If indeed some type of joined-primes does stop at some point, why would it?

[Kyriakos]

In the context of stuff about primes, such as the famous question about whether some type [eg so-called 'twin primes', primes that differ from each other only by a dyad (2), as in 11 and 13, 17 and 19 etc ], i have been occasionally thinking of this question.
Intuitively i would guess (no proof) that no kind of type of tied-primes ever stops in the infinite progression of positive integers. But supposing that some type or types would, Why would that happen? And what would it mean for such a progression?

Feel free to discuss, you may end up helping me get the much needed 1 million dollar prize for this math issue

(and i will share it with you in such a case, i am not a miser )

 

[Kyriakos]

Recently i heard of Zhang's (a relatively unknown mathematician) paper which (supposedly?) proves a very weakened tied theory on the twin primes. Namely it is reported to show that "there are infinitely many pairs of primes that differ more than a number of 70 million between them". That means that if prime A and prime B are a pair of primes, prime B is prime A + 70.000.000, and infinite such pairs form for numbers larger than that.

Of course the twin prime conjecture, going all the way back to Euclid, is not about such a vast number linking prime pairs. It is about showing whether or not there are infinite prime pairs that differ just by a number value of 2.

*

I tried googling for a version of Zhang's paper, cause i want to see how he went on about this. Anyone know where that paper may be found online?

(fwiw i am intuitively of the view that indeed infinite twin prime numbers exist; in fact i suppose than any prime pair has infinite number of cases)

 

[Agent327]

If indeed some type of joined-primes does stop at some point, why would it?

An If does not necessarily imply a Why. IF life came about doesn't imply there was a reason for it. (Although likely there would have been a cause.)

 

[Kyriakos]

An If does not necessarily imply a Why. IF life came about doesn't imply there was a reason for it. (Although likely there would have been a cause.)

Yeah. Of course math is a system of thought, and not an actual being, so properties in parts of math (such as properties of primes) always can be attributed to something in that system and its axiomatic basis. Indeed if they could not then the system would have inherent gaps and chasms, and would no longer make sense as a system.
Unlike with phenomena which are from the start distinct from the observer. Eg '1' exists if i think of it, i don't have to see a '1' in the physical world. A rock exists in my imagination not as a rock but as an imagination of a rock, cause it is an object with those attributes sensed as external to myself.

(but this is not the topic of this thread tldr: in math it is always valid to ask a 'why' for any property found in a set object of any type).

 

[Borachio]

I've absolutely no idea whether "joined-primes" stop or not.

But, if they did, instead of asking why would they, why on Earth wouldn't they?

 

[Kyriakos]

I've absolutely no idea whether "joined-primes" stop or not.

But, if they did, instead of asking why would they, why on Earth wouldn't they?

To stop means that after a number x (last joined prime) there is nothing of that kind again, in the endless oceans of infinity. So in that sense it is not that intuitive that there is some last such tied prime.

I mean they obviously can be hugely rarer as numbers get vaster and vaster, and it seems they do. But to stop entirely is not that intuitively probable, no?

 

[Atticus]

Namely it is reported to show that "there are infinitely many pairs of primes that differ more than a number of 70 million between them".

That should probably be "less", since with the word "more" it's trivial given that there are infinitely many primes. (exercise #1)

(fwiw i am intuitively of the view that indeed infinite twin prime numbers exist; in fact i suppose than any prime pair has infinite number of cases)

Even the pairs with gap of 3? What about 7? Why not? (ex. #2)

 

[Kyriakos]

That should probably be "less", since with the word "more" it's trivial given that there are infinitely many primes. (exercise #1)

No. There are of course infinitely many primes. But not infinitely many pairs of primes proven (nitpick: of course this means infinite cases for any given type or paired prime, in this case paired with a difference of at least 70 million). The twin prime conjecture is about there being or not infinitely many twin primes. It seems that this chinese mathematician's paper 'proves' that there are infinitely many pairs of primes, for which the smallest distance between the one and the other prime is the number 70 million.
(and it should go without saying that it is meant not that the pairs one can make are infinite, for if primes are infinite as proven then it is trivial there are infinite named pairs since you just need to show there are infinite odd or even numbers there, which also axiomatically follows)

Even the pairs with gap of 3? What about 7? Why not? (ex. #2)

Sure, that is not known either, obviously. Intuitively i also suspect those do not end ever, neither does any other gap of paired primes.

 

[Atticus]

No. There are of course infinitely many primes. But not infinitely many pairs of primes proven.

I know that. But from the knowledge that there are infinitely primes it is easy to prove that there are infinitely many pairs of primes at least 70 million numbers apart from each other, try it!

Thus he must've proved that there are infinitely many primes at most 70 million numbers apart from each other. Since he's famous for it, I assume it's not an easy thing.

Sure, that is not known either, obviously. Intuitively i also suspect those do not end ever, neither does any other gap of paired primes.

Think again. Mention some prime pairs with the gap 3. Note the plural "pairs", one pair isn't enough for me.

 

[Kyriakos]

Well, by gap i assume you mean 3 numbers which therefore cause prime A to be prime B- 4?

As in 3---7, 7---11 and so on. It is not proven if the obviously many cases of such a gap are infinite or not, but they are very many at any rate.

For i have to suppose you don't mean an actual numerical difference of 3, 7, or any other odd number, for if two numbers differ by an odd amount they can't both be odd, and thus aren't both primes in the first place (only, and sole special case there, is 2 and 5, both primes, but obvious one-offs).

*

As for the first part of your post, the thing is that this mathematician does not mean a difference of 70.000.000 between all primes examined, ie a pairing of p1,p2 where p2= p1+70.000.000. He means that in his work it becomes proven that for any pairing of numerical difference over 70 million, that type of pairing keeps on appearing forever. That is not just one pair type, but up to infinite pairs, which always are over 70 mil of difference between p2 and p1.

*

And an article on that work: https://www.quantamagazine.org/20130519-unheralded-mathematician-bridges-the-prime-gap/

 

[Agent327]

Yeah. Of course math is a system of thought, and not an actual being, so properties in parts of math (such as properties of primes) always can be attributed to something in that system and its axiomatic basis. Indeed if they could not then the system would have inherent gaps and chasms, and would no longer make sense as a system.
Unlike with phenomena which are from the start distinct from the observer. Eg '1' exists if i think of it, i don't have to see a '1' in the physical world. A rock exists in my imagination not as a rock but as an imagination of a rock, cause it is an object with those attributes sensed as external to myself.

(but this is not the topic of this thread tldr: in math it is always valid to ask a 'why' for any property found in a set object of any type).

You missed the point, which was not about math, but about logic. (Which incidentally is used in mathematics.) But since you seem insistent in not comprehending, 'why' does not apply in mathematics. It is a set of rules, which operates under the condition of logic. (Such as 'if... then'...') One might ask 'Why is this so?', but the question wouldbe immaterial. Either you accept the rules of mathematical logic, or you don't. 'Why?' is a philosophical questyion by nature and does not belong in mathematics. (So Why? is not a valid question in mathematics.)

Interestingly, there are certain truths in mathematics that can only be induced, not proven. In its simplest form this would be something like 1+1=2. There is no reason why 1 plus 1 should be 2 - this is just a rule agreed upon -, but we accept it as truth. So while mathematics, like theology, has its dogmas, it also has its undeniable truths. Equally interestingly, the question 'Why is there God?' is as pointless as 'Why does 1 plus 1 give 2'? The question 'Why?' really belongs in philosophy - and should well do to stay there.

(But seriously, you can ask 'Why would some type of joined-primes stop at some point? ', but then If is redundant, since that would be part of the answer.)

 

[Kyriakos]

^Must have missed it.

 

[Atticus]

For i have to suppose you don't mean an actual numerical difference of 3, 7, or any other odd number, for if two numbers differ by an odd amount they can't both be odd, and thus aren't both primes in the first place (only, and sole special case there, is 2 and 5, both primes, but obvious one-offs).

Exercise #2 done!

I'm not going to read that article, now at least, so won't comment on that.

You missed the point, which was not about math, but about logic. (Which incidentally is used in mathematics.) But since you seem insistent in not comprehending, 'why' does not apply in mathematics. It is a set of rules, which operates under the condition of logic. (Such as 'if... then'...') One might ask 'Why is this so?', but the question wouldbe immaterial. Either you accept the rules of mathematical logic, or you don't. 'Why?' is a philosophical questyion by nature and does not belong in mathematics. (So Why? is not a valid question in mathematics.)

You can ask for example: "Why continuous functions are integrable" or "why differentiable function is continuous". That's another way to ask for proof actually. In this case where proof isn't available, it may be asking for some kind of intuitive incomplete reasoning, although it sounds odd.