Elias Daher wrote:
>(2^2-1) is prime! (3)
>
>(2^[2^2-1]-1) is prime! (7)
>
>(2^[2^(2^2-1)-1]-1) is prime! (127)
>
>(2^[2^(2^[2^2-1]-1)-1]-1) is prime! =
>(170141183460469231731687303715884105727)
Chris Caldwell's webpage on Mersenne numbers has some information on
this old (and most likely untrue) conjecture, due to Catalan (1876):
http://www.utm.edu/research/primes/mersenne.shtml
>So why wouldn't: (2^[2^(2^[2^(2^2-1)-1]-1)-1]-1) be prime?! =
>(2^170141183460469231731687303715884105727-1)
This number has no known factors (and I believe Tony Forbes and Co.
have done a fair amount of sieve-based factoring on it), but is so
large that we can currently only try simple sieveing for small factors.
Choose any reasonably large prime exponent p, and roughly half of the
corresponding Mersenne numbers 2^p - 1 will also fail to have a factor
small enough to be found by a reasonable amount of sieving. Most of
them (at least based on the ones small enough to be tested by
nonfactorial means, i.e. Lucas-Lehmer test) aren't prime, either.
The above conjecture reminds me of a similar one made many years
ago by an obscure fellow named Fermat:
Conjecture: F_m := 2^(2^m) + 1 is prime for m = 0,1,2,3 and 4,
so must in fact be prime for all integers m >= 0.
In this case the first *five* terms of the sequence are prime,
but F_5 is not, neither are F_6, F_7, F_8, F_9, F_10, F_11, F_12,
F_13, F_14, F_15, F_16, F_17, F_18, F_19, F_20, F_21, F_22, F_23,
F_24, F_25, F_26, F_27, F_28, F_29, F_30, F_31 and F_32 - F_33 is
currently the smallest such number whose character is unknown, and
(based on heuristics regarding the possible form of factors of such
numbers and the distribution of candidate exponents) it is quite
likely (though unproven) that there are no prime Fermat numbers
besides the first five. About as disastrous as a conjecture as
can be. Interestingly, Fermat could have quite easily shown F_5
to be composite using the "little" theorem that now bears his
name and a few hours of hand computation, but for unknown reasons
failed to do so. Fortunately for him, he made a few other conjectures
that proved more successful. :)
Cheers,
Ernst
(whose earliest number-theoretical conjecture, made at age 6,
namely that all numbers of the form 10^n + 7 (for n > 0) are prime,
isn't true, either.)
