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proof that $4^x$ exceeds the product of the primes up to $x$

Major Section: 
Reference
Type of Math Object: 
Proof

Mathematics Subject Classification

11A41 no label found11A25 no label found11N05 no label found

Comments

I am aware that my initial strategy for this proof is flawed. At about x = 30, even [x/2]!! overtakes 2^(2x - 1). I'm still trying to figure out some high-ball for the primorial pi(x)# that can be proven to stay below 4^x.

What's wrong with using the binomial coefficients to prove this? They're nothing terribly advanced, and if you make sure to include the word "binomial coefficient" it will get linked, in case anyone finds the parentheses notation obfuscating. Besides, binomial coefficients were good enough for Erdős.

i think that in view of the failed proof the article should be withdrawn.
if the author manages to construct a correct proof without
using binonial coefficients then that would in my view be a new proof that does not duplicate what we already have in PM and can be a new article.

Don't confuse my willingness to admit mistakes with a readiness to give up. I probably wouldn't be able to prove this in an olympiad-setting, but that's for the young anyway. A solution could present itself to me when I'm working on something else and this is at the fringe of my memory.

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