The concept of prime number and the Legendre conjecture

become a convention. For example, if we decide that 16 is conventionally prime, we have and each number can be written according to 16 and its rational exponent instead of 2. If we decide conventionally that each Fermat number is prime, and it is possible by the fact that they are coprime two by two, then each new prime (new primes=bricks with rational exponents in the writing) replaces anther one in the list of the old primes (old primes=bricks with integral exponents in the writing).


Introduction
The prime numbers are called primes because they are the bricks of the numbers : Each number n can be written as where are primes and are integers. This writing is called the decomposition in prime factors of the number n. In fact, this definition is a very particular case of a much more general one.
Indeed, if are rationals, everything changes.
Considering that the decomposition in prime factors of an integer n when are rationals . In this writhing, then, the have no reason to be the same than before and they become a convention. For example, if we decide that 16 is conventionally prime, we have and each number can be written according to 16 and its rational exponent instead of 2.
If we decide conventionally that each Fermat number is prime, and it is possible by the fact that they are coprime two by two, then each new prime (new primes=bricks with rational exponents in the writing) replaces anther one in the list of the old primes (old primes=bricks with integral exponents in the writing).
Example: If by convention, the fifth Fermat number is prime, we can decide that it replaces 641 which becomes compound when 6700417 remains prime or 641 remains prime and it replaces 67004147 which becomes compound. In all cases, the advantage is that we have a formula which gives for each n a prime. And we can see the the primes are infinite. There is another interesting result: Let Ulam spiral. The Fermat numbers are all situated in the same line.

The Legendre conjecture
The Legendre conjecture states that there is always a prime number between the squares of two consecutive integers. So where p is prime. What does it become with our new definition ? It remains true ! Effectively :

Proof :
We have But we will prove now that (CD : common divisor) : 2015-01-01 ISSN: 2349-4484, Vol. 3, pp 16-18 doi:10.18052/www.scipress.com/IFSL.3.16 2015 It is true for the two first assertions and for the third, let us suppose d dividing both the two equations, we have because d is odd. And And And and can be taken primes simultaneously with our definition of the primes, for example, the first divisible by 5 is for m=4, and then is no more prime and 65 is prime, the second is for m=6 and then is no more prime and 145 is prime, etc… until infinity. By the same way, the first ivisible by 3 is for m=2 and then is no more prime and 33 is prime, etc… until infinity ; but The Legendre conjecture is true for the news definition of the primes, we have proved it.

Back to the traditional definition of primes
Let now the Legendre conjecture, we have found that for the new definition Impossible ! It means that for all x, there exists p old prime number for which b=0 and the conjecture is also true for the old definition !

Conclusion
We have generalized the definition of the primes and proved the Legendre conjecture for the generalization of the definition of primes, a reasoning which leaded to absurdity allowed to prove that this conjecture is true for the old definition too.