Re: Conjecture Related to Goldbach's Conjecture


Thanks, Gerry. I see there is an interesting graph of numbers for Levy's Conjecture--doesn't actually look much like Goldbach's Comet.

Levy's Conjecture isn't exactly the same as the one I described, since he apparently doesn't require that the "twice a prime" number be twice an odd prime. I also guess that his conjecture allows the "twice a prime" to be twice the prime that is added to it, to get the odd integer. I don't use that, because it is, of course, simply a multiple of three, which just repeats the odd number rather than being a total with new primes.

My idea was to formulate a conjecture that used only the odd primes as factors of the "twice a prime" term, since the same primes would then be available for that conjecture as for Goldbach's--of course, it's impossible to use integer two as one of the primes in Goldbach's strong conjecture, but many odd integers are four more than a prime.

Looks like quite a bit of research has been done on Levy's Conjecture, so maybe there is, in the research, some analytical method that I can use.

Another interesting "sums of primes" investigation, is the problem of using only primes that are twin primes--which have some unusual properties--as pairs of primes that equal even integers. The two primes, of course, don't have to be the two successive twin primes of the pairs they belong to.

I guess the proposition would be, that, at some magnitude, most even numbers above that magnitude are not the sums of two primes that are both twin primes. It's rather interesting that, among small even integers, only three even numbers between 90 and 100 are not the sum of two twin primes. Since twin primes thin out faster at large magnitudes than primes in general do, I have to think that, above a certain number, very few integers are the sum of two twin primes. Would be interesting to see how the diminishing of sums works out.
.