Mountain climbing is a beloved metaphor for mathematical research. The comparison is almost inevitable: the frozen world, the thin air and the implacable harshness of mountaineering reflect the unforgiving landscape of numbers, formulas and theorems . And just as a climber pits his abilities against the unyielding object in his case- a sheer wall of stone- a mathematician often finds himself engaged in an individual battle of human mind against the rigid logic.
In mathematics, the role of the highest peaks is played by the greatest conjectures – sharply formulated statements that are most likely true, but for which no conclusive proof has yet been found. These conjectures have deep roots and wide ramifications. The search for the solution entails a large chunk of mathematics. Eternal fame awaits those who conquer it first.
Goldbach Conjecture Is one of the oldest and the best known unsolved problems in number theory and all of mathematics . It states that:
“Every even whole number greater and 2 is a sum of 2 prime numbers”
A modern version of the marginal conjecture is :
“Every integer greater than 5 can be written as a sum of 3 primes”
And a modern version of Goldbach’s older conjecture of which Euler reminded him is :
“Every even integer greater than 2 can be written as a sum of 2 primes”
Its very easy to state, but it seems very difficult to prove.
If you try it out via a small program you will very soon convince yourself that it works for any integer you care to use as a test case. However, this is not a mathematical proof and there may just be an integer out there waiting to be found that needs three or more primes in the sum.
There are a number of variations of the basic Goldbach’s conjecture that are often studied as a sort of warm up to the real thing. For example, the Odd Variant postulates that every odd number greater than 7 is the sum of three odd primes. This was proved, but the proof relied on a modification of the Riemann hypothesis – which is as yet another unproven, well-known, challenge.
To date there have been a smattering of proofs that set bounds on the size of a number that disobeys the conjecture – every sufficiently large odd integer is the sum of three primes and almost all even integers are the sum of two primes. You might think that “sufficiently large” would mean that we could take a program and test the numbers up to the upper bound where the proof takes over.
Unfortunately the upper bounds are very “upper” , for example bigger than 106846168, which makes computational help still out of reach.
Now we have another proof that places a different sort of upper bound on the conjecture. Terence Tao, a Fields medalist, has published a paper that proves that every odd number greater than 1 is the sum of at most five primes. This may not sound like much of an advancement, but notice that there is no stipulation for the integer to be greater than some bound. This is a complete proof of a slightly lesser conjecture, and might point the way to getting the number of primes needed down from at most five to at most 2.
Notice that no computers where involved in the proof – this is classical mathematical proof involving logical deductions rather than exhaustive search.
So does this have any application to real world problems?
I doubt it, but number theory is so central to coding and cryptography. On another note, it is a demonstration that mathematics hasn’t fallen to computer-based constructive proofs just yet and perhaps there is still a neat one-page proof of the four-color theorem waiting to be written.
Goldbach’s Comet One interesting phenomenon generated by the conjecture is the so called Goldbach’s comet which is a visual representation of the number of possible Goldbach partitions of an even integer n. The comet-like structure can probably be explained as a consequence of how the number of partitions varies between different congruence classes. Now this figure just expresses the number of partitions up to n = 5 · 104 , but we can clearly see that the number is steadily increasing. This would make us assume that the conjecture holds, but this has nothing to do with a proof as we can not know that there is no exception for a large n where the number of partitions will be zero.
Goldbach’s Comet, Goldbach partitions up to the integer n = 50000 on the x -axis, and number of partitions on the y-axis. This was generated by a Python script using a modified version of the code from [Sci].
Now, the original Goldbach conjecture can be proven. The original conjecture is as follows:
Every even integer greater than 2 can be written as the sum of two primes.
The mathematical derivation above caters for all even integers greater than or equal to 6. So, the only even integer greater than 2 not catered for is 4. And 4 can be expressed as 2 + 2. 2 is an even prime number but the Goldbach conjecture places no restrictions on the parity of the prime number. In other words, the prime number need not be odd. Therefore, it is indeed true that all even integers greater than 2 can be expressed as the sum of two prime numbers.
The point of such conjectures are often not the result themselves ; more often than not the knowledge of the veracity of such statements is completely uninteresting. The interest lies in the methods of proof to argue against such statements ; they often lead to developing new theories and inspire mathematicians to create new kinds of arguments which can then apply to other situations where they may hopefully be useful. It can be considered as a training ground for mathematicians.