Gauss once said, "Mathematics is the queen of the sciences and arithmetic is the queen of mathematics." The "arithmetic" mentioned here is actually what we now call number theory. Since I was very young, I have had a special fondness for number theory. Although I later encountered many more interesting branches of mathematics, my passion for number theory remains undiminished. I think this is probably due to a childhood connection. In elementary school, at a young age, I had just learned concepts like prime numbers, composite numbers, divisors, and divisibility, and I was particularly interested in numbers. At that time, I think the only mathematical problems I could understand were in the field of number theory. For example, Fermat’s Last Theorem, x^n + y^n = z^n, which has no positive integer solutions for n > 2, is very easy to understand in terms of what it is saying. Another example is Goldbach’s Conjecture: every even number greater than 4 can be split into the sum of two odd primes. It is also very simple to understand. Therefore, after understanding these problems, my younger self played with numbers with great interest. Perhaps it is precisely because of this that I developed a deep love for numbers and mathematics.
Goldbach’s Conjecture is undoubtedly a bright pearl in number theory, but at present, it remains out of reach. A conjecture that seems so simple has puzzled mathematicians for hundreds of years and remains unsolved. Nevertheless, I am willing to study it carefully and savor it slowly. While "demonstrating" or verifying it, I appreciate the mysterious beauty of mathematics. This article mainly studies the "Goldbach partition number" of a given even number—that is, the number of different ways to split each even number into the sum of two primes through actual calculation. For example, 6 = 3 + 3 has only one partition method; 8 = 3 + 5 = 5 + 3 has two partition methods; 10 = 3 + 7 = 5 + 5 = 7 + 3 has three partition methods, and so on. The partition number of an even number 2n is denoted as G_2(2n).
(Note: The definition of "partition number" here is different from the definition found online and in general literature. Here, 3+5 and 5+3 are considered two different partition methods, whereas the general definition online considers them as one. The advantage of my definition is that the number of partition methods actually represents the total count of primes involved in the partitions.)
It is true that Goldbach’s Conjecture is difficult, but in fact, Goldbach’s Conjecture is a very weak proposition. It states that "every even number greater than 4 can be split into the sum of at least two odd primes." In the terminology used above, this means that the "Goldbach partition number" of every even number is greater than or equal to 1. However, actual calculations show that the larger the even number, the larger its Goldbach partition number; the two are generally positively correlated. For example, G_2(100) = 12, G_2(1000) = 56, G_2(10000) = 254, \dots Therefore, in terms of strength, this is similar to saying "there is at least one prime number less than n" (though the difficulty is worlds apart).
Why is such a weak proposition so difficult? In fact, there is a very basic idea for proving Goldbach’s Conjecture, which is the sieve method, somewhat similar to the Sieve of Eratosthenes. We can reasonably estimate the Goldbach partition number of each even number, but we cannot strictly prove it in detail. This is where the difficulty lies. As Hardy said, "We have not failed in principle, but in detail."
The following table shows the Goldbach partition numbers I obtained using Python (due to space limitations, only even numbers up to 100 are listed; the attachment contains a table of partition numbers up to 10,000):
| Even | G_2(2n) | Even | G_2(2n) | Even | G_2(2n) | Even | G_2(2n) |
|---|---|---|---|---|---|---|---|
| 6 | 1 | 32 | 4 | 58 | 7 | 84 | 16 |
| 8 | 2 | 34 | 7 | 60 | 12 | 86 | 9 |
| 10 | 3 | 36 | 8 | 62 | 5 | 88 | 8 |
| 12 | 2 | 38 | 3 | 64 | 10 | 90 | 18 |
| 14 | 3 | 40 | 6 | 66 | 12 | 92 | 8 |
| 16 | 4 | 42 | 8 | 68 | 4 | 94 | 9 |
| 18 | 4 | 44 | 6 | 70 | 10 | 96 | 14 |
| 20 | 4 | 46 | 7 | 72 | 12 | 98 | 6 |
| 22 | 5 | 48 | 10 | 74 | 9 | 100 | 12 |
| 24 | 6 | 50 | 8 | 76 | 10 | ||
| 26 | 5 | 52 | 6 | 78 | 14 | ||
| 28 | 4 | 54 | 10 | 80 | 8 | ||
| 30 | 6 | 56 | 6 | 82 | 9 |
Generally speaking, the larger n is, the larger G_2(2n) becomes, but this is not a global law. For example, the partition number of 96 is 14, but 98 drops to 6, and 100 rises again to 12. It fluctuates and is elusive, seemingly following no simple pattern.
In fact, the partition number is related to the prime factors contained in the even number 2n. Generally, the fewer the prime factors, the relatively larger the partition number. Is there an "indicator," or an approximate formula, that gives the upper or lower bound of G_2(2n)? After verification and research, I have derived an empirical formula:
G_2(2n) > \frac{2n}{\ln^2(2n)} \quad (n > 3)
We might remember that the formula \frac{2n}{\ln(2n)} is the formula given by the Prime Number Theorem. If the factor \frac{1}{\ln(2n)} is the result of a single sieve (filtering out primes less than 2n), then \frac{1}{\ln^2(2n)} is actually equivalent to performing the sieve twice. Therefore, Goldbach’s Conjecture can be roughly understood as the fact that after natural numbers are filtered through two sieves, some "essence" still remains.
If there is no way to find the general law of partition numbers, can we first study the law of partition numbers for a certain class of even numbers? Since the partition number is related to prime factors, I naturally thought of the sequence 2^n. Below is the partition table for this class of numbers:
The predicted value is \frac{2^n}{\ln^2(2^n)}. When n is relatively large, the prediction effect of \frac{2^n}{\ln^2(2^n)} stabilizes, basically staying at about 2/3 of the partition number. Of course, I haven’t verified the later ones because as n increases, it exceeds the calculation range of the program I wrote, mainly because I have just started learning programming and the program efficiency is still very low.
However, some clues can be found from this data. For a certain class of numbers, as long as its prime factors are fixed, the variation law of its partition number becomes more obvious, especially for the class of numbers k^n, where k is an even number. Perhaps a simple future exploration could be in this direction.
Attachment: Goldbach Partition Number Table.zip
To be continued...
Original address: https://kexue.fm/archives/1727
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