On Feb 2, 10:46 am, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote: > In article <04113ffd-b6f8-4092-a939-55aff8c18...@z31g2000vbt.googlegroups.com>, > > Paul <pepste...@gmail.com> wrote: > >I conjecture that, for all integers N > 1, there exists an integer E > >such that E can be expressed as the sum of two primes in more than N > >different ways. > > To convince yourself (rather than prove) that this is true, consider > > http://www.cogsci.ed.ac.uk/~richard/goldbach.html > > Since creating that graph, I have learnt that it is known as Goldbach's > Comet.
Your graph is interesting, but it may be possible to show that Goldbach's Comet can mislead the intuition. For example, if there is a very large x (say > 10^18) which can be expressed as the sum of two primes in very few ways (for example < 10 ways), then the limitations of such graphs would be shown.
Does anyone know of any large near counter-examples to Goldbach's conjecture? For example, what's the largest known even number that can be expressed as the sum of two primes in only 1 way?
Does anyone know any results/tables for the largest known even number E_k that can be expressed as the sum of two primes in <= k ways?