RE Vinogradov's Theorem & Warings Prime Number Conjecture
- Hi Bob:
I'm far from being the expert you requested, but I think I can help you a bit:
Waring's PNC states that:
"Every odd integer is a prime or the sum of three primes".
Look carefully: the condition explicitly says "every odd integer". It must be fulfilled
explicitly for 3 as well as for 21, 7863 and 2*10^27+1.
Instead, Vinogradov's theorem states that:
"Every sufficiently large odd number is a sum of three primes." (Vinogradov 1937).
The key to the difference are the words "sufficiently large". Vinogradov didn't find a way
to assure explicitly that every odd integer is the sum of three primes, but he found an
asymptotic way to prove it.
The words "sufficiently large" can mean two things:
a) "In the limit". The theorem implies some limit calculus and the desired statement can
be inferred from limits equivalences. It's a quite common situation, favoured by the
application of real or complex analysis to number theory problems.
b) Sometimes this type of conjectures are proved with a specific (but very very high)
bound, and then they usually said things like "every number bigger than 10^10^738
sastisfies that...", but they can instead say the words "sufficiently large", since the
bound is pretty hard to reach in "usual" applications. In Vinogradov's theorem, the bound
number was 3^(3^15). Today it has been reduced to e^(e^11.5) by Chen&Wang. This is also a
consequence of applied analysis.
To answer your other question, I will say that proving GC means that you must prove "every
even number is the sum of two primes". It isn't enough to prove it with "every
sufficiently large even number" or with "the sum of 3 primes"(that's the so called weaker,
ternary, or former GC ), although they would be great theorems themselves (indeed, we
could say the first one is already proven by Estermann: "allmost all even numbers are the
sum of two primes").
Regards. Jose Brox.