[seqfan] Re: Goldbach-Yamada conjecture: a(n) > 0 for n >= 6.

[M. F. Hasler]

Yamada's result from 2015 is the explicit bound
M = e^e^36 ≈ 3.4 • 10 ^ 1 872 344 071 119 348
for Chen's theorem from 1966 (full details of proof published in 1973) that
all sufficiently large even integers are the sum of a prime and another
prime or semiprime.
There is an obvious link between Chen's theorem and the proposed sequence:
if there aren't enough decompositions then there can't be such products
(but it's an obfuscation to speak of products rather than pairs of such
primes, it actually simply replacing the first "prime" by "twin prime").
So, if a(n)>0 for all n>=6, then Yamada's bound could be replaced by 2*6.
But
1) without any doubt Juri is not the first to observe that Chen's theorem
*seems* to hold for all even N > 12 (and that the number of these
decompositions grows rapidly with N), and
2) a conjecture (or wild guess) can't be considered as a strengthening of a
rigorous proof (even if experimental data appears overwhelmingly convincing
as in RH or Legendre's conjecture). The point in Yamada's work was not
to*propose* a value, but to*prove* it.

Finally I'd like to say (in defense of JJ) that we do injustice of we
attribute more value to words than to acts and intentions. JJ used harsh
words but with good intentions. Others use nice words but insult thousands
of valuable subscribers by implicitly saying "Hey all of you, (please) take
your time to work out the details of my ingenious idea, my great mind can't
waste it's time for such basic work, correct wording and elementary
consistency checks ; I'm already producing another great idea."
And seem not to care about the fact that having thousands of others lose 1
minute to understand their badly explained thoughts is wasting more time
than if they invested 3-4 afternoons to work out their idea and its
explanation with references etc.

But maybe that's a matter of personal taste and/or perception. Let's hope
that such incident will not prevent the whole community of SeqFans to spend
an Excellent New Year 2019 !
--
Maximilian

On Tue, 1 Jan 2019, 07:56 юрий герасимов <2stepan at rambler.ru wrote:

> Where a(n): 0, 0, 1, 1, 0, 1, 2, 2, 1, 2, 2, 1, 4, 4, 1, 1, 2, 3, 2, 4,
> ... is
> the number of products of 2 successive primes of the form 2*n-(some prime)
> and
> 2*n-(some semiprime). This conjecture represents the strengthening of
> Goldbach's
> conjecture and the explict version of Yamada's teorem (2015). Dear
> SeqFans, the
> proposed reformulation is useful or useless? Thanks You.
>
> P.S. a(3) = 1 because if 2*3-3(some prime)=3 and 2*3-4(some semiprime)=2
> then
> 3*2=6;
>
> a(4) = 1 because if 2*4-5(some prime)=3 and 2*4-6(some semiprime)=2 then
> 3*2=6;
>
> a(6) = 1 because if 2*6-7(some prime)=5 and 2*6-9(some semiprime)=3 then
> 5*3=15;
>
> a(7) = 2 because if 2*7-11(some prime)=3 and 2*7-9(some semiprime)=5.
>
> 2*7-7(some prime)=7 and 2*7-9(some semiprime)=5 then 3*5=15, 7*5=35, ...
>
> EXAMPLE
>
> Triangle begins:
>
>