+1=2x^2.$$
Brute force calculations suggest that the only solutions are $(n,x)=(0,1)$ and $(n,x)=(3,78)$. I haven’t proven it algebraically…
Kieren MacMillan
- 8,199
10
votes
2 answers
Collatz Conjecture: For a cycle where the maximum odd integer is $x_{max}$, does it follow that $x_{max} < 3^n$
I am working on understanding the upper limit in the case where a non-trivial cycle exists for the Collatz Conjecture.
Is the following reasoning valid for establishing that the maximum odd integer in a cycle consisting $n$ odd integers is less than…
Larry Freeman
- 10,189
9
votes
2 answers
Solve : $\space 3^x + 5^y = 7^z + 11^w$
Solve the diophantine equation $3^x + 5^y = 7^z + 11^w$,here $x,y,z,w$ are all non-negative integers.
I find three solutions by force algorithm use Mathematica: (0,0,0,0)(1,1,1,0)(1,3,1,2).And there is no else when $y,z,w<100$.
Thanks in advance.…
lsr314
- 16,048
8
votes
1 answer
Show that there are an infinite number of positive integers that cannot be represented in the form of $a^{bc} - b^{ad}$.
this is a problem from the 2024 National Ukrainian Mathematics Olympiad. None of contestants got more than 0 out of 7 points. Here it is:
Show that there are an infinite number of positive integers that cannot be represented in the form of $a^{bc} -…
Pashtet671
- 151
- 7
8
votes
6 answers
A very interesting question: intersection point of $x^y=y^x$
I've been investigating the Cartesian graph of $x^y=y^x$. Obviously, part of the graph is comprised of the line $y=x$ but there is also a curve that is symmetrical about the line $y=x$. (We can prove this symmetry by noting that the function…
A-Level Student
- 6,834
8
votes
1 answer
Exponential Diophantine equation $7^y + 2 = 3^x$
Find all positive integer solutions to $$7^y + 2 = 3^x.$$
ATTENTION: MY SOLUTION HAS A TERRIBLE MISTAKE WHICH I HAVE OVERLOOKED!
Obviously, $x > y$. Then, we have $3^x = 7^y + 2 \equiv 0 \pmod {3^y}$. Also, $$7^y = (6 + 1)^y = \sum_{k = 0}^{y} {y…
user98186
7
votes
1 answer
Finding every triplet $(n,a,b)$ such that $n!=2^a-2^b$
Question : Let $n,a,b$ be positive integers. Are there infinitely many triplets $(n,a,b)$ which satisfy the following equality?$$n!=2^a-2^b$$
If Yes, then how can we prove that? If No, then how can we find every such triplet $(n,a,b)$?
The…
mathlove
- 151,597