Goldbach’s conjecture

Even numbers expressed as sums of prime numbers: equal numbers are highlighted with equal colors

The proof of the Goldbach’s Conjecture is one of the biggest still unsolved problems regarding prime numbers. Originally expressed by the mathematician Christian Goldbach, from whom the conjecture takes its name, it was cited by Euler for the first time in 1742, in the form in which we know it today:

Every even number greater than 2 can be expressed as a sum of two prime numbers.

The empirical evicence in favour of the conjecture is overwhelming: not only every even number greater than two can be expressed as a sum of two prime numbers, but it can be so expressed in different ways. This can be seen even stating from the smallest numbers:

4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7 = 5 + 5
12 = 5 + 7
14 = 3 + 11 = 7 + 7
16 = 3 + 13 = 5 + 11

In particular, in practice we can see that the number of different ways in which an even number can be expressed as a sum of two primes tends to grow as the considered number increases, as you can see from the following picture, the so called Goldbach’s comet:

Number of different ways (y axis) in which an even number (x axis) can be expressed as a sum of two prime numbers

In spite of the empirical evidence and the simplicity of its statement, the conjecture is resisting to every attempts of proof since almost three centuries. Several mathematicians have proved some weaker versions of the conjecture. Essentially these weaker versions can be grouped into two categories, according to how they differ from the conjecture:

  • In the Goldbach’s conjecture there is a sum of two primes; in some weaker versions there is a sum of a greater number of primes, or a sum of a prime and a semiprime, that is the product of two primes.
  • The Goldbach’s conjecture, if proved, would be valid for every even number greater than 2; some weaker versions are valid for “almost” every even number greater than two.

Some weaker versions can be classified into both categories, as they state that almost every even number greater than two can be expressed as a sum somehow more complex than a sum of two primes.

There’s also the so-called weak Goldbach’s conjecture, which states that every odd number greater than 5 can be written as the sum of three primes (for example 7 = 2 + 2 + 3, 9 = 3 + 3 + 3, 11, = 3 + 3 + 5, …). It’s called “weak” conjecture because, if Goldbach’s conjecture (also known as “strong” Goldbach’s conjecture, in order to distinguish it from the other) was proved, the former would be a simple consequence. In fact, if every even number could be expressed as the sum of two primes, then, given an odd number d greater than 5, d - 3 would be an even number greater than 2; so, due to strong Goldbach’s conjecture we’d have d - 3 = p + q, where p and q are two primes, so d = 3 + p + q, i.e. d would be the sum of three primes.
Then, the strong Goldbach’s conjecture implies the weak one, but the opposite is not true. In fact, the weak conjecture has been proved by the Peruvian mathematician Harald Andrés Helfgott nel 2013, but, nevertheless, the strong conjecture continues to resist against all attempts to prove it.

The fact that several weaker versions of the conjecture have been proved, without ever arriving to prove the original statement, lets us think that behind the Goldbach’s conjecture there is some deep mechanism that has yet to be understood, and it could require new proof techniques. For this reason we are sketching out the proof on the basis of a new theory, specifically built for studying the specifical problem stated by the conjecture: the dashed line theory.

2 Risposte a “Goldbach’s conjecture”

  1. I would like experts to expose the errors in my simple proof, “The Stepladder Proof of the Goldbach Conjecture.” I posted it on Academia.edu.

    In short, prime,prime pairs = total pairs – prime,composite pairs – composite,composite pairs.
    Subject to known adjustments, every even number, n, can be expressed as n/4 unique and mandatory even number pairs. Each pair sum = n. Primes are embedded. (prime + 1) + (prime + 1) = n. For example, 200 has 50 unique and mandatory even number pairs. My excess pairs algorithm isolates 37 pairs with at least one composite. Prime,prime pairs = 50 – 37 = 13 exactly. The power and simplicity is in n/4.
    Also email to [the email address has been omitted for avoiding spam]
    Thanks for questions and comments.
    Gregory Mazur

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