135 research outputs found
A Note on Goldbach Partitions of Large Even Integers
Let be the set of all partitions of the even integers from the
interval into two odd prime parts. We show that
as . We also assume that a
partition is selected uniformly at random from the set . Let
be the size of this partition. We prove a limit theorem which
establishes that converges weakly to the maximum of two random
variables which are independent copies of a uniformly distributed random
variable in the interval . Our method of proof is based on a classical
Tauberian theorem due to Hardy, Littlewood and Karamata. We also show that the
same asymptotic approach can be applied to partitions of integers into an
arbitrary and fixed number of odd prime partsComment: 8 page
Every sufficiently large even number is the sum of two primes
The binary Goldbach conjecture asserts that every even integer greater than
is the sum of two primes. In this paper, we prove that there exists an
integer such that every even integer can be expressed as
the sum of two primes, where is the th prime number and . To prove this statement, we begin by introducing a type of double
sieve of Eratosthenes as follows. Given a positive even integer , we
sift from all those elements that are congruents to modulo or
congruents to modulo , where is a prime less than .
Therefore, any integer in the interval that remains unsifted is
a prime for which either or is also a prime. Then, we
introduce a new way of formulating a sieve, which we call the sequence of
-tuples of remainders. By means of this tool, we prove that there exists an
integer such that is a lower bound for the sifting
function of this sieve, for every even number that satisfies , where , which implies that can be expressed as the sum of two primes.Comment: 32 pages. The manuscript was edited for proper English language by
one editor at American Journal Experts (Certificate Verification Key:
C0C3-5251-4504-E14D-BE84). However, afterwards some changes have been made in
sections 1, 6, 7 and
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