Show that for each a < b a, b ∈ N we have the following

The statement "If d > 2, d ∈ N, then d does not divide both 3^(2^a) + 1 and 3^(2^b) -1" cannot be proven with the given information. It is possible for d to divide both expressions, as shown in the example you provided. Without further clarification or conditions, this statement cannot be proven.
  • #1
Missing template due to originally being posted in different forum.
1) 3^(2^a) + 1 divides 3^(2^b) -1

2) If d > 2, d ∈ N, then d does not divide both 3^(2^a) + 1 and 3^(2^b) -1

Attempt:

Set b = s+a for s ∈ N

m = 3^(2^a). Then 3^(2^b) - 1 = 3^[(2^a)(2^s)]-1 = m^(2^s) -1

Thus, m+1 and m-1 divides m^(2^s) -1 by induction.

If s = 1, then m^(2^s) -1 = m^2 - 1 = (m+1)(m-1)

For s>= 1, m^(2^s) = (m^(2^(s-1))+1)(m^(2^(s-1))-1). The induction hypothesis approves.

I'm confused on how to prove with the second condition.
 
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  • #2
Are you sure the second problem statement is correct?
3^(2^1)+1 = 10, 3^(2^2)-1 = 80.
d=5 divides both 10 and 80.
 
  • #3
Yes. This is why it was confusing to me.

I tried,

d = m+1>2 which shows that it can divide both m+1 and m^(2^(s-1)) -1.
 
  • #4
There is something missing in the problem statement then.
 

1. What does "a < b" mean in this statement?

In this statement, "a < b" means that a is less than b. This notation is commonly used in mathematics to compare the values of two numbers.

2. What does "a, b ∈ N" mean?

The notation "a, b ∈ N" means that both a and b are natural numbers. Natural numbers are the positive integers (1, 2, 3, ...).

3. What does it mean to "show" something in a mathematical context?

In mathematics, "showing" something means to prove or demonstrate its truth or validity. This often involves using logical reasoning and mathematical principles to support a statement or equation.

4. How can this statement be proven?

This statement can be proven using mathematical induction, which is a technique for proving statements about natural numbers. It involves showing that the statement holds for a base case (usually a = 1) and then showing that if the statement holds for some arbitrary value of a, it also holds for the next value of a (a+1).

5. Can you provide an example to illustrate this statement?

Yes, an example of this statement would be: if a = 2 and b = 5, then we can show that 2 < 5, since 2 is less than 5. This follows the pattern of the statement "for each a < b a, b ∈ N", as a is less than b and both a and b are natural numbers.

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