Fair enough. Though, if you do introduce modular arithmetic at all, even if only in the most superficial way, I don’t think it needs any “extension” to handle this problem. It’s obvious that every odd number is either 1, 3, 5, or 7 (modulo 8), so, like I said (and ultrafilter and yabob above), just test those four cases and you’re done.
Also, incidentally, returning to the general theorem, note that L(a) divides 2 if and only if a divides 24, which would explain why the OP didn’t find any other similar patterns, if he was restricting himself to squaring and not higher exponents. [I.e., the “If a and b have no common prime factors, then a^2 - 1 is a multiple of b” rule only applies for divisibility by 1, 2, 3, 4, 6, 8, 12, and 24]. Of course, at higher exponents, similar rules abound; e.g., if a is not a multiple of 5, then a^4 - 1 is. If a isn’t divisible by 3 or 7, then, a^6 - 1 is divisible by 63. And so forth…
Most of the answers seem correct (I haven’t really checked) but unnecessarily complicated.
p^2-1 = (p-1)(p+1) and the factors are two consecutive even numbers one of which must be divisible by 4 (or a higher power of 2) and the other by 2. So it is divisible by 8. Of the three consecutive numbers p-1, p, p+1 one must be divisible by 3 and it isn’t p, so it is p-1 or p=1.
And I said it, too. They wrote their responses while I was typing mine, which was a pretty bog standard modular arithmetic argument. I took a bit more time to point out that this wasn’t something that had anything to do with primes. It’s not like I don’t know about the idea.
As for using modular arithmetic to demonstrate this problem, I think you’re not noticing how much algebra there actually is. For example, saying that any odd number has remainder 1, 3, 5 or 7 is fine, and most people will probably believe it. On the other hand, saying that any odd number can be written 8k+i for some remainder i involves (elementary) algebra. How hard is it to get your mind around? Well, I’m teaching an introduction to proofs class right now, and when I first said something similar, I had a room full of kids who want to be math majors looking at me cockeyed. They thought about it for a second and then they got on board. None of them had ever seen modular arithmetic, and the idea that n[sup]2[/sup] = 1 (mod) 8 when n is odd was… well let’s just say they were surprised.
I’ve taught math for elementary school teachers. In that class we talked about this kind of problem. For those students, the kind of numerical approach was much more convincing because it made sense to them. After you had convinced them that it was true, you could go and try to prove it to them algebraically, but if you started right in with algebra their eyes would glaze over. Moreover, a lot of the students weren’t comfortable with manipulating equations algebraically, except for the purpose of “solving for x”. Trying to use algebra outside that context smells of sophistry and is somehow unconvincing.
Your experience obviously differs from mine, but I’ve dealt with students both at big public schools and small private liberal arts colleges, and across the board I’ve seen students of every age from 18 on up balk at the kind of argument you’re proposing.
Sorry, I forgot to mention that; in fact, credit where credit is due, you were the most explicit and thorough about it.
I never thought or meant to imply otherwise.
These are different statements? What do people think of “remainder” as meaning?
Not that I would try to communicate the idea in terms of writing “8k + i” in the first place. I would imagine it to perhaps be more intuitively expressed as “last digit arithmetic”, only in arbitrary bases. That is, I would first introduce mod 10 arithmetic by observing how the last digit of an arithmetic operation such as multiplication depends only on the last digits of its inputs, so that we can carry out calculations pretending nothing exists except the last digit. This would allow us to piggyback on all the intuitions students already have from years of calculation. Then, for generality, I would simply point out that the particular base 10 plays no special role here, and that this can be done for any base.
I wouldn’t expect them to know offhand that n[sup]2[/sup] = 1 (mod 8) when n is odd; that’d be silly. I’d expect them to begin to see it after they worked out examples, for all the possible remainders/“last digits”. (As above, I would think of modular arithmetic as just “last digit arithmetic”, rather than via explicitly writing out, manipulating, and reducing “8k + i” terms). But I defer to your experience.
Aye, but I didn’t intend to write it out as algebra and equations. I intended to present it the same way as I might present “Hey, did you realize that the square of any odd number indivisible by 5 always ends in 1 or 9?” “Really? Hm, let me try all the possibilities. My number might end in 1, 3, 7, or 9, so its square must end in… 1, 9, 9, or 1, yeah, you’re right.” Not a variable in sight, much less an equation or any other explicit mark of algebra.
I think you may be taking me to propose a different argument (or different presentation) than the one I am. I am just thinking in terms of intuitive “last-digit arithmetic”, not explicit manipulation of polynomials and indeterminate variables and so forth.
Also, I mentioned Carmichael’s theorem, but I wouldn’t propose explaining that in full to the sixth graders; that’s just for the edification of the readers of this thread.