A proof of the Legendre and Goldbach conjectures via a generalization of the concept of prime number

In this paper, we generalize the concept of prime number and define new primes. It allows to apply the new concept to the Legendre and Goldbach conjectures and to prove them. Introduction The prime numbers are called primes because they are the bricks of the numbers : Each number n can be written when where j p are primes and nj 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 nj are rationals, everything changes. Considering that the decomposition in prime factors of an integer n when nj are rationals . In this writhing, then the j 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 : RE TR AC ED International Frontier Science Letters Online: 2014-10-01 ISSN: 2349-4484, Vol. 2, pp 24-27 doi:10.18052/www.scipress.com/IFSL.2.24 2014 SciPress Ltd, Switzerland SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/


Introduction
The prime numbers are called primes because they are the bricks of the numbers : Each number n can be written when where j p are primes and nj 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 nj are rationals, everything changes.
Considering that the decomposition in prime factors of an integer n when nj are rationals . In this writhing, then the j 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 : International Frontier Science Letters Online: 2014-10-01 ISSN: 2349-4484, Vol. 2, pp 24-27 doi:10.18052/www.scipress.com/IFSL.2.24 2014 Proof : We have But we will prove now that It is true for the two first asserstions and for the third, let us suppose d divising both the two equations, we have And 4m 2 +1 and 4m 2 +8m+1 can be taken primes simultaneously with our definition of the primes, for example, the first 4m 2 +1 divisible by 5 is for m=4, and then 4m 2+ 1=65= 13.5  5= 65.13 -1 is no more prime and 65 is prime, the second is for m=6 and then 4m2+1=145=29.529=145.5-1= 145.13.65-1is no more prime and 145 is prime, etc… until infinity. By the same way, the first 4m 2 8m1divisible by 3 is for m=2 and then is no more prime and 33 is prime, etc… until infinity ; but

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The Goldbach conjecture With the new definition are always primes. Effectively So each even number is the sum of two primes.

Back to the traditional definition of primes
Let Goldbach true for the new definition and false for the old one, so it means that in one side there is an odd number, in the other an even one ! It is impossible : b=0. So the Goldbach conjecture is true !

Conclusion
We have generalized the definition of the primes and proved the Legendre and the Goldbach conjectures for the generalization of the definition of primes, a reasoning whicgh leaded to absurdity allowed to prove that those conjectures are true for the old definition too.