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 in 1742 by the mathematician Christian Goldbach, from whom the conjecture takes its name, it was rephrased by Euler 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.

Tha fact that several weaker versions of the conjecture have been proved, without ever arriving to prove the original statement, lets us think thar behind the Goldbach’s conjecture there is some deep mechanism that has yet to be understanded, 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.

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