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Is there a particular theorem or name defining the property/behavior of primes such that all primes (greater than 3) are congruent to 1 or 5 (mod 6)? I could have sworn I saw one years ago, but I haven't been able to find it again. I don't have a formal background in number theory, so that could be part of the reason, since I don't know the proper terms to search. I have been doing some work with Cuban primes. and I would like to know the name if there is one before speaking with a professor from a different country. The more we are on the same page the better.

Mara
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    Welcome to Maths SX! I don't think there's a special name. Simply, it' obvious that the other numbers, congruent to $0,2,3,4,$ can't be prime (except of course $2$ and $3$) since $6$ is divisible by $2$ and $3$. – Bernard May 14 '19 at 21:49
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    Cf. this question – J. W. Tanner May 14 '19 at 21:55
  • This observation was surely made shortly after primes have been "invented" ! – Peter May 14 '19 at 21:56
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    The term you're looking for may be "relatively prime", as in "$1$ and $5$ are the residues relatively prime to $6$". – Ethan Bolker May 14 '19 at 22:04
  • I am mostly looking for a term to make conversation/explanation more simple, like "and because we know theorem x to be true" or "and from theorem x" within the context of conversation. I already know how it works and how I am using them, but I am changing a one variable equation for a type of prime into a three variable problem. The visual presentation (being the most simple) is complicated enough, so I am going to try to find any ways I can to make the written/verbal part more simple and math friendly. – Mara May 14 '19 at 22:45

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https://en.m.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n list just a few possibilities, depending on use. These include:

  • Multiplicative group of integers mod n
  • Group of primitive residue classes modulo n
  • Group of units of the ring of integers modulo n.

Specifically, n is 6 in this case.

  • Thanks, I found the link very interesting. Unfortunately it wasn't able to solve my problem, but I always like to add more knowledge to my arsenal, since I wasn't able to finish my degree. I just need a term or phrase to define the concept, if one exists. – Mara May 16 '19 at 23:03
  • how does this fail ? $$1\cdot 1\equiv1 \bmod 6\5\cdot 5\equiv 1\bmod 6$$ showing multiplicative inverse( last term). Furthermore: $$\gcd(1,6)=1\gcd(5,6)=1$$ Showing coprimality ( fitting second term). Lastly, they fit all the group axioms with multiplication ( fit the first term) –  May 16 '19 at 23:23
  • I understand, however I was never looking to prove coprimality, so I am unclear of your direction. I am working with cuban primes and extending their range, and in doing so I use mod 6 a lot and depend on the fact that all primes are congruent to 1 or 5 mod 6 for the foundation of the work. I was just trying to see if there was a particular theorem that stated that they are all congruent to 1 or 5 mod 6 to clean up what I am writing. I am already very familiar with using modular arithmetic with multiple variables within this work. I am just looking for a word/phrase. – Mara May 17 '19 at 20:56
  • and any of the above can work. primes need to be coprime to a modulus or a prime factor of it. –  May 17 '19 at 23:21