PRIME NUMBERS AND GOLDBACH'S CONJECTURE
In 1742, Christian Goldbach wrote to Leonard Euler speculating that every even number greater than four can be expressed as the sum of two prime numbers, a prime number being such that it can only be divided by one and itself. Non-prime or composite numbers have various factors. While this conjecture has been shown to be correct up to very large numbers there has been no proof, so far, that it is always true, except when half the even number is a prime number. With the exception of 2, all even numbers are composite numbers. Odd numbers can be either prime or composite. While the number one is not considered to be a prime number, in this study it will used as a substitute for two since only odd numbers are considered. .
Odd numbers can be derived from one of three equations, y = 3 + 6x, y = 5 +6x and y = 7 + 6x, where x is an integer. While the first equation only gives composite numbers the other two give a mixture of composite and prime numbers. While prime numbers do not show any regularity composite numbers do. Prime numbers occur where there is a void in the composite network. For most, if not all, even numbers there are multiple prime-prime pairs. Using a simple example, it is seen that when an even number is not divisible by 3 the prime-prime pairs that survive a filtering process can in one case be defined by the second equation and in another by the third equation. When an even number is divisible by 3, the prime-prime pairs that survive have one member from the second equation and one from the third. This suggests that, since primes are not random but are controlled by the above equations, prime-prime pairs could survive at all values of even numbers. However, there is no way to prove or disprove this.
The number of primes up to N is defined as pi(N). This paper shows that for the number 2N the upper limit of prime-prime pairs is pi(2N) – pi(N) and the lower limit of prime-composite pairs is 2pi(N) – pi(2N). Graphs, derived from the data, suggest that the upper Iimit of prime-composite pairs is also pi(2N) – pi(N) which could mean that the lower limit of prime-prime pairs is the same as that of prime-composite pairs, namely 2pi(N) – pi(2N). However, an example is given where the lower limit for prime-prime pairs is less than 2pi(N) – pi(2N) which raises the possibility that it could, in some cases, be zero.
Odd numbers can be derived from one of three equations, y = 3 + 6x, y = 5 +6x and y = 7 + 6x, where x is an integer. While the first equation only gives composite numbers the other two give a mixture of composite and prime numbers. While prime numbers do not show any regularity composite numbers do. Prime numbers occur where there is a void in the composite network. For most, if not all, even numbers there are multiple prime-prime pairs. Using a simple example, it is seen that when an even number is not divisible by 3 the prime-prime pairs that survive a filtering process can in one case be defined by the second equation and in another by the third equation. When an even number is divisible by 3, the prime-prime pairs that survive have one member from the second equation and one from the third. This suggests that, since primes are not random but are controlled by the above equations, prime-prime pairs could survive at all values of even numbers. However, there is no way to prove or disprove this.
The number of primes up to N is defined as pi(N). This paper shows that for the number 2N the upper limit of prime-prime pairs is pi(2N) – pi(N) and the lower limit of prime-composite pairs is 2pi(N) – pi(2N). Graphs, derived from the data, suggest that the upper Iimit of prime-composite pairs is also pi(2N) – pi(N) which could mean that the lower limit of prime-prime pairs is the same as that of prime-composite pairs, namely 2pi(N) – pi(2N). However, an example is given where the lower limit for prime-prime pairs is less than 2pi(N) – pi(2N) which raises the possibility that it could, in some cases, be zero.
At this point it has not been possible to prove or disprove Goldbach's Conjecture
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