There is a conjecture that the nth Mersenne exponent resulting in a prime
is approximately (3/2)^n. Consider Mersenne primes through M37. (I don't
know exactly what M38 is yet, and there may be other small ones. Also, the
double checks of the range through M37 haven't been completed.)
You can see the basis of this conjecture if you use semi-log graph paper
and graph the index of the Mersenne exponents along the X axis and the
exponents on the Y axis. If you don't have semi-log graph paper, graph the
log of the exponent. (Or you can use software!) The data is pretty close
to a straight line.
If you do a linear regression of the log of the exponent vs. the index, you
get a correlation coefficient of 0.996 - which indicates a very strong
linear relationship. The linear regression parameters yield the relation
M(n) = 1.4796^n + c, where c is a small constant. 1.4796 is pretty close
to 3/2.
This is one reason why we need to have a complete list of Mersenne primes
up through some value.
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| Jud "program first and think later" McCranie |
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