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August 11

Definition of modulo congruence on rational numbers

The previous question elicited a link Wolstenholme_prime#Definition via harmonic numbers that seems to use the concept of modulo arithmetic applied to rational numbers without defining the concept or providing a suitable link to clarify this (the cited reference Zhao (2007) is opaque to me in this regard). Intuitively it does not seem unreasonable to map rational numbers Q → Z/Z_n Q → Z/nZ : p/q ↦ [p]_n/[q]_n where gcd(p,q) = 1. Could someone point me to a suitable article that treats this topic? — Quondum^☏ 13:36, 11 August 2012 (UTC)

No, there is no such map. In the case of the harmonic numbers, the numbers $1,2,\dots ,p-1$ are units modulo $p^{3}$, so the quotients $1,1/2,1/3,\dots ,1/(p-1)$ are in $Z/p^{3}Z$. Sławomir Biały (talk) 13:41, 11 August 2012 (UTC)

I may be missing something here. I understand what you are saying as defining the harmonic numbers in modulo arithmetic as

$[H_{n}]_{m}=[1]_{m}+[2]_{m}^{-1}+[3]_{m}^{-1}+\cdots +[n]_{m}^{-1}=\sum _{k=1}^{n}[k]_{m}^{-1}$

for the chosen modulus m (in this case m = p³). Without the map I suggested (which, as you say, may not exist) or the explicit indication of a prior mapping of integers Z → Z/mZ, this interpretation is less than obvious. Am I wrong? — Quondum^☏ 16:32, 11 August 2012 (UTC)

This only makes sense if the residues [1],...,[n] are units modulo m. If so, then the harmonic number H_n is uniquely defined. Sławomir Biały (talk) 16:40, 11 August 2012 (UTC)

The closest thing to the map you want uses localizations of the integers, which are contained in Q . The map you give doesn't make sense over all of Q , e.g. for p/q if when p & q are reduced to lowest terms, when q & n have a gcd > 1. Particularly: 1/n or 1/q when q divides n, "reduced mod n" doesn't make sense. If R is the subring of Q formed by adjoining inverses of all integers relatively prime to n, then this is the localization of Z at the ideal nZ, and is the largest subring where the "reduce mod n" map makes sense.John Z (talk) 21:06, 11 August 2012 (UTC)

Perhaps I should rephrase my question. My map was an attempt at interpretation, which perhaps we should forget (and thus can ignore its being a partial function). How am I to interpret the first-linked section, which appears to refer to reducing a non-integer modulo an integer? — Quondum^☏ 21:22, 11 August 2012 (UTC)

Well as Sławomir Biały indicated, these particular non-integers are reducible mod m = p³, as they are in the "reduce mod m" partial function's domain, this localization subring of Q. Put all the fractions in lowest terms & they all make sense in, are calculable in Z/mZ. Not much else to say.John Z (talk) 23:18, 11 August 2012 (UTC)

Thanks. It seems I was not so far off in my initial interpretation, though there seems to me Wikipedia is a bit weak on this aspect. — Quondum^☏ 00:01, 12 August 2012 (UTC)

Right, you weren't so far off.John Z (talk) 03:28, 12 August 2012 (UTC)