Integer solution to exponential diophantine equation

In summary, the conversation discusses finding out if there are existing integers A and B that satisfy the equation C = 2^A * 3^B for an arbitrary integer C. It is suggested to test if C is divisible by 2 or 3 and then divide and iterate to find the values of A and B. It is also mentioned that this can be solved with a computer or with paper and mental calculations.
  • #1
adoado
72
0
Hey everyone!

I was recently scribbling on paper, and after a series of ideas, I got stuck with a problem. That is, can I find out if there exists some integers A and B such that

[itex]C=2^{A}3^{B}[/itex]

For some integer C?

For an arbitrary C, how do I know whether some [itex]A, B \in \textbf{Z}[/itex] exist?

Cheers for reading!
Adrian
 
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  • #2
Hi, Adrian,
it shouldn't be harder than testing if the number is divisible by 2 or by 3; and, in that case, if you are interested in the actual values of A and B, just divide and iterate. Unless you mean really big numbers.
 
  • #3
Code:
Solving with a computer:

Factor C  givin a list of it's prime factors and their occurences;
if there i9s a factor > 3 then 'No, C is not of the required form'
else 'Yes, A and B are the number, the factor 2 (resp 3) occurs

Solving with paper and your head:

Set A and B to zero
Loop2:
If C is even replace C by C / 2 andd add 1 to A
   loop until C is odd
Loop3:
If C is multiple of 3 (add the digits modulo 3)
   replace C by C / 3 andd add 1 to B
   loop until C is not a multiple of 3
Test: it the remaining C is one, then 'Yes' else 'No'
 

1. What is an exponential diophantine equation?

An exponential diophantine equation is an equation in which the unknown variables appear as exponents. It is in the form of ax^m + by^n = cz^k, where a, b, and c are constants and m, n, and k are positive integers.

2. How do you find integer solutions to an exponential diophantine equation?

There are various methods to find integer solutions to an exponential diophantine equation, including the use of modular arithmetic, factorization, and the theory of Pell equations. In some cases, trial and error may also be used to find solutions.

3. Is there a general formula for finding integer solutions to an exponential diophantine equation?

No, there is no general formula for finding integer solutions to an exponential diophantine equation. The methods used to find solutions are specific to each equation and may vary depending on the form of the equation.

4. Are there any applications of exponential diophantine equations in real life?

Exponential diophantine equations have various applications in mathematics, physics, and computer science. They are used in cryptography, number theory, and the study of exponential growth and decay in natural phenomena.

5. Can exponential diophantine equations have infinite solutions?

Yes, exponential diophantine equations can have infinite solutions. For example, the equation 2x^2 + 3y^2 = 5z^2 has an infinite number of solutions in integers, known as Pythagorean triples.

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