Difference in Powers of Odd Primes

In summary, the conversation discusses the possibility of proving the equation p^x - d^y = p - d for odd primes p and d and natural numbers x and y, where x and y are not equal to one. Some examples were given, but ultimately the conversation concludes that the equation is not true for these conditions.
  • #1
omalleyt
15
0
I'm curious, can anyone think of a way to prove whether or not p^x - d^y = p - d, for any odd primes p,d and natural numbers x,y where x,y are not equal to one? This would be useful for a proof I am trying to work on.

So far, I have found that 3^2 - 2^3 = 3 - 2, but for this proof I am interested only in situations where p and d are both odd primes. I haven't found any examples that satisfy the equation with odd primes, but I haven't found a way to prove this equation impossible under these conditions. Ideally I would like to prove this impossible.
 
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  • #2
omalleyt said:
... can anyone think of a way to prove whether
or not p^x - d^y = p - d, for [itex] > > [/itex]any odd primes p,d [itex]< < [/itex]and natural numbers x,y
where x,y are not equal to one?

omalleyt,

can you give more restrictive parameters?

As it is, if p = d = an odd prime, and x, y > 1, and x = y, then

p^x - d^y =

p^x - p^x =

0 =

p - d =

p - p =

0
 
  • #3
13^3-3^7=2197-2187=10=13-3.
 
  • #4
Thanks, that saved me a lot of time trying to prove something that isn't true
 
  • #5


There are a few points to consider in response to this question. First, it is important to clarify the question and specify the domain of the variables. Are we looking at all possible combinations of odd primes p and d, or are there any restrictions on the values of x and y? Also, it would be helpful to define what is meant by "useful" in this context and what the goal of the overall proof is.

That being said, there are a few potential approaches to proving or disproving this equation. One possibility is to use mathematical induction, starting with the base case of p = 3 and d = 2, and then assuming the equation holds for some p and d and trying to prove it for p+2 and d+2. Another approach could be to use number theory and properties of prime numbers to analyze the equation and see if there are any patterns or contradictions that can be derived.

Regardless of the specific approach, it is important to carefully consider the properties of odd primes and how they interact with each other. For example, we know that all odd primes are congruent to either 1 or -1 mod 6, which could provide some insight into the equation. Additionally, it may be helpful to look at the prime factorizations of both sides of the equation and see if there are any relationships or cancelations that can be observed.

In summary, while there may not be a straightforward proof or disproof of this equation, there are certainly avenues to explore and techniques to apply in order to gain a better understanding of its validity. As a scientist, it is important to approach this problem with an open and analytical mindset, considering all possible angles and utilizing various mathematical tools to arrive at a conclusion.
 

What is the definition of "Difference in Powers of Odd Primes"?

The difference in powers of odd primes refers to the difference between the exponents of two odd prime numbers in a given mathematical expression or equation.

Why is the difference in powers of odd primes important in mathematics?

The difference in powers of odd primes plays a crucial role in prime factorization, which is the process of breaking down a number into its prime factors. It also has applications in cryptography, number theory, and other areas of mathematics.

Can you give an example of the difference in powers of odd primes?

Sure, let's take the expression 3^5 - 5^3. The difference in powers of odd primes in this case would be 5 - 3, which is equal to 2.

How is the difference in powers of odd primes calculated?

The difference in powers of odd primes can be calculated by finding the prime factorization of the numbers and subtracting the exponents of the corresponding prime factors.

Are there any patterns or properties associated with the difference in powers of odd primes?

Yes, there are several patterns and properties associated with the difference in powers of odd primes, such as the fact that it is always an even number, and that the magnitude of the difference increases as the numbers get larger.

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