numeri primi congetture

It is not true until it is demonstrated: what mathematical truths are

The 3 and the 7 are two Prime numbers – that is, positive integral numbers that are divisible only by 1 and for themselves – and on this, it does not rain. However, if we support the two numbers, we get the number 37he too first. What if we reversing the figures between them? We get the number 73always a first number. We then add a 7 to the beginning: 773still a first number. AND 337? He too first! Well at this point, it would seem that each number that is the combination of 3 and 7 is consequently a first number, and therefore it will also be 3337 … and instead not! The number 3337 is divisible by 47 and 71, as well as for 1 and 3337.

This example helps us to understand a fundamental mathematical concept: a scheme or numerical evidence cannot be used as mathematical truth, It is always necessary to prove it.

A numerical evidence does not indicate a mathematical truth, but a conjecture

The first 6 numbers we considered – 3, 7, 37, 73, 773 and 337they are all first and all formed by combinations of figures 3 and 7. This can hypothesize, on the basis of our experience, that any number formed by a combination of 3 and 7 is first. But that’s not the case! It was enough to consider the number 3337 For Disassemble our hypothesis. In fact, as we said, 3337 can be written as 47 x 71 and therefore it is not a first number.

This simple example, we need to understand a fundamental concept in the mathematical world: you don’t have to do trick from appearances! Any fact, in mathematics, can be considered true only if it is demonstrable, that is, if it is consistent with the mathematical properties already demonstrated and therefore deductible by axioms, that is, those (few) mathematical principles that are considered true in themselves and on which the construction of the whole is based mathematical theory known.

mathematical demonstration

A fact that instead appears true to us, If it has not been shown only one hypothesis remains or conjecture, And it is our job – or rather, of mathematicians – try to verify whether this fact is consistent with mathematical laws.

The unsolved problems of the history of mathematics

The history of mathematics is full of so -called “open problems”, also these problems unresolved, that is, conjectures that seem true but that nobody is able to solve. In truth, throughout history, some open problems have been solved after years, such as the famous Last stop theorem which was formulated in 1637 by the French mathematician Pierre de Ferrat and that It was resolved three centuries laterin 1995 from the English mathematician. Another example is the problem of the four colors, formulated in 1852 and resolved in 1976.

Periodically the major world mathematicians take stock of open and solved problems. Of the century in century, new conjectures are born that commit the minds of the most fine scholar. But There are unsolved problems for centurieswhich even seeming intuitive cannot find a formal demonstration. An example is the Goldbach conjectureformulated in 1742, which states that Each equal numbers equal 2m You can write as the sum of two prime numbers P+q. If we base ourselves on our experience, it seems to be true:

  • 4 is equal to 1+3
  • 6 is equal to 1+5
  • 20 is equal to 3 + 17
  • 340 is the same as 3 + 337
  • …and so on

it does not exist However, to date, one demonstration that ensures that each equal number N = 2m It can be written as the sum of two prime numbers P+q whatever the value of m.