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In various different places, I've seen the notation $$a \equiv b \equiv c \pmod n,$$ with the intended meaning obviously being $$a \equiv b \pmod n \quad\land\quad b \equiv c \pmod n \quad\Longrightarrow\quad a \equiv c \pmod n,$$ as in

$$x \equiv x - 1 + 1 \equiv 1 \pmod{x - 1}.$$

However, others do not share my views on this matter, saying it's ambiguous and notation abuse, among other things, with their primary problem being that multiple equivalence signs are corresponding to a single mod. As such, I am looking for a notable source, preferably a scientific paper, that uses (or discusses) this kind of notation to educate myself and/or the others in question on the matter.

Maya
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    I have never seen anyone object to such "chaining". – Eric Wofsey Nov 28 '18 at 22:47
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    Do they give any examples where it is ambiguous? I can't think of any, and certainly similar chaining of transitive relations is very common, as for example in $a=b<c<d≤e=f$, therefore $a<f$. – MJD Nov 28 '18 at 22:51
  • @MJD I don't exactly follow their logic, but it has something to with how there is just one $\pmod n$, and it is apparently not clear that it applies to all congruence relations. – Maya Nov 28 '18 at 22:54
  • @EricWofsey This is also a first for me. – Maya Nov 28 '18 at 22:54
  • It might be something that a teacher wants students to be careful about in a first course on modular arithmetic. I'm sure that checking that modular arithmetic gives an equivalence relation is a useful exercise. But you won't find a paper about it because once you've checked that equivalence relation everything else is easy. – Chessanator Nov 28 '18 at 23:29
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    If it's good enough for Hardy and Wright, surely, it should be good enough for anybody - and I find an instance of this kind of "chaining" on page 57 of their An Introduction to the Theory of Numbers (fourth edition 1960): $$(vn'^2+v'n^2)\bar{H} \equiv vn'\bar{h} + v'n\bar{h'} \equiv K \pmod{nn'}.$$ I also found it on page 36 of H. E. Rose, A Course in Number Theory (second edition 1994): $$x \equiv c_im_i'm_i'' \equiv x c_i \pmod{m_i}.$$ Those were the first (indeed only) books I looked at. Instances of this convention in respectable mathematical literature don't seem to be hard to find. – Calum Gilhooley Nov 28 '18 at 23:58
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    From page 50 of the third book I looked at, H. Davenport's The Higher Arithmetic (eighth edition 2008): $$(ab)^{lk} \equiv (a^l)^k(b^k)^l \equiv 1 \pmod{p}.$$ Page 108 of the fourth book, Tom M. Apostol's Introduction to Analytic Number Theory (1976): $$23{,}716 \equiv 640 \equiv -1 \pmod{641}.$$ Unscientific conclusion: this convention is found everywhere. :) – Calum Gilhooley Nov 29 '18 at 02:41
  • @jgon I did try skimming through a pseudorandom sampling of number theory papers (some of them even used modular arithmetic itself) without success, but it didn't occur to me to look in textbooks. This is different from most transitive relations in that a part of the information is expressed to the right of the statement, and not strictly "inline". – Maya Nov 29 '18 at 11:47
  • See also here on chaining for any (transitive) relation. $\ \ $ – Bill Dubuque Nov 26 '24 at 10:41

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