Primes

[armstrongm]

@Indistinguishable

Sure. And I apologize for any harshness in my tone.

It seemed, from reading armstrongm’s posts, that they were specifically interested in the question of whether, up to any particular finite limit, there were always more primes of the form 6n - 1 than of the form 6n + 1. This is clearest in their second post in the thread.

They phrased this at times as the question of whether there were more primes of the one form than the other, without explicitly mentioning what was meant by more primes of the one form than the other. For example, in the OP. But no matter; later context revealed what they meant by this. (They also explicitly said they were not interested in density, which is in fact the right thing to say, because this question about whether there are always more of the one form than the other is not affected by the fact that the two have equal density)

Because of the OP, however, others (never armstrongm themself) introduced into the thread the idea of cardinality and the observation that the two forms gave rise to sets of equal cardinality. Which is true, but irrelevant to the notion of “more” that armstrongm was interested in. And I was distressed to see that armstrongm seemed to be getting flak simply for expressing their interest in studying a different notion of “more” than the one others were noting.

I think those of little mathematical sophistication are often unduly pummeled for not expressing themselves in orthodox mathematical language, even when the concepts they are grasping for (however inchoately) are perfectly sensible (and the language being used by the amateur to describe it is perfectly reasonable, and indeed is drawn from a perspective with which it highlights important analogies, but just happens not to match some arbitrary (and often relatively recent) tradition of how mathematicians speak). And so I am perhaps quick to “defend” them.

Finally, I will note that, though armstrongm apparently has a proof they seek to publish, which is very likely to eventually take this thread into crank-territory, all the above still applies. Whatever further crankery comes in, armstrongm did make a legitimate observation and have a legitimate question, which had nothing to do with cardinality even though they used the language of “more” and so on, and I think it was basically fine of them to ask about the things they were interested in using the language that they did.

The proof that will be published soon is essence says this.

Lets say that we have a difference of four between two primes such as 7 and 11. (7+11)/2 = 6n-3. This is true for differences between primes whose difference is a multiple of 4 but not a multiple of 6. So differences of 4;8;16;20 etc. Let’s call this prime difference group 2.

Now let’s say we have a difference of 2 between two primes such as 5 and 7. (5+7)/2 =6n. This is true for all differences that are not a multiple of 4 or 6. SO the differences are 2;10;14;22;26 etc. Let’s call this prime difference group 1.

The proof I sent in for publication does not address multiples of 6. I can create a proof for what happens with these differences though.

You end up with 4 groups. The last two groups being when the difference between primes is a multiple of 6 and not a multiple of twelve and when the prime difference is a multiple of 12.

What the proof shows is that if we assume that the differences between any two primes is infinite. The proof shows this assumption is incorrect the potential differences between any two primes are not infinite.

It all depends on whether the smaller prime can be written as 6n+1 or 6n-1. The reasons for this are quite simple when you read the proof sections of the proof imply that the potential differences are limited.

The question I ask is, “If we go all the way up to infinity which group will have the largest number of differences?” This includes when differences are repeated. such as if 2 I repeated 5 times then prime difference group 1 has 5 differences.

[Exapno_Mapcase]

@armstrongm

What the proof shows is that if we assume that the differences between any two primes is infinite. The proof shows this assumption is incorrect the potential differences between any two primes are not infinite.

C’mon, Indistinguishable. Will you now admit that armstrongm does not have any understanding of infinity? None.

[Indistinguishable]

@Indistinguishable

Finally, I will note that, though armstrongm apparently has a proof they seek to publish, which is very likely to eventually take this thread into crank-territory, all the above still applies. Whatever further crankery comes in, armstrongm did make a legitimate observation and have a legitimate question, which had nothing to do with cardinality even though they used the language of “more” and so on, and I think it was basically fine of them to ask about the things they were interested in using the language that they did.

armstrongm is a crank, absolutely. And armstrongm’s lack of mathematical sophistication means no one need waste time trying to decipher their claims, much less providing futile corrections, unless one for whatever reason took pleasure in it.

But there really do tend to be more primes of the form 6n - 1 than of the form 6n + 1, (in a natural sense), armstrongm really did notice this (by all indication), and it was perfectly fine for them to ask about it without caring about cardinality.

That’s all I was saying. On everything else, I’m perfectly happy to throw armstrongm to the wolves.

[Indistinguishable]

There’s a stray comma I’d like to strike but now must live with…

[armstrongm]

@Indistinguishable

armstrongm is a crank, absolutely. And armstrongm’s lack of mathematical sophistication means no one need waste time trying to decipher their claims, much less providing futile corrections, unless one for whatever reason took pleasure in it.

But there really do tend to be more primes of the form 6n - 1 than of the form 6n + 1, (in a natural sense), armstrongm really did notice this (by all indication), and it was perfectly fine for them to ask about it without caring about cardinality.

That’s all I was saying. On everything else, I’m perfectly happy to throw armstrongm to the wolves.

I’ll be the first to admit that I am left field but crank is a bit far. I just see things differently to you is all.