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This question is about any numbers that seemed unusually close to disproving Goldbach's conjecture. Meaning any large numbers (say above 100) that had very few sets of primes which satisfied the conjecture. I imagine the number of sets increases as the numbers get larger. Has there ever been found a number that suddenly shot down to only a few solutions out of nowhere?

Juel Herbranson
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    "any large numbers (say above 100)" -- woah. I would have understood "large" as much larger in this context. – Torsten Schoeneberg Jul 01 '22 at 07:27
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    What means "close to disprove Goldbach" ? We can surely find even numbers $n$ , such that there is no prime $p<10^5$ (for example) such that $n-p$ is prime as well. But if $n$ has several hundred digits, this is extremely far from close to be a counterexample. In fact, the smallest prime doing the job is usually much smaller than $n$, so that there are probably no such examples. – Peter Jul 01 '22 at 07:31
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    To formulate it in another way : No even number that can be expected as a counterexample is known. – Peter Jul 01 '22 at 07:33
  • I can imagine that "8, 12, etc only have one solution" might be a quick response which I'm trying to avoid. After a couple dozen there seem to be multiple solutions which just keeps growing. I'm wondering about any numbers that suddenly plunge back down to just one solution. – Juel Herbranson Jul 01 '22 at 07:35
  • To clarify. The graphs I've seen are very compelling that there is no way we will ever find a solution that disproves Goldbach's conjecture. The way the graph explodes as the integers get large seem to have a very well defined lower limit to the number of sets of primes that satisfy the conjecture, so it seems that it's basically true, just not provable. I'm curious if there is some known example that might bring that confidence into question. That some people actually believe there is reason to hold out belief that the conjecture is almost certainly true. – Juel Herbranson Jul 01 '22 at 07:41
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    It could be a nice project to discover cases with "unusual few" solutions. Not sure whether someone has done this already. – Peter Jul 01 '22 at 07:41
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    In fact we can safely assume that Goldbach's conjecture is true. The heuristical evidence is overwhelming. We just cannot prove it. Same for the twin prime conjecture. I do not think that any mathematician seriously doubts about those conjectures. – Peter Jul 01 '22 at 07:44
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    If the smallest counterexample has , say , $50$ digits , we are screwed because we have no chance to prove this to be a counterexample in practice. Even if it had $25$ digits, this would be almost impossible. Considering the checked range, there is no hope that a counterexample is found (which does not rule out that someone disproves the conjecture nonconstructively) – Peter Jul 01 '22 at 07:48
  • I found a reference of 4x10^14 has been checked. But was from 2001. Edit 10^18 is current. – Juel Herbranson Jul 01 '22 at 07:59
  • $10^{18}$ should be realiable , but there are claims that much larger ranges have been checked. – Peter Jul 01 '22 at 08:05
  • They'd be 1 or 2 mod 3 would be my guess – Roddy MacPhee Mar 17 '23 at 15:12

1 Answers1

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Maybe what you are looking for is continaed in OEIS A002375 or OEIS A045917. For example, the former has a list of the first 20000 even numbers and how many (unordered) decompositions into odd primes it has, like 2·15 has 3 different decompositions:

1 0
2 0
3 1
4 1
5 2
6 1
7 2
8 2
9 2
10 2
11 3
12 3
13 3
14 2
15 3
...

So you can run scripts on this to find entries with small number of representations. There are also nice plots that go up to 200th resp. to 20000th.

Using a small Python script, we can use that data to find the largest even $n\leqslant 40000$ that has a specific number of representations. The 1st column contains the number of unordered representations as sum of two odd primes, and the 2nd column lists the largest even $n\leqslant 40000$ with that number of representations. The number of representations up to 20:

0 4
1 12
2 68
3 128
4 152
5 188
6 332
7 398
8 368
9 488
10 632
11 692
12 626
13 992
14 878
15 908
16 1112
17 998
18 1412
19 1202
20 1448

For example, 128 is the largest number amongst them that has just 3 representations, all larger numbers have at least 4 representations. Here is the Python3 script for reference. It mostly deals with retrieving the data from the URL from above.

#!/usr/bin/env python3

from urllib.request import urlopen
import re


URL = "https://oeis.org/A002375/b002375.txt"


line_pat = re.compile (r'\s(\d+)\s+(\d+)\s')


count_to_n = {}


with urlopen (URL) as data:
    for line in data:
        match = line_pat.match (line.decode ("ascii"))
        if match:
            n = 2 * int(match.group(1))
            count = int(match.group(2))
            if count <= 20:
                count_to_n[count] = n


for c in sorted(count_to_n):
    print (c, count_to_n[c])
emacs drives me nuts
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  • As an example, when looking at the plot graph, just below 5000 there is an entry that dips below the rest of the data. Other than that, the lower floor seems very well defined. But it's that one entry that makes me wonder how large of a deviation might exist from that common floor of lower values somewhere hidden in the very large integers. If we could find anomalies like that and see if they trace out some other curve, or if it becomes erradic and might throw a strange number that dips way down. I hope my comment is making sense. – Juel Herbranson Jul 01 '22 at 17:04
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    That dot could relate to 9866 = 2·4933 with 86 representations or to 9602 = 2·4801 with 77 representations, if that helps your research. – emacs drives me nuts Jul 01 '22 at 17:42