A Proof for the Twin Primes Conjecture

This video is an overview of my proof for the Twin Primes Conjecture, a long-standing open question in mathematics. I hope that it will give a decent understanding of my method and a measure of confidence in my result.

The paper I wrote to explain in more detail:

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[twilightstar]

Hello! I wanted to make this a relatively simple presentation, otherwise I never would have finished this. I might make an improved version in the future, but for now I hope it suffices :)

I admit that I am nervous posting this; I worry that after all my work, I have made a mistake somewhere. Nevertheless, I am at a point where I am as confident as I can be that I have gotten it right, and if not, then perhaps someone else can pick up where I left off.

[philippwei]

Hi, I read your paper. The first gap that I spotted, (which might be fixable) is that in the end, just because a sequence (in this case your lower bounds) is increasing, doesn't mean it goes to infinity, e.g. 1/2, 3/4, 7/8, ... is increasing but stays bounded by one.

The more likely non-fixable gap seems to be in your claimed bounds of H_n(x)+-n for the number of valid rows below x. The heuristic H_n(x) only becomes the true number for x being the product of the first n prime hexas, which is a huge number compared to your error term n, so that seemed wrong to me. I didn't understand the "proof" since you never actually explained what exactly your IN function is supposed to count, but I wrote a python script to try to find a counterexample for the bounds. Here are the first few. coprime_count counts the number of valid rows:


Failed for n = 5 and x = 134 . coprime_count = 30 and H(n) * x = 35.08080155138979
Test failed for n = 5
Failed for n = 12 and x = 1809 . coprime_count = 277 and H(n) * x = 289.1043308634691
Test failed for n = 12
Failed for n = 14 and x = 2507 . coprime_count = 355 and H(n) * x = 369.12998667052705
Test failed for n = 14
Failed for n = 15 and x = 2264 . coprime_count = 307 and H(n) * x = 322.050708996898
Test failed for n = 15



And below you can find my script for you to check in case I did something wrong.

Cheers,
Philipp


def generate_primes_up_to(n):
sieve = [True] * (n + 1)
sieve[0] = sieve[1] = False
for start in range(2, int(n**0.5) + 1):
if sieve[start]:
for multiple in range(start * start, n + 1, start):
sieve[multiple] = False
return [num for num in range(n + 1) if sieve[num]]


primes =
# print(primes)


def H(n):
p = 1.0
for i in range(0, n):
p *= (primes[i] - 2) / primes[i]
return p


def coprime(y, n):
for i in range(0, n):
referent = (primes[i] + 1) // 6
if y % primes[i] == primes[i] - referent or y % primes[i] == referent:
return False
return True


def test(n):
H_n = H(n)
coprime_count = 0

for x in range(1, primes[n - 1] ** 2):

if coprime(x, n):
coprime_count += 1
# print("x =", x, "is coprime with {5, ..., ", primes[n - 1], "}")

if coprime_count < H_n * x - n: # or coprime_count > H_n * x + n:
print(
"Failed for n = ",
n,
" and x = ",
x,
". coprime_count = ",
coprime_count,
" and H(n) * x = ",
H_n * x,
)
return False
return True


for i in range(len(primes)):
if not test(i + 1):
print("Test failed for n = ", i + 1)
# break

[Monkala]

Your ideas are fun and creative, but ultimately they do just amount to a restatement of the sieve of Eratosthenes and the heuristic argument.

Your arguments about the "critical area" are probabilistic and can be boiled down to "since there are infinite primes of the form 6k + 1 and 6k - 1, there must exist infinite k's such that 6k + 1 and 6k - 1 are prime." In fact, that's what's at the heart of what makes this problem so compelling to many mathematicians, that it appears so obvious and yet is so dubious when trying to prove rigorously. Don't take this video down, and keep exploring math! It's a good record of your journey and how far you will come.

Remember that when proving something, you have to show a condition definitively. It doesn't suffice to just state that "this pattern looks true and it makes sense to me so it must be true, " you need to adhere specifically to the formal principles of mathematical proofs. If there is even a single permutation where, even with a narrowed critical area, that the hexas end up being the exact primes / nonprimes ad infinitum that would prevent twin primes from appearing, then the proof is not considered rigorous, and so heuristics aren't sufficient to validate the claim.

[caspermadlener]

Happy to see you are both really humble and quite smart, which made this video a joy to watch!

Two other people have already made the remarks I had about the mathematics, but I also think your use of "hexa", "referent" and "anchor" is unnecessary, and just make things a bit harder to read.
You should have called these "6n±1", "n", and "6n", respectively, and consistent used n as the variable.

If I give a name to something in mathematics, it's because I can't be bothered to write out something all the time.
For example, instead of "a number that can be written as the sum of two squares", I used the term "happy number", in an essay necessary to finish high-school.

[Skeleman]

Email Matt Parker!
He did a video involving the "all primes greater than 3 must be one more or less than a multiple of six" so he might be open to discussing it. He is also friends with many academic mathematicians sothis could be a way into getting more eyes on it.

[OutbackCatgirl]

I've commented on a recent numberphile video in hopes someone there spots it and humours taking a look, and even if it turns out to be an incomplete proof it will be a great opportunity to learn some funky weird fun mathematics quirks!

[agranero]

*"all hexas bigger than 3 must be hexas"* is a tautology: you may want to say all hexas greater than 3 must be primes. If so he is pathetically wrong: 385 (6*64+1) is not. *If by hexadjacent you mean 6n+1 OR 6n-1* you are right is fairly simple to prove in a way far easier than you try (and your proof of this not valid for that part).
Your language is flawed (see above) and you claim lemmas that you don't prove despite of being true.
Sorry the rest does not worth reading the rest. Life is too short.
Sorry for being blunt but I am not being 10% as blunt as a mathematician will be.
Don't despair a lot of proofs Ramanujan sent to Hardy were wrong too.