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This question is probably dumb but I can't find anyone else on the internet who seems to be troubled by this.

The Legendre symbol is denoted as $\left(\dfrac37\right)$, which is very useful for number theory and quadratic residues. On the other hand, this very obviously looks like a fraction in parentheses. While fractions are not always necessarily in parentheses, I find this notation rather unsettling.

Is the identification of a Legenfraction in a context such as $5\left(\dfrac37\right)$ determined solely by the general context that it appears in? Is there an alternate notation (besides the plaintext (3 | 7)) that leaves no room for creeping doubts?

greenturtle3141
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    There are a lot of notations in mathematics that serve more than one purpose; and, yes, we could rely on the context. – Leonard Blackburn Feb 22 '19 at 17:06
  • Well, nothing to do with it. The only solution is don't write parentheses on fractions when you work with Legendre symbols. There are not a lot of fractions in modular arithmetic anyway, so it's hard to get confused. – Mark Feb 22 '19 at 17:08
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    I also find this notation confusing, and have been burned by it on occasion. If Legendre had asked my advice, he would have used something different. Unfortunately, he didn't, and we're stuck with it. The upshot is that number theorists have to be careful not to use parentheses around fractions when there is any possibility of misinterpretation. – Robert Israel Feb 22 '19 at 17:11
  • @RobertIsrael That's when \cdot is useful – ə̷̶̸͇̘̜́̍͗̂̄︣͟ Feb 22 '19 at 20:50

1 Answers1

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Parentheses around a single fraction are typographically unnecessary (see for example the comment about the use of \cdot ) - most probably influencing Legendre's choice, see extract below of the definition in his book $\underline{\text{ Essai sur la Théorie des Nombres}}$.

I do not find the notation confusing at all - but I admit I may be influenced by my habit of avoiding parentheses as much as possible.

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Maestro13
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  • What about expressions such as (5/3)^1000? – greenturtle3141 Feb 27 '19 at 23:42
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    Why would you ever want to write $(\frac{p}{q})^{any power}$ for a Legendre symbol if you know that depends only on the parity of $any power$ so simply equals $1$ or $(\frac{p}{q})$? Hence no confusion as you will never write an expression like this. It does take some discipline but not much, to simply avoid any confusion. – Maestro13 Feb 28 '19 at 10:35