論説A Weak Proof of the Goldbach Conjecture

Hiroyoshi Kozu＊

Abstract
We attempt to prove the Goldbach conjecture by using Chebyshev's theorem, as well as the properties of inequalities and the harmonic mean. TheGoldbach conjecture is revealed when the smallest sum of two primes is calculated, and the even number derived from the harmonic mean is the largest sum of two primes. We also find that the Goldbach conjecture holds
when there is a large gap between the prime numbers, or when the twoprime numbers are the same. Much of this content is based on Kozu [4].

1. Introduction
Most of the research on prime numbers is devoted to the Riemann andGoldbach conjectures, as these are difficult, unsolved problems. TheGoldbach conjecture is that "all even numbers greater than or equal to 4 canbe expressed as the sum of two prime numbers". Tao [6] proved that odd＊

Aichi University, Faculty of Business Administration, Japan (e-mail: kozu@aichiu.ac.jp)
4-60-6, Hiraike-cho, Nakamura-ku, Nagoya-city, Aichi 453-8777 JAPAN― 137 ―

numbers above 3 can be represented as the sum of five prime numbers, andHelfgott [2] attempted to prove Goldbach's weak conjecture with a large
gap between prime numbers. Meanwhile, the weak conjecture was verifiedcomputationally (Helfgott and Platt [3]). The gap between prime numbers
was studied by Maynard [5] and Ford, Green, Konyagin, and Tao [1].
Considering these studies, we attempt to prove the Goldbach conjectureby using Chebyshev's theorem (or the Bertrand-Chebyshev theorem), as
well as the properties of inequalities and the harmonic mean. The Goldbach
conjecture is displayed when the smallest sum of two primes is calculated,and the even number derived from the harmonic mean is the largest sum oftwo primes.

2. Weak Proof of the Goldbach conjecture
If Chebyshev's theorem holds for the real number axis, the range ofprime numbers is(1)

a ＜ Pk ＜ 2a ,where a (2  a) is a natural number and Pk is a prime number. Then, another prime number on the same axis is similarly represented by(2)

b ＜ Pm ＜ 2b ,where b (2  b) is a natural number and Pm is a second prime number.
Then, adding (1) and (2), the sum of the two primes is given by(3)a ＋ b ＜ Pk ＋ Pm ＜ 2 (a ＋ b) .

The range is reduced using the geometric and harmonic means on the thirdterm in (3). Indicating both equality and inequality signs at the harmonic― 138 ―

A Weak Proof of the Goldbach Conjecturemean as the final mean, we obtain(4)

a ＋ b ＜ Pk ＋ Pm 8ab＜ 4 √ab ＜ 2 (a ＋ b) ,a＋b

where a ≠ b. Then, (4) simplifies to(5)a ＋ b ＜ Pk ＋ Pm 8ab.
a＋b

There must be at least two prime numbers in (5), and because 3  a and5  b, 8  a ＋ b. Then, (5) can be rewritten as(6)

8  P k ＋ Pm 

8ab.
a＋b

The value of the harmonic mean in (6) suggests that the sum of two primenumbers is the highest even number. We consider that this formula maximizes the sum of two prime numbers while minimizing the range. Furthermore, upon transforming the third term of (6), it can be written as(7)

8  P k ＋ Pm 

8ab8a＝.
a＋ba＋1bHere, if b is much larger than a, thena 〜0 . Substituting this into theb 〜

fourth term of (7), we obtain(8)8  Pk ＋ Pm  8a .

Furthermore, (8) can be rewritten as(9)8  Pk ＋ Pm  2a ,where 4 a. When the minimum value of (9) is 8, a ＝ 4. This equation will
yield an even number of 4 or more. There is no problem provided that eachprime range has only one prime number. Since there is an odd number― 139 ―

immediately preceding each of the even numbers 2a in (1) and 2b in (2), ifthat odd number is a prime number, or if we use this method to determine
the range of prime numbers, then from (9), we obtain(10)

Pk ＋ Pm ＝ 2a .

Since each of a and b is a natural number, of which there are an infinity,
there is an infinite range of prime numbers. Therefore, since Pk and Pm are
any prime numbers, any sum of two primes is an even number.
Further, if a ＝ b, then the minimum sum of two primes is 4. Substituting
a ＝ b into (5) gives(11)

4  Pk ＋ Pm  4a .

Using the same interpretation as above, (11) can be written as(12)

Pk ＋ Pm ＝ 2a ,where 2  a. When there is one prime number in each range, (12) does nothold, from (1) and (2). However, if Pk-1  a and Pm-1  a, this is satisfied by(13)Pk-1 ＋ Pm-1 ＝ 2 Pk-1 ＝ 2 Pm-1 ＝ 2a .

From (13), we have(14)

Pk-1 ＝ Pm-1 ＝ a .

According to Chebyshev's theorem, this indicates that the prime number
immediately preceding a is Pk-1 and Pm-1 . By contrast, when there are twoprime numbers in the same range, from (1) and (12), we must have Pk ≠Pm . In addition, the Goldbach conjecture holds for the sum of two prime
numbers when there is a large gap between the larger and smaller prime― 140 ―

A Weak Proof of the Goldbach Conjecture

numbers, with the range given by a large value of a. The interpretation is
the same as when there is a considerable difference between a and b.

3. Conclusion
We attempted to prove the Goldbach conjecture, using the properties ofinequalities under the existence of prime numbers derived from Chebyshev'stheorem. We found that the properties of the harmonic mean can be applied
to the properties of prime numbers, and considered cases where there is adifference in the range of each prime number, two prime numbers arewithin the same range, or there is an extremely large range. Finally, weclarified that the Goldbach conjecture holds when there is a large gap between the prime numbers, or when the two prime numbers are the same.

Reference[1] K. Ford, B. Green, S. Konyagin and T. Tao, Large gaps between consecutive primenumbers, Annals of Mathematics, vol. 183 (2016), pp. 935-974.
[2] H. A. Helfgott, The ternary Goldbach conjecture is true, (2013) arXiv: 1312. 7748
[math. NT][3] H. A. Helfgott and D. J. Platt, (201 Numerical verification of the ternary Goldbachconjecture up to 8.8755・10 30, Experimental Mathematics, vol. 22 (2013), pp. 406-409.
[4] H. Kozu, A proof of the Goldbach conjecture, The Keiei Sogo Kagaku (Journal ofManagerial Research), Institute of Managerial Research, Nagoya, Japan, vol. 113(2020), pp.149-153.
[5] J. Maynard, Large gaps between primes, Annals of Mathematics, vol. 183 (2016), pp.
915-933.
[6] T. Tao, Every odd number greater than 1 is the sum of at most five primes, Mathematics of Computation, vol. 3 (2014), pp. 997-1038.

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