Discussion Overview
The discussion revolves around proving that no number of the form $3k-1$ is a perfect square. Participants explore various approaches, including modular arithmetic, specifically mod 3, and the implications of certain theorems related to perfect squares.
Discussion Character
- Exploratory, Technical explanation, Mathematical reasoning
Main Points Raised
- Some participants propose using the theorem that states if a positive integer $n$ is such that $n \mod(4)$ is 2 or 3, then $n$ is not a perfect square.
- Others suggest examining the expression $3k-1$ under mod 3, noting that $(3k-1) \mod 3$ results in $-1$ or $2$.
- One participant questions the possible squares modulo 3, specifically asking for the values of $0^2$, $1^2$, and $2^2$ mod 3.
- It is noted that $0^2 \mod 3 = 0$, $1^2 \mod 3 = 1$, and $2^2 \mod 3 = 1.
- Some participants argue that since $-1$ and $2$ are not perfect squares, this suggests that $3k-1$ cannot be a perfect square, while others clarify that this alone does not suffice as proof.
- Further discussion highlights that any square must be of the form $(3r)^2$, $(3r + 1)^2$, or $(3r + 2)^2$, leading to the conclusion that any square is either $0$ or $1 \mod 3$, indicating a mismatch with $3k-1$.
- Participants also consider whether demonstrating the result in $\mathbb{Z}_3$ is sufficient for a complete proof.
- One participant explains the utility of using integers mod n in number theory to simplify the proof process by reducing the number of cases to check.
Areas of Agreement / Disagreement
Participants generally agree on the approach of using modular arithmetic to analyze the expression $3k-1$, but there is no consensus on whether the arguments presented constitute a complete proof. Some participants believe it suffices, while others express uncertainty.
Contextual Notes
The discussion includes various assumptions about the properties of perfect squares and modular arithmetic, but these assumptions are not universally accepted or resolved within the thread.
