a result that shouldn't be called a result since it's practically a restatement of the definitions?

[Mike South]

Hi,

I recently sent this feedback in to the mathworld project.  I was only there because I was looking up stuff about how pi shows up in the distribution of primes, and I was only looking that up because of our discussion today about pi.

So, it's not necessarily directly relevant to anything we discussed, but I thought you might be interested, and would be interested to see if you think I was too hard on Bungus' 1599 "result".

Begin submission:

On http://mathworld.wolfram.com/PrimeNumber.html it cites a "result":

With the exception of 2 and 3, all primes are of the form [congruent to 1 or 5 mod 6].  

I don't understand how that's worth citing a source for, or even commenting on, at all.  With the exception of p_x and p_y, all primes are relatively prime to p_x*p_y...so what?  Why pick 2, 3 and 6?  Just because they are the lowest?  By selecting the congruence families of 6 as your search space, you're thick with the multiples of 2 (0, 2, and 4) and multiples of 3 (0, and 3). So what if there are only two congruence families left?  Six is a small number!  And even then, one third of the possible congruence families qualified as possible primes.  That's a huge fraction, not a remarkably small one. 

So all you're really saying is that there are no primes other than 2 or 3 that are multiples of 2 or 3.  Was that a fancy result in 1599?  It's just a restatement of the definition of a prime number.  It's weird to me that you would want to point this out in the article as if there were some significance to it.

It just seems utterly obvious to me, along the lines of people thinking 2 is cool because it's the only "even prime" (people do this).  Three is the only "threeven" prime!  Five is the only "fiveven" prime!  It's just a restatement of the definition.

Imagine another universe where there are different primes, and someone multiplies the first two primes together, and says:

Hey, all numbers of the form N*P_1*P_2 + M*P_1 are divisible by P_1 (because of this cool thing called FACTORING!).  And all the numbers of the form N*P_1*P_2 + M*P_2 are divisible by P_2 (also because of the same cool thing called FACTORING).  Holy crap!  That means that, out of the congruence families denoted by 0...P_1*P_2, we have to THROW OUT the ones with factors of P_1 or P_2, and only the stuff that's left can contain any non P_1, non P_2 primes!  Because those are the only ones without P_1 or P_2 ALREADY IN THEM BY DEFINITION! WHOA!

I don't think it would be interesting in that universe, either.

End submission.

lemme know what you think,

mike