** Next:** Computing signatures of logarithms
** