The proof provided is invalid due to a flawed argument, particularly in the attempt to prove the contrapositive and how specific cases are handled. Here's a breakdown of the logical errors:

Claim Restatement

The claim asserts:

"For all natural numbers $n$, if $2n + 1$ is a multiple of 3, then $n^2 + 1$ is a multiple of 3."

Strategy Used in the Proof

The proof attempts to prove the contrapositive of this claim:

"If $n^2 + 1$ is not a multiple of 3, then $2n + 1$ is not a multiple of 3."

While proving a contrapositive is a valid technique in logic, the execution in this proof contains several errors.

Logical Errors in the Proof

1. Contrapositive Misapplied for $n = 3a + 1$:

   • The second case of the proof assumes $n = 3a + 1$, leading to: $2n + 1 = 2(3a + 1) + 1 = 6a + 3$ This results in $2n + 1$ being a multiple of 3, but the proof asserts that this implies "false implies everything." This reasoning is incorrect because, in this case, we are supposed to check whether $n^2 + 1$ is a multiple of 3. The conclusion drawn from "false implies everything" doesn't correctly address whether the contrapositive is true in this instance.

   • For $n = 3a + 1$: $n^2 + 1 = (3a + 1)^2 + 1 = 9a^2 + 6a + 2$ This expression is not a multiple of 3, so $n^2 + 1$ is not divisible by 3 in this case. The reasoning provided in the proof fails to correctly connect the contrapositive logic for this case.

2. Misinterpretation of the Contrapositive:

   • In the third case, where $n = 3a + 2$, the proof correctly calculates: $n^2 + 1 = (3a + 2)^2 + 1 = 9a^2 + 12a + 5$ and shows that $n^2 + 1$ is not a multiple of 3. However, it fails to link this to whether $2n + 1$ is not a multiple of 3, which is the core of the contrapositive.

3. Failure to Prove the Contrapositive Completely:

   • The logic behind proving the contrapositive involves ensuring that when $n^2 + 1$ is not a multiple of 3, we conclude that $2n + 1$ is also not a multiple of 3. The proof fails to establish this in all cases, particularly for $n = 3a + 1$, where it prematurely concludes without properly checking the contrapositive.

Conclusion: Key Logical Error

The main issue lies in the handling of the case where $n = 3a + 1$. The proof fails to properly link $n^2 + 1$ being not a multiple of 3 to $2n + 1$ also being not a multiple of 3. It misapplies logical reasoning and prematurely claims success with "false implies everything," without properly concluding the contrapositive argument.

Would you like further clarification or detailed examples?

Here are 5 related questions:

1. What is the correct way to prove a contrapositive for a conditional statement?
2. How can we rework the proof to correctly handle the case $n = 3a + 1$?
3. What role does modular arithmetic play in analyzing the divisibility properties in this proof?
4. How can we check if $n^2 + 1$ is divisible by 3 using different values of $n$?
5. Why is proving a contrapositive a valid method for proving a statement?

Tip: When proving statements involving divisibility, modular arithmetic can simplify the analysis by breaking down numbers into cases based on their remainders.