Conjecture 1:
Every odd number n >= 11 can be
expressed as the sum of three (higher order) primes (each of
order 1, 2 and 3 respectively) in at least one way.
That is, n can be expressed as p(i) +
p(p(j)) + p(p(p(k))) for positive integers i, j, and k,
where i > 1.
Conjecture 2:
Every even number n >= 6 can be
expressed as the sum of two (higher order) primes (each of
order 1 and 2, respectively) in at least one way.
That is, n can be expressed as p(i) +
p(p(j)) for positive integers i and j, where i > 1.
NOTES ON THE CONJECTURES:
If it can be proven that Conjecture 1 holds
true, then if this is coupled with the results that 7 = 3 +
2 + 2 and 9 = 3 + 3 + 3 and 9 = 5 + 2 + 2 then this is
enough to also prove that the Weak Goldbach Conjecture holds
true.
If it can be proven that Conjecture 2 holds
true, then if this is coupled with the result that 4 = 2 + 2
then this is enough to also prove that the Strong Goldbach
Conjecture holds true.
Just for the record, I’ve written a computer
program to check my conjectures. Conjecture 1 holds for all
odd n greater than or equal to 11 and less than or equal to
15,001. (Values beyond this are unchecked)
Conjecture 2 holds for all n greater than or
equal to 6 and less than or equal to 70,000. (Values beyond
this are unchecked).
Obviously, I have not found any
counterexamples
Later, on my request,
Mark added the following arguments in favour of his
conjectures;
The conjectures are stricter, more
constrained versions of the classical ones which is why mine
have, in general, fewer solutions than their Goldbach
counterparts.
-
By constraining a more
general result (like Goldbach) it is sometimes easier to
see what is happening overall. Applying constraints is
often a good way to see the important data, because it
gets rid of other trivial results. I have yet to see a
pattern with my conjectures, but it might enable someone
else to see one.
-
Looking at Goldbach from a
new perspective or new viewpoint might help people to
think about the problem from a different angle that they
haven’t yet considered. The given conjectures are not so
obvious.
-
I found it surprising that
even with all those constraints it’s still possible to
find at least one solution for every given value of n.
Intuitively, I would have guessed that their might be no
solutions for some values of n, but this is not the
case. The lowest number of solutions is at least 1.
There are indeed solutions for all n (or at least it
seems to be the case for the values of n I have checked
so far).
Q1.
Can you find a proof for these two conjectures or
counterexamples?
Q2. Are these conjectures an alternative
(easier/worst) approach to prove the Goldbach original ones?
Q3. Any comments related to the arguments 1-3
by Mark?
