Mersenne Primes and Benford's law - Page 2

T.Rex · 2021-09-15, 04:23

Quote:

Originally Posted by slandrum

That's due to logarithmic bias. ... It get scarcer as the exponents get larger.

Hummmm Right.
My 2nd comment was stupid based on the first one.
First one seems to say that there is no rule enabling to find the next Mersenne prime based on the knowledge of previous known Mersenne primes exponents.

Dobri · 2021-09-15, 12:14

The Benford's law could indirectly be linked to a tendency toward maximum entropy.

The Gaussian distribution corresponds to maximum entropy on the entire axis from negative to positive infinity.

The uniform distribution has maximum entropy on a finite interval.

Then the exponential distribution has maximum entropy on the positive half line from zero to infinity.

The convolution of several exponential distributions results in a hypoexponential distribution on the positive half line.

The maximum entropy curve of the exponential distribution could potentially be what the decaying Benford's law curve tends to in terms of maximizing the entropy for some number theory cases.

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2021-09-15, 12:14	#13
Dobri "刀-比-日" May 2018 320₁₀ Posts	The Benford's law could indirectly be linked to a tendency toward maximum entropy. The Gaussian distribution corresponds to maximum entropy on the entire axis from negative to positive infinity. The uniform distribution has maximum entropy on a finite interval. Then the exponential distribution has maximum entropy on the positive half line from zero to infinity. The convolution of several exponential distributions results in a hypoexponential distribution on the positive half line. The maximum entropy curve of the exponential distribution could potentially be what the decaying Benford's law curve tends to in terms of maximizing the entropy for some number theory cases.