# 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.

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.

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.

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.

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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