Mathematical mysteries: the Goldbach conjecture

Share this page
Leonhard Euler

Leonard Euler (1707-1783) corresponded with Christian Goldbach about the conjecture now named after the latter.

Here is one of the trickiest unanswered questions in mathematics:

Can every even whole number greater than 2 be written as the sum of two primes?

A prime is a whole number which is only divisible by 1 and itself. Let's try with a few examples:

  • 4 = 2 + 2 and 2 is a prime, so the answer to the question is "yes" for the number 4.
  • 6 = 3 + 3 and 3 is prime, so it's "yes" for 6 also.
  • 8 = 3 + 5, 5 is a prime too, so it's another "yes".

If you keep on trying you will find that it seems that every even number greater than 2 can indeed be written as the sum of two primes. This is also the conclusion that the Prussian amateur mathematician and historian Christian Goldbach arrived at in 1742. He wrote about his idea to the famous mathematician Leonhard Euler, who at first treated the letter with some disdain, regarding the result as trivial. That wasn't very wise of Euler: the Goldbach conjecture, as it's become known, remains unproven to this day.

In 1938 Nils Pipping showed that the Goldbach conjecture is true for even numbers up to and including 100,000. The latest result, established using a computer search, shows it is true for even numbers up to and including 4,000,000,000,000,000,000 — that's a huge number, but for mathematicians it isn't good enough. Only a general proof will do.

There is a similar question, however, that has been proven. The weak Goldbach conjecture says that every odd whole number greater than 5 can be written as the sum of three primes. Again we can see that this is true for the first few odd numbers greater than 5:

  • 7 = 3 + 2 + 2
  • 11 = 3 + 3 + 5
  • 13 = 3 + 5 + 5
  • 17 = 5 + 5 + 7.

Until very recently the result had only been verified for odd numbers greater than 2 x 101346 — that's a number with 1,347 digits! But then, in 2013, the Peruvian mathematician Harald Helfgott closed the enormous gap and proved that the result is true for all odd numbers greater than 5.

The weak and strong Goldbach conjectures are just two of many questions from number theory that are easy to state but very hard to solve. See here to read about some more, and here to find out more about the Goldbach conjecture and our Goldbach calculator.



Since July 2016, the following statements are established : 1- " Any even number is a difference of two odd prime numbers " 2- "Any odd number is a prime, or a sum or a difference of an odd prime number and the even prime number 2 ".
Therefore, taking into account the classic Goldbach conjecture* stating that "An even number greater than 2 may be expressed as the sum of two prime numbers ", the following general statement is established: " Any integer number is a prime, or a sum or difference of two prime numbers ". More info will be available shortly. in the meantime, the above statements may be checked and double checked.
*also established.


How come it remains unproven? What is considered to be proof? And how is it the weak conjecture has been proven? What will settle as proof?


Has anybody ever seen mentions of the following conjectures, very similar in appearance to Goldbach's or Lemoine's?

Lemoine’s conjecture is ”All odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime”
We know that it can be can be rewritten:
2n + 1 = p + 2q always has a solution in primes p and q (not necessarily distinct) for n > 2.

It appears that the following very similar conjectures (also applying to the relations between an odd number, a prime and a semi-prime) can also be verified for values greater than 2 millions.
Additional conjecture 1: O = 2n + 1 = p - 2q always has a solution in distinct odd primes p and q for o >= 5
Additional conjecture 2: O = 2n + 1 = 2q - p always has a solution in p and q with q < p, p being an odd prime and q being 1 or an odd prime and o >= 1 (For every odd positive integer O there exists two distinct odd elements from PRIMES&1 p0 and p1 such that p1 is the average of O and p0 )

In addition, the Lemoine conjecture can be split into two different conjectures:
Additional conjecture 3: 2n + 1 = p + 2q always has a solution in p and q where p > q, p odd prime and q in PRIMES&1 for n > 2
Additional conjecture 4: 2n + 1 = p + 2q always has a solution in p and q where p <= q , q odd prime and p in PRIMES&1 for n > 2

Similarly the Goldbach conjecture:
E = 2n = p + q with E > 2 has a solution in primes p and q
Has a twin:
E = 2n = p – q with E > 2 has a solution in primes p and q
… but this conjecture, I believe, follows from the Goldbach conjecture and the fact that p+q which is even can always be rewritten as p-q


I would suggest that the Goldbach Conjecture is merely a reflection of the persistently preserved balance inherent in arithmetic number theory. For example, every integer has an additive inverse; every operation has an inverse; the number line maintains symmetry about a zero; etc., etc. And of course the fact that every integer is either prime or is a product of unique prime factorization - this is also a reflection of the symmetric balance maintained throughout the number theoretic "rulebook".
Goldbach pairings also reflect a balance about a midpoint and thus the conjecture appears to me as really just a translated restatement of the governing axioms and conventions. Perhaps instead of calling numbers even and odd we might try calling them balanced or unbalanced - maybe then Goldbach might seem more readily sufficient and necessary. Of course, you can't "prove" axioms - you can only conjure up equivalent statements.

  • Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.