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User:Marcos Librán López

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I'm a mathematician and number theorist.

Has anybody read about Goldbach and Legendre conjectures?

Goldbach: Any even number is the sum of two primes or 1.

2 = 1 + 1 4 = 2 + 2

Let be n= p1 + p2 with p1, p2 any prime number excepts 2. So by Euclid's division algorithm, there exists q1 and q2 natural numbers such that p1=2 q1+ 1 and p2 = 2 q2 +1.

Then n = p1 + p2 = 2 q1 + 1 + 2 q2 + 1 = 2 (q1 + q2 +1), so we are working with all evens (excepts 2 and 4, verified before this).

Legendre: there always exists at least one prime between two consecutive squares.

So, for any n at N there exist a prime at P, n^2 < p < (n+1)^2

the opposite proposition is at follows:

not Legendre: there exists an n at N, for any p at P, n^2>= p or p>=(n+1)^2.

if p<= n^2 is true, then p>=(n+1)^2 is false because (n+1)^2 = n^2 + 2n + 1 > n^2 >=p. The same for p>=(n+1)^2.

So we can descompose not Legendre on the two following propositions: A: there exists an n at N, for any p at P, n^2>=p B: there exists an n at N, for any p at P, (n+1)^2<=p

And not Legendre is equal to A or B.

A is a false statement because P has no supreme element. B is a false statement because with Bertrand's Theorem.

So not Legendre is true, so Legendre is true.