
Mathematicians will never stop proving the Prime Number Theorem - theafh
https://www.quantamagazine.org/mathematicians-will-never-stop-proving-the-prime-number-theorem-20200722/
======
elcomet
What's interesting about new proofs for existing theorems is that they can
give new insights about the problem and create new questions to be solved.
Sometimes they can link two previously unrelated ideas, and create entirely
new fields of study.

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Koshkin
Indeed, this is only natural. Mathematics may be unique in that a proof is
often just as valuable an asset (and sometimes more so) than the "fact" that
it proves. Another aspect of this is that in the best case an attempt to prove
something or to find another proof could result in emergence of a new theory.
(According to Grothendieck's viewpoint, this should be taken as a matter of
principle - see
[http://www.landsburg.com/grothendieck/mclarty1.pdf.](http://www.landsburg.com/grothendieck/mclarty1.pdf.))

~~~
JadeNB
> Mathematics may be unique in that a proof is often just as valuable an asset
> (and sometimes more so) than the "fact" that it proves.

This is the HoTT point of view: a 'fact' is just a 0-truncated (or maybe I've
got my indexing off) type, where all we know is whether it's inhabited
(proveable) or uninhabited (unproveable). We get much more information from
considering not just the bare fact itself, and not just even one particular
proof of it, but the space of all possible proofs.

Also, _is_ math really unique here? Maybe it is in the sense of having
absolute proofs; but I'd imagine that, in disciplines where reproducible
experiment is the substitute for absolute proof, it is also true that the
experiment is often as valuable as what it establishes, and that, the more
kinds of experiments (not just the more trials of the same experiment) there
are to establish the same thing, the better.

~~~
giraj
Your indexing is off :) Propositions (or 'facts' as you call them) are the
(-1)-types, whereas 0-types are usually referred to as sets. A way to remember
this is that the path-spaces (i.e. equalities) in a set are propositions,
bringing you one level down.

While you're right that "proof relevance" is one of the key features of HoTT,
I don't agree that what's being described here "is the HoTT point of view".
Any mathematical proof bears on techniques that may be interesting beyond the
proven result itself; HoTT doesn't clarify or elucidate that, nor does it aim
to.

~~~
JadeNB
> I don't agree that what's being described here "is the HoTT point of view".

I am no expert in HoTT, so I certainly defer to what sounds like your
expertise here. I should probably have said something closer to the weaker
statement "this reminds me of HoTT", which hopefully is not incorrect—although
you're right that what I really had in mind was proof relevance.

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jupp0r
> Mathematicians Will Never Stop Proving the Prime Number Theorem

Entropy in the universe is finite (as far as we know). At some point
mathematicians will not be able to continue trying to prove the prime number
theorem. qed.

~~~
gmfawcett
For extra rigour (!), replace "as far as we know" with one of the classic
weaselly proof introductions: "Clearly, ..." , "It can easily be shown
that...", "It is left as an exercise to the reader to show..."

~~~
jupp0r
Thanks for the peer review!

It can be easily shown that entropy in the universe is finite. Thus, at some
point mathematicians will not be able to continue trying to prove the prime
number theorem. qed.

~~~
brutt
> It can be easily shown that entropy in the universe is finite.

Finite Universe can get stuck in a "dead" state. Giving infinite time, it will
definitely will do so. The only solution for this problem is to have infinite
Universe. Infinite Universe will never stop.

(non native speaker)

~~~
gmfawcett
This reminds me of the "Boltzmann universe" theory! "Proof by appeal to
authority: see Boltzmann." :)

[https://en.wikipedia.org/wiki/Boltzmann_brain#Boltzmann_univ...](https://en.wikipedia.org/wiki/Boltzmann_brain#Boltzmann_universe)

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bryanrasmussen
Surely this is only so if there were an infinite number of proofs of the Prime
Number Theorem (assuming of course that new mathematicians are created every
generation for eternity.)

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cl3misch
I first thought that the occurences of "Math processing error" were a
mathematical joke, showing that some quantity can not be computed. But reading
the sentences again I think there should have been an equation...

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gweinberg
If you believe in "The Book", it follows that someday mathematicians someday
will stop proving the prime number theorem, and the fact that people keep
proving it indicates that the Book proof hasn't been found yet. I don't
believe in The Book myself, I think which of multiple proofs is most elegant
is largely subjective. But I'm not really a mathematician, just a guy who is
interested in math.

