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In general, if we want to handle several levels of nested
exponentials, we will have to precompute several primes
n1, n2, ... so that
ni+1 = 4kini+1.
This will make it possible to represent i at all levels.
The values of ki should be chosen as small as possible,
as we want to keep the ratio between the largest and smallest
The following Maple code generates a sequence of 5 primes
with this property.
It should be noticed that the ratio between
n5/n1 = 1536.
The minimum ratio, for all ki=1, or 256 is not attainable.
n1 := 10^9:
maxn5 := n1*10^10:
to 10000 do
n1 := nextprime(n1);
if modp(n1,4) = 1 then for n2 from 4*n1+1 by 4*n1 while 64*n2+21 < maxn5 do
if isprime(n2) then for n3 from 4*n2+1 by 4*n2 while 16*n3+5 < maxn5 do
if isprime(n3) then for n4 from 4*n3+1 by 4*n3 while 4*n4+1 < maxn5 do
if isprime(n4) then for n5 from 4*n4+1 by 4*n4 while n5 < maxn5 do
if isprime(n5) then
maxn5 := n5;
fi od fi od fi od fi od fi od:
The best set of primes found with the above is
1000077157 4000308629 48003703549 192014814197 1536118513577.