Numeri idonei

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Introduction

Since we have already commented one day, Leonhard Euler is the most prolific mathematician of the history. We can find his name in almost all the branches of the mathematics, from algebra up to complex analysis, happening for geometry and topology. But the more it investigates one in his works the more it is surprised. However much we think that we know the Euler works it always appears by surprise with a new topic that us was foreign. This itself is what it has spent to me a few days ago. And, how not, am I going to tell it to you.

Numeri idonei

In a letter directed to the Swiss physicist Nicolas Béguelin, Euler he was commenting on the following thing:

All the suppressed numbers of only one form prime in sound or cousins' doubles where and they are prime between themselves. I have observed that other similar expressions of the form enjoy the same property giving to the letter suitable values.

This is, any number that can express itself of the only form how, for and relative cousins, it is prime or the double of a cousin. In particular, any odd number that could express itself of the only form in the previous sense is prime.

But there is still more. Not only it serves an expression of the type, but certain values exist of such that an expression of the type fulfills the same property. To these values of es to which there are called they numeri idonei (suitable numbers or suitable numbers in Spanish and suitable numbers or idoneal numbers in English).

At least this was the initial definition of suitable number. But this way of defining this type of numbers presents some problems. For example, it is a suitable number (we will see it further on) and for him it is fulfilled that:

it is the only representation of the number 9 as. But since we all know 9 it is not prime, although yes it is a potency of a cousin, since. Therefore we should say that it is a suitable number if any odd number that could express itself of the only form as it is prime or promotes of a cousin, but it is possible to play in tune a little more to eliminate this new possibility, this is, that the number is a potency of a prime number (in the first linkage of the sources you can see some of the conditions that we can add to him to the definition to avoid this).

Knowing a little the way of working of any Euler he can imagine that it did not remain there, that his investigations on this topic did not end in the establishment of the definition of this type of numbers. Knowing about his character investigator one he tends to think that it tried to study in depth more the matter. And having little information about his achievements it is not difficult to become convinced of that it did it, and very deeply. Since yes, this way it was. Euler prepared a list of suitable numbers. It is the following one:

In whole 65 numbers that Euler verified that they were suitable (in the sense commented previously). In fact it investigated more: it used it is ready to construct prime numbers even of eight numbers.

Come to this point the most logical thing is that we rise the following question to ourselves: is the set of suitable numbers infinite? The answer is not. In 1934, the mathematician Sarvadaman Chowla demonstrated that the set of suitable numbers is finite.

Knowing this us another question arises: are there more suitable numbers apart from the found ones by Euler? Unfortunately for this question there is no answer yet, although yes information is had. Specifically it is known that like much one more suitable number exists, apart from that they are in the list. And that if the latter suitable number in fact exists, must be major than 100000000.

Major prime number met the suitable numbers

We have mentioned before Euler that it used these numbers to find prime numbers relatively grades (up to eight numbers). The biggest prime number that Euler found with this téctica was. To demonstrate that this number of eight numbers is prime it would be necessary to verify that the only solution of the equation