A Generalised Form of Goldbach’s Conjecture
Jackson Capper
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Goldbach’s conjecture states that:
Every even integer greater than \(2\) can be expressed as the sum of two prime numbers.
It’s apparent that the recurring theme of \(2\) is expressed thrice in this statement for good reason: they are related. Even refers to divisibility by \(2\). So, suppose we could extend this to any arbitrary divisibility and call it \(x\).
Thus, a more generalised and complete conjecture would be expressed as:
Every integer divisible by \(x\) greater than \(x\) can be expressed as the sum of \(x\) prime numbers.
This seems to hold true for any measured value. Simple examples include (\(x=3\)),
\(6=2+2+2\)
\(9=3+3+3\)
\(12=2+3+7\)
In fact more generally, it seems to be true that an integer (\(i\)) divisible by \(x\) can be expressed as the sum of a number of primary terms (\(y\)) where the integer is equal to, or greater than, \(xy\).
Thus a generalised form of Goldbach’s Conjecture,
Every integer \(i\) can be expressed as the sum of \(x\) primary terms where \(i\) is divisible by \(y\) and \(i \geq xy\).