How is 4 a quadratic residue of 7?

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In short, $2^2 \equiv 5^2 \equiv 4 \pmod 7$. Since $4$ appears as a square, we see that $4$ is a quadratic residue.

In fact, $4$ is always a square mod primes. When looking mod $2$, we see that $4$ is zero (so it's the trivial square). Otherwise, $4$ will always appear as the square of $2$, regardless of whatever prime we are modding by.

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Arthur Collé
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Arthur Collé

I'm an analyst at Goldman Sachs in Interest Rate Products (IRP Frontline), within Fixed Income, Currencies and Commodities (FICC Technology). I work on the FICC trading floor of their Manhattan headquarters at 200 West Street in New York City. My team builds, extends and maintains the real-time trading systems (trade booking, risk mgmt, inquiry management) used by the US dollar interest rate sales & trading franchise. While I have a special focus on OTC derivatives as my first team was a cross-asset derivatives lifecycle management engineering team, I wanted to gain the asset-class specific business perspective of working with the desk, moving into rates after my first year with the firm. Mostly worked with the exotics and volatility traders at the beginning of my time with the new (current) team and learned about how Goldman represents financial instruments like swaps, swaptions and structured products in its systems (having gained a tremendous amount of experience with Slang development and SecDb engineering as a result). My role was recently expanded to directly work with the US Treasury bond traders and the systems used by them which is interestingly very distinct from what is used for the derivatives business. My team works extensively with sales and trading to further the business objectives of the Securities division and I'm very keen to continue to expand my skill set in these areas, expanding my knowledge of financial market dynamics and operating a profitable market making business, at-scale, in order to drive revenue growth and profits for the firm's public side. I'm very interested in developing new trading algos for automated market-making or perhaps prop trading if that becomes a thing again for big banks. We'll see - I'm having a lot of fun these days. contact info: alias set=export set FIRST=arthur set LAST=colle set OMG=hello set CORP=group set AT=@ set DOT=. set TLD=com echo "business inquiries: "$FIRST$AT$LAST$CORP$DOT$TLD #for business inquiries echo "personal inquiries: "$OMG$AT$FIRST$LAST$DOT$TLD #for personal inquiries

Updated on August 01, 2022

Comments

  • Arthur Collé
    Arthur Collé over 1 year

    On Wolfram's dictionary, it shows that the quadratic residues of 7 are 1,2,4. It shows that the quadratic residues of 5 are 1,4. I tested 1 and 4, and as you can see: $$1^2 = 1 \pmod 5$$ and $$ 4^2= 16 \pmod 5 = 1 \pmod 5$$ since 5*3 = 15

    If $4^2 = 16 \pmod 7 = 2 \pmod 7$

    Doesn't this mean it would fail the criterion that a quadratic residue must be congruent to a perfect square modulo p (here, p = 7) ? Doesn't it need to always be congruent to $1 \pmod p$ ?

    • DonAntonio
      DonAntonio over 9 years
      $\;2^2=4=4\pmod 7\;$...in fact, $\;4\;$ (and any other natural square) is a quadratic residue modulo any prime...though not always a non-zero one.
    • Arthur Collé
      Arthur Collé over 9 years
      Right, just realized that any time you get another perfect square as the remainder then the number that yielded that is a quadratic residue
    • robjohn
      robjohn over 8 years
      I believe you mean "$4$ is a quadratic residue mod $7$." This means that there is an $x$ so that $x^2\equiv4\pmod7$.