Mathematics always seems difficult, but the most complicated problems are easy to understand, even if the solution occupies 200TB.
It is very curious when one realizes that the most complicated problems and those that have been unresolved for years, these are problems that any ordinary citizen can understand. The problem and the real difficulty are fully concentrated in the search for the solution. It is the same thing that happens with several of the problems of the millennium that have become so famous in the mathematical world, among other things for the associated million dollar prize.
The problem that has solved a computer today It is not of this group, although it also has a rather interesting history. We owe this problem to Graham and it is based on the equality of pythagoras for the right triangles a2 + b2 = c2. All numbers that meet this equality they are called Pythagorean triplets and they are the basis of our problem.
Graham’s problem
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Graham imagined the following scenario: if I paint all the integers with two colors onlyCan I get all python triplets to be one color? The question may seem confusing, but it is not, let’s go with an example. The numbers 3, 4 and 5 are a triplet of Pitgoras (9 + 16 = 25) so both would have the same color to fulfill Graham’s premise. The problem comes when we realize that the 5 also belongs to the triplet 5, 12, 13.
Here things get complicated because we start to see dependencies between colors that can make the task difficult that Graham proposes. Solving this positively involves demonstrating mathematically that s can be colored that way. But in mathematics there are tricks to answer the problem, for example, the mathematical counterexamples that consist of finding a case in which it is not fulfilled what the problem asks for.
When the solution is a mathematical counterexample
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This is the easiest option to devise, but the most complicated many times to check. You just have to start trying color combinations for all the numbers and see if you can or not. This would be easy if there weren’t infinite numbers, so humans stopped doing this, normally, and looked for a theoretical solution. But computers they are better than us checking things So what they have done now is create a computer that is capable of searching for counterexamples for Graham’s problem.
The program is trying combinations and if it is possible to square the numbers with the right colors he goes on to larger numbers until he finds a mathematical counterexample that shows that Graham’s problem cannot be solved with just two colors. The result is that the data necessary to support said mathematical counterexample occupy 200TB, which is not turkey mucus for a trial and error program.
But what we must not forget is that these computers, however smart and powerful they may seem they do nothing but very concrete operations very quickly.If we want a solution to more complicated problems like How many colors do we need so that all the python triplets are of a single color? So we need Artificial Intelligence or a good mathematician (and if he is Russian and crazy, better)