~~~
throwaway2048
what is "The Book"

~~~
iambrj
The mathematician Paul Erdős often referred to "The Book" in which God keeps
the most elegant proof of each mathematical theorem. He once said "You don't
have to believe in God, but you should believe in The Book."

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jeffreyrogers
Isn't most math that is published in journals just new proofs of already known
mathematics? I have a hobbyist's interest in advanced mathematics, so someone
else may correct me, but my understanding is that truly new mathematics is
pretty rare, and usually depends on all the little advances in understanding
that these new proofs and reorganization of mathematical knowledge provide.

~~~
omaranto
Nope, that's completely wrong. A new proof of a known theorem has to pretty
damn cool to be publishable. It is much easier to publish theorems that were
not previously known and the result is that most research papers are about new
theorems not new proofs of old theorems.

I wish it were actually a little easier to get a new proof of a known result
published.

~~~
webmaven
Huh. Seems like that would be a viable (indeed, valued and interesting) niche
for a mathematics journal.

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welfare
I get a feeling that this applies to a lot of sciences.

I feel like in the field of computational complexity theory, people seem to
find new ways of proving that a problem is NP-complete even though it's
already been proved before.

Isn't this just about general curiosity, people just want to understand and
explore a certain problem from different perspectives?

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jzer0cool
> This amount is only off by about 4% from the actual answer, 37,607,912,018.

This doesn't seem very good? Is this the best we can do today?

~~~
pmiller2
Not even close. The prime number theorem itself gives Li(x) as an
approximation to pi(x), where Li(x) = \int_2^x dt / (ln t). (If you don't read
LaTeX, this says "integral from 2 to x of dt / ln t". Li(x) can be numerically
evaluated with pretty good accuracy.

Here is the approximation to pi(10^12) using Li(x) in Wolfram Alpha:
[https://www.wolframalpha.com/input/?i=Round%5BLogIntegral%5B...](https://www.wolframalpha.com/input/?i=Round%5BLogIntegral%5B10%5E12%5D+-+LogIntegral%5B2%5D%5D)

The numerical value is 37607950280. Compared with the true value of
37607912018, this gives a relative error of around 1.0 * 10^-6, or 5 correct
digits.

Edit: Here is an algorithm to approximate pi(x) more directly:
[https://en.wikipedia.org/wiki/Meissel%E2%80%93Lehmer_algorit...](https://en.wikipedia.org/wiki/Meissel%E2%80%93Lehmer_algorithm)

