Prime numbers, one of the most celebrated numbers in number theory, are the natural numbers greater than 1 and whose only factors are 1 and itself e.g. 2 = 1 ´ 2, 3 = 1 ´ 3, 5 = 1 ´ 5, 41 = 1 ´ 41, etc. Equivalently, one can say that a prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. There are several unsolved mysteries related to prime numbers e.g. Riemann Hypothesis, Twin Primes Conjecture, Goldbach Conjecture and many more. Though this article is focused on Goldbach Conjecture, the others are of equal importance.

Goldbach’s Conjecture is one of the oldest and best-known unsolved problems in number theory. In 1742, Christian Goldbach in his letter to Euler proposed the following conjecture:

Every integer greater than 2 can be written as a sum of three primes.

This is because he considered 1 to be a prime, but this convention is now abandoned. So, the Goldbach

conjecture can be restated as:

Every integer greater than 5 can be written as a sum of three primes.

7 = 2 + 2 + 3

8 = 2 + 3 + 3

12 = 2 + 5 + 5

53 = 3 + 19 + 21

Euler gave an equivalent version of the conjecture (Binary Goldbach Conjecture):

Every even integer greater than 2 can be written as a sum of two primes.

6 = 3 + 3

8 = 3 + 5

12 = 7 + 5

52 = 5 + 47

Although stating Goldbach’s Conjecture is easy, yet it is an unsolved problem. Several efforts have been made to solve this problem which led to the formulation of a “weaker” version of Goldbach’s conjecture which is a solved problem.

Weak Goldbach Conjecture: Every odd integer greater than 5 can be written as a sum of three primes.

15 = 5 + 5 + 5 = 3 + 5 + 7, 17 = 3 + 3 + 11 = 3 + 7 + 7, etc.

Goldbach’s Conjecture implies the aforementioned statement, which is why it is called weak.

In 1923, Hardy and Littlewood showed that assuming the generalized Riemann hypothesis, the weak Goldbach conjecture is true for all sufficiently large odd numbers. Ivan Matveevich Vinogradov, in 1937, proved directly that all sufficiently large odd integers can be written as a sum of three primes. For more history, interested readers may refer to [1,6] and references within. However, Vinogradov didn’t specify how large the number should be. Borodzin (A student of Vinogradov) proved that 314348907, a 6,846,169 digit number, is large enough. In 2002, Ming-Chit and Tian-Ze [3] lowered this to approximately e3100, yet this is too large to check on computers.

Olivier Ramare (who visited IISER Berhampur in 2018) proved that every even number greater than or equal to 4 is, in fact, the sum of at most six primes in his paper, “On Schnirelmann’s constant” (interested readers may refer to [4]), from which it follows that every odd number greater than 4 is the sum of at most seven primes.

In 2013, Harold Helfgott [2] uploaded a proof of Goldbach’s weak conjecture on arXiv but it is not published yet.

Now let us focus on an interesting fact regarding the existence of primes. Bertrand’s postulate was conjectured by Joseph Bertrand in 1845. It states that if n > 1 is a natural number, then there always exists at least one prime between n and 2n. This conjecture was proved by Chebyshev in 1852 and hence this is now known as the Bertrand-Chebyshev theorem or simply Chebyshev theorem.

Interestingly, Bertrand’s Postulate also follows from the Goldbach Conjecture if we assume (Euler’s version) to hold true. In 2005, H. Ricardo [5] observed a simpler proof to Chebyshev theorem. We end this article with his proof.

Theorem: For n Î ℕ and n > 1 then there always exist a prime p between n and 2n.

Proof:

Let n Î ℕ and n > 1. By Goldbach’s Conjecture,

2n = p + p',

where p and p' are two primes.

If n is composite, then either p > n or p' > n. If not, p < n and p' < n implies

2n > p + p'

which is a contradiction.

Therefore n < p < 2n or n < p' < 2n proving the statement for composite n.

If n is a prime, then it may also happen that n = p = p’. If n is even, n = 2, but we know that 3 is the prime between 2 and 4. If n is an odd prime, by Goldbach’s conjecture,

2(n + 1) = q + q',

where q and q' are primes.

By the above argument, since n + 1 is even, and hence composite,

n + 1 < q < 2(n + 1) or n + 1 < q' < 2(n + 1).

 Without loss of generality, let n + 1 < q < 2(n + 1) (what if q = 2?), which implies

n < q < 2(n + 1).

Also, q ≠ 2n + 1 because this implies q' = 1 (which is not a prime). Hence n < q < 2n.

Therefore Bertrand’s Postulate follows from Goldbach conjecture.

[1]  L. E. Dickson. History of the theory of numbers. Vol. I: Divisibility and primality. Chelsea Publishing Co., New York, 1966. [2] H. Helfgott, “arXiv:1312.7748 [math.NT],” 30 12 2013. [Online]. Available: https://arxiv.org/abs/1312.7748v2. [Accessed 12 04 2019]. [3] M.-Ch. Liu and T. Wang. On the Vinogradov bound in the three primes Goldbach conjecture. Acta Arith., 105(2):133–175, 2002. [4] O. Ramare, “On Schnirelmann’s constant,” Annali della Scuola Normale Superiore di Pisa, vol. 22, no. 4, pp. 645-707, 1995. [5] H. Ricardo, “Goldbach’s Conjecture Implies Bertrand’s Postulate,” The American Mathematical Monthly, p. 492, 2005. [6] N. Tchudakoff, “On Goldbach-Vinogradov’s Theorem,” Annals of Mathematics, pp. 515-545, 1947