RE: [math-fun] Modular notation
Fred wrote: << At 03:43 PM 6/7/2007, Schroeppel, Richard wrote: < I don't know the history: I don't think the extended notations are used in Hardy & Wright, except maybe for (mod 1) to mean fractional part, or for stuff with algebraic integers.
Actually Hardy & Wright do use this notation, and in exactly the same context as the Monthly article that Dan cited. Theorem 116 says "If p>3, and 1/i is the associate of i (mod p^2), then 1 + 1/2 + 1/3 + ... + 1/(p-1) == 0 (mod p^2)." (By "associate" they mean "multiplicative inverse".)
Wait a sec -- the quote from Hardy & Wright looks as though it has no immediate connection with harmonic numbers H_n. Rather, it seems to be saying that if, in the ring Z/(p^2), we add the multiplicative inverses* of 1,2,...,p-1, then we get 0. (Otherwise, what does "i" mean here?) Maybe there is some clever way to connect this with harmonic numbers. But my first impression is that Hardy & Wright are not using the "mod" notation to refer to anything but integers (or integers mod N). --Dan __________________________________________________________ * Of course for p prime, not all members of Z/(p^2) have multiplicative inverses, but 1,2,...,p-1 do.
Dan Asimov <dasimov@earthlink.net> wrote:
Fred Helenius wrote:
Richard Schroeppel wrote:
I don't know the history: I don't think the extended notations are used in Hardy & Wright, except maybe for (mod 1) to mean fractional part, or for stuff with algebraic integers.
Actually Hardy & Wright do use this notation, and in exactly the same context as the Monthly article that Dan cited. Theorem 116 says "If p>3, and 1/i is the associate of i (mod p^2), then
1 + 1/2 + 1/3 + ... + 1/(p-1) == 0 (mod p^2)."
(By "associate" they mean "multiplicative inverse".)
Wait a sec -- the quote from Hardy & Wright looks as though it has no immediate connection with harmonic numbers H_n. Rather, it seems to be saying that if, in the ring Z/(p^2), we add the multiplicative inverses of 1,2,...,p-1, then we get 0. (Otherwise, what does "i" mean here?)
Maybe there is some clever way to connect this with harmonic numbers. But my first impression is that Hardy & Wright are not using the "mod" notation to refer to anything but integers (or integers mod N).
This is best understood in terms of localization theory. See any modern textbook on (commutative) algebra, or see the paper [1] below which presents an elementary exposition for p=3 by casting 9s from *fractions* in the subring of Q with denoms coprime to 9 (similarly, one casts out 11s, etc). Be sure to grok the *universal* property of localizations (cf. [1] Appendix). That is the heart of the matter here. As for the history, fractional ideals and modules go way back to Dedekind, but general localizations were constructed only much later (Grell 1927, Chevalley 1944, Uzkov 1948). I don't know who first thought to extend casting 9s etc to fractions. It must be very old. Nowadays it's obvious. I'd be indebted for any early historical references. --Bill Dubuque [1] Hilton, P; Pedersen, J. Casting Out Nines Revisited Mathematics Magazine, Vol. 54, No. 4 (Sep., 1981), pp. 195-201 http://home.comcast.net/~billwgd/Casting_nines_from_fractions.pdf http://links.jstor.org/sici?sici=0025-570X(198109)54%3A4%3C195%3ACONR%3E
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