The discussion revolves around the application of mathematical induction to prove divisibility properties, specifically regarding the expression \(5^n - 1\) and its divisibility by 4 for all natural numbers \(n \geq 0\). Participants are reviewing and critiquing proposed proofs, addressing both the structure of the proofs and the clarity of the arguments presented.
Participants generally agree on the need for clarity and structure in proofs by induction, but there is no consensus on the specific phrasing and conclusions to be drawn from the inductive process. Some participants express differing views on how to articulate the final statements of the proof.
Limitations include potential misunderstandings regarding the notation and terminology used in mathematical induction, as well as the varying levels of familiarity with the proof structure among participants.
Yes. The problem statement says, "prove that $4\mid(5^n-1)$ for every $n\ge0$". The smallest $n$ for which a proof is required is 0, so the base case consists of proving that $4\mid (5^0-1)$. Of course, $5^0-1=0$, which is divisible by everything (except another 0).outkast32 said:1 thing, I am confused on the "base case" i understand it supposed to be "easy" and simple, do i just say n = 0?
Evgeny.Makarov said:Yes. The problem statement says, "prove that $4\mid(5^n-1)$ for every $n\ge0$". The smallest $n$ for which a proof is required is 0, so the base case consists of proving that $4\mid (5^0-1)$. Of course, $5^0-1=0$, which is divisible by everything (except another 0).
Okay i can label them,Evgeny.Makarov said:At the end of the inductive step, you write "Therefore, $5^k-1$ is a multiple of 4". Could you explain why you used the word "therefore"? How does it follow from what comes earlier? In fact, what is the status of this claim: were you proving it, or did you accept it for granted?
The rest seems OK, but I would tweak the style a little. In my opinion, every proof by induction should have the following clearly marked elements.
(1) What I call the induction statement $P(n)$, a propety of natural numbers that you are supposed to prove for all $n$. Writing this property explicitly is important because it makes it easier writing the induction hypothesis and the claim that needs to be proved in the induction step. Besides, in more complicate proofs by induction, the induction statement $P(n)$ may be different from the property $Q(n)$ that appears in the problem statement: "Prove $Q(n)$ for all $n\ge0$". It may happen that $Q(n)$ is too weak for the induction to go through, and one needs to find a suitable strengthening, or generalization. This is a nontrivial step and should definitely be recorded.
(2) The base case (you did this correctly).
(3) The induction step (you have not marked it).
(4) The induction hypothesis (you wrote it, but did not label it as such).
(5) The content of the induction step, i.e., a proof of $P(k+1)$ from $P(k)$. The application of the induction hypothesis $P(k)$ should be clearly marked (you have not done this).
P.S. As a non-native English speaker, I am never sure whether to use "inductive: or "induction" as an adjective. Sorry for any grammatical errors.
I could write what I think you meant, but I would prefer you re-reading it, thinking about it and answering the questions above.Evgeny.Makarov said:How does it (i.e., the claim that $5^k-1$ is divisible by 4) follow from what comes earlier? In fact, what is the status of this claim: were you proving it, or did you accept it for granted?
Yes (and please use parentheses in 5^(k+1) because exponentiation is performed before addition). It is this that you have shown in the inductive step and not $4\mid(5^k-1)$. The latter fact you assumed as the inductive hypothesis. It would be funny to switch what you assume with what you need to prove, don't you think?outkast32 said:Should it have been therefore 5^k+1 -1 is a multiple of 4.. from the inductive process i was showing above.
The phrases "$5^{k+1}-1$ is divisible by 4" and "4 divides $5^{k+1}-1$" mean exactly the same thing, so there is no need to use both. I would conclude the step as follows. "Therefore, $5^{k+1}-1$ is divisible by 4, which concludes the inductive step. By induction, it follows that $5^{n}-1$ is divisible by 4 for all $n\ge0$.outkast32 said:So it should be "Therefore, 5^k+1 -1 is a multiple of 4 because 4 divides (5^k+1 -1) and it is true for every n >=0.
Evgeny.Makarov said:Yes (and please use parentheses in 5^(k+1) because exponentiation is performed before addition). It is this that you have shown in the inductive step and not $4\mid(5^k-1)$. The latter fact you assumed as the inductive hypothesis. It would be funny to switch what you assume with what you need to prove, don't you think?
The phrases "$5^{k+1}-1$ is divisible by 4" and "4 divides $5^{k+1}-1$" mean exactly the same thing, so there is no need to use both. I would conclude the step as follows. "Therefore, $5^{k+1}-1$ is divisible by 4, which concludes the inductive step. By induction, it follows that $5^{n}-1$ is divisible by 4 for all $n\ge0$.