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From Gauss' Disquisitiones Arithmeticae §131:

Sī p est numerus prīmus fōrmae 4n+1, erit +p, sī vērō p fōrmae 4n+3, erit -p residuum vel nōn-residuum cuiusvīs numerī prīmī quī positīvē acceptus ipsīus p est residuum vel nōn-residuum.

(I have taken the liberty to add macrons.)

From what I know about this theorem, that is called quadratic reciprocity nowadays, the translation should roughly be:

If p is a prime number of the form 4n+1, let p* denote +p; otherwise if p is of the form 4n+3, let p* denote -p; then p* is a [quadratic] residue (resp. non-residue) of whichever prime number q that is a [quadratic] residue (resp. non-residue) of this very p.

(I have added p* as an intermediate definition to make the sentence flow better in English.)

The last sentence can also be rephrased as:

p* is a quadratic residue of q iff q is a quadratic residue of p.

However, I do not understand where the phrase positīvē acceptus fits in this translation.

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    It seems to me that, after fiddling with the sign of p at the start of the sentence, Gauss is now saying that q is to be taken positive. (Caution: I claim no expertise in Latin, but I'm a mathematician, and this reading makes sense to me.) – Andreas Blass May 1 at 0:11
  • @AndreasBlass As a bit of both mathematician and latinist, I agree. I tried to argue that in my answer too. I didn't manage to find whether Gauss in general required primes to be positive, but no other reading seems to make sense to me. – Joonas Ilmavirta May 1 at 14:54
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I think the "any other prime" is taken to be positive. I was unable to find an answer by a quick search, but perhaps Gauss is allowing negative prime numbers. Quadratic reciprocity does not work quite the same if you flip signs, and that is why Gauss also used ±p for the first prime.

Most are taught in school that the prime numbers are positive (2, 3, 5, 7, 11…), but it makes sense to regard their negative counterparts (-2, -3, -5, -7, -11…) as primes as well. It depends on how you define a prime; if you only refer to divisibility, then negative numbers make perfectly valid primes. And sometimes this is convenient in modern mathematics, as there are cases where one can speak about divisibility and primes but not positivity.

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