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webnrrd2k
So the question is: is there a largest prime number _proof_?

~~~
Sohcahtoa82
Quite the opposite.

There's proof that there's an infinite number of prime numbers.

EDIT: It's actually a pretty simple proof.

First, recall that any composite number can be made by multiplying primes. If
a factor is not prime, then it can be broken down further until you end up
with only primes. For example, 60 = 12 * 5, but 12 is not prime and can be
broken down further into 3 * 4, but 4 is 2 * 2. So the prime factorization of
60 is 2 * 2 * 3 * 5.

Second, assume that there is a finite number of prime numbers, and label them
p1, p2, p3, etc, all the way up to pn, where pn is the highest prime number.

Multiply all the prime numbers together and add 1. Since this number has the
form p1 * p2 * p3 * ... * pn + 1, if you try to divide it by any prime number,
you will still have a remainder of 1. Therefore, this number will be prime.

Therefore, because you can always create a new prime by multiplying all primes
and adding 1, there are an infinite number of prime numbers.

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thomasqm
Why is it that posts from quantamagazine consistently appear on the front
page, just to be debunked (though this one literally does not contain any
substance!). Why do you upvote this?

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AnotherGoodName
Interesting fact about the Prime Number Theorum. It is NOT an estimate of the
number of primes below a certain N but instead it is an upper bound on the
number of primes. The classic formula derived from it, n/log(n), is always
above the actual number of primes (except for very small 'n' (<10) where the
approximation has significant error).

You can compare the actual counts of primes below 10^x (sequence A00688) to
the estimates provided by the prime number theorum to see this (sequence
A057835) for yourself.

Explanation: At the risk of once again proving the Prime Number Theorum we can
use a sieve to illustrate the upper bound. Above 2 there's an upper bound of
1/2 numbers possible being prime, since only 1/2 numbers are not a factor of
2. Above 6 there's an upper bound of 1/3 numbers being prime (only 1/3 numbers
above 6 are not a factor of 2 or 3) and the cycle of multiples of 2&3 repeats
every multiple of 6 so only 2 number remain that aren't a factor of 2 or 3
every multiple of 6. We could say above 30 (which is 2 * 3 * 5) only 8/30
numbers are possibly prime since only 8/30 numbers are not a factor of 2,3 or
5 on a repeating cycle. We could again create a sieve at 2 * 3 * 5 * 7 (210)
with a repeating cycle of those factors (with 48/210 numbers remaining that
aren't factors).

The prime number theorum can be derived from this upper bound.

If you're sieving out factors of a prime each time one is discovered you would
be checking 1/2 * 2/3 * 4/5 * 6/7 * 10/11... etc. numbers for primality. This
is essentially the left hand side of the Euler product formula you can see
here -
[https://en.wikipedia.org/wiki/Proof_of_the_Euler_product_for...](https://en.wikipedia.org/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function#Proof_of_the_Euler_product_formula)
Also see
[https://en.wikipedia.org/wiki/Euler%27s_totient_function#Eul...](https://en.wikipedia.org/wiki/Euler%27s_totient_function#Euler's_product_formula)

Now that same totient function is a good way to derive the prime number
theorum -
[https://en.wikipedia.org/wiki/Prime_number_theorem#Prime_num...](https://en.wikipedia.org/wiki/Prime_number_theorem#Prime_number_theorem_for_arithmetic_progressions)

So the prime number theorum is essentially defining the counts resulting from
a sieve of primes. A sieve is an upper bound on the number of primes. Eg.
there's numbers of the form 6n+1 or 6n+5 that aren't prime (eg. 25) that won't
be filtered by that sieve but also aren't prime and as we've seen the prime
number theorum can be dericed from this sieve. This is why the prime number
theorum is an upper bound that always overestimates the actual number of
primes below a certain number.

I suspect there are possibly ways to refine the prime number theorum further
so that it's not always over estimating but i don't know how to do this. I can
just understand why it's always an overestimate.

~~~
gjm11
I'm afraid there are quite a lot of errors here. (Not everything in the
following is pointing out errors, but most of it is.)

1\. Yes, the quantity in the PNT _is_ an estimate of the number of primes
below a certain N. As usually stated among mathematicians, PNT is the
statement that the number of primes up to x is asymptotic to Li(x), the
logarithmic integral, and this is _known_ to be sometimes over and sometimes
under.

2\. If you use x / log x instead of Li(x), then indeed it is known that once x
isn't very small this is always an overestimate. But, again, PNT isn't the
statement that it's an overestimate, it's the statement that it's a _pretty
good_ estimate. If Chebyshev or someone of the sort had proved that x / log x
is an overestimate for the number of primes up to x, it would have been hailed
as a fine achievement but _not_ as a proof of PNT.

3\. But your argument about "sieving out factors of a prime each time one is
discovered" is not anywhere near to being a proof of PNT, let alone the
stronger theorem that x / log x is an overestimate (which requires some
concrete bounds on the error term in the approximation pi(x) ~ Li(x); the way
you prove that x / log x is an overestimate is to show that x / log x is
bigger than Li(x) by a certain amount, and then show that the difference
between pi(x) and Li(x) is smaller than that certain amount; the latter is a
_strengthened_ version of the Prime Number Theorem).

4\. The Wikipedia link you give is _not_ saying that the Euler totient
function is "a good way to derive the prime number theorem". It is stating a
nice generalization of the prime number theorem that has the Euler totient
function in its statement.

5\. When you say "the prime number theorem is essentially defining the counts
resulting from a sieve of primes. A sieve is an upper bound on the number of
primes. [...] This is why the prime number theorem is an upper bound", I'm
afraid this is completely wrong. If there is a proof of the prime number
theorem that amounts to saying "look at this sieve, which easily gives an
upper bound that's easily seen to be x / log x", then I don't believe anyone's
found it yet.

6\. It's "theorem", not "theorum".

