Whatever, but nonces appear uniformly distributed contradicting the arbitrary claims of some people that pretend to be knowledgeable.
HAH. You seem to forget even your own posts so quickly "(except slighty off at the very end of the range)". And I provided specific, concrete examples of the very software and hardware that produces those effects.
I also countered your points _assuming_ uniformity, only noting that the distribution was known to be non-uniform and pointing out why; to point out that this assumption was known to be approximate. You might note that I never attempted to characterize the actual non-uniformity, and all the examples I gave were based on uniform numbers.
I note, several more days have passed and no more blocks have again matched your criteria--- I was sad to see that you declined to further discuss the "optimization" you discussed after I pointed out that midstate handling is universally used in all mining systems and has for the life of the system.
Why stopping? It is fun and educational read about how to become a crackpot. It isn't very interesting technically or mathematically, but it is quite educational from the psychological point of view. The cryptocoin community will soon probably need an equivalent of the well-know "crackpot index" from the physics community.
http://math.ucr.edu/home/baez/crackpot.htmlActually this was already done a long time ago;
https://www.wpsoftware.net/andrew/crackpot.html Dear Mr Maxwell,
Let's try to not make it personal and focus on Technical Discussion for the benefit of the Technical Discussion. I will do my part.
The non-uniformity of the nonce distribution at the end of the range is expected because many miners increase the nonce from 0 to 2^32-1 stopping when they find a match, thus it is expected that the end of the range is less frequent. This is what we already pointed out from the beginning jl2012 and myself:
Let's look closer at nounces:
We assume that nounces are uniformly distributed (not exactly true since if we start increasingly with nounce 0 they follow a Poisson law, but taking into account that nounce cycles many times before finding the solution it is well approximated by the uniform distribution).
But this non-uniformity at the end of the range on which you focussed is irrelevant for the question discussed.
What is relevant for the discussion is to know if the nonce distribution is uniform in the small range where the nonces of these blocks do cluster, which is not at the end near 2^32, but at the middle range around 2^31.
This is the reason why I carried further analysis in the neighborhood of this middle point at 2^31
Nounce of block 354641 is within 0.048% of the total range from 2^31.
There are 159 occurences of nounces within an interval of length 2x0.048=0.096% centered at 2^31 which count for 0.11% of the total of 140411.
Nounce of block 354642 is within 0.20% of the total range from 2^31.
There are 577 occurences of nounces within an interval of length 2x0.20=0.4% centered at 2^31 which count for 0.41% of the total of 140411.
There are 14194 nonces within an interval of length 10% from 2^31 whcih count for 10% of the total of 140411.
Nounces appear to be uniformly distributed, in particular around 2^31.
I hope this clarifies things.