Some Notes on the Goldbach Conjecture

The Goldbach Conjecture is a very simple conjecture to understand providing you have some basic knowledge about primes. It is one of those conjectures which is very easy to comprehend, but fiendishly difficult to solve. It was first conceived during a letter written by Goldbach to Euler in 1742. There is currently a strong and weak version of the Goldbach Conjecture, with the weak conjecture actually being the original conjecture.

goldbach_conjectures

xkcd: Goldbach Conjectures

The original conjecture asks to prove that all positive integers greater than 2 are equivalent to the sum of three primes, the only problem with the original conjecture, is that Goldbach considered the unit “1” to be prime, which has now been dismissed in Mathematics. However, the conjecture has been restated by Euler to the following modern equivalent, which is known as the strong Goldbach Conjecture (sometimes referred to as the binary Goldbach Conjecture):

Strong Goldbach Conjecture: Every positive even integer greater than or equal to 4, can be expressed as the sum of two primes.

The conjecture can also be expressed using the totient function:

\phi (p) + \phi (q) = 2m,

where m is a positive integer and p and q are both prime numbers. The totient function can be defined as the following (p is prime):

\phi(p) = p - 1

The weaker version of the Goldbach conjecture asks a slightly different question, and is as follows:

Weak Goldbach Conjecture: Every odd integer greater than or equal to 9, can be expressed as the sum of three odd primes.

Similarly, Levy’s Conjecture claims that all odd integers greater than or equal to 7, can be expressed the sum of prime and twice a prime.

p + 2q = n, where n is a positive integer greater than or equal to 7, and p and q are both primes.

Goldbach Number:

Goldbach numbers are any positive even integers which are the sum of two odd primes.

Goldbach Partition:

The Goldbach Partition is the number of ways in which a positive even integer n, can be written as the sum of two primes. The function which produces these partitions is called the Goldbach Partition function G(n). It is defined in the following manner:

G(n) = #{(p, q) | n = p + q, p \le q}

 Methods for proving the Goldbach Conjecture:

The two most common methods which have been used for solving the Goldbach Conjecture involve the Hardy-Littlewood Method along with a number of different sieving methods which have been used to produce greater asymptotic bounds on the distribution of which natural numbers can be written as Goldbach Partitions.

References:

Goldbach Conjecture – Wolfram MathWorld

Goldbach Partition – Wolfram MathWorld

Goldbach Number – Wolfram MathWorld

On Partitions of Goldbach’s Conjecture (Woon)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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About 0x14c

I'm a Computer Science student and writer. My primary interests are Graph Theory, Number Theory, Programming Language Theory, Logic and Windows Debugging.
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One Response to Some Notes on the Goldbach Conjecture

  1. Goldbach simply cannot be broken using “just even numbers” en masse or even
    if separated into five Base 10 even columns. However, when converted to
    Nature’s Base 6 there are only THREE even columns and it turns out that
    there are THREE Quantum solutions, which are different for each of the three columns.

    In a pseudo generated Base 6 format — c2 = (6n+2) = 2,8,14,20….. to infinity
    with solution set c2 = c7+c7′ = (6m+1)+(6p+1);
    and c7; c7’=(6q+1) = 1,7,13,19,25,31……to infinity

    Similarly for c4 = (6n+4) = 4,10,16,22,….. {{ (6n+10) {mod 6}}
    giving c4=(6n+10) = c5+c5′ = (6m+5)+(6p+5); c5=5,11,17,23,29……

    Similarly for c6=6,12,18,24,30……. As can be seen in the attached article, c6 has twice
    the solution set to c2 and c4, which is why there appears to be two separated comets.

    In Base 10 —- 2,4,6,8,10 are spread across five different even columns which
    merges all THREE TYPES and makes finding any solution essentially impossible.

    Separating all the even numbers into three columns demonstrates
    “WHOLE COLUMN SOLUTIONS” some of which are not prime.
    Once the STRUCTURE is known the composites are then extracted
    leaving only Prime Solutions.

    The methodology of the construction can be shown to
    have continuation to infinity and thereby prove Goldbach
    since the expansion of the Prime Solution set for any X=even
    can be shown to be growing faster.

    See a little more at http://primenumbers.eu.pn

    Like

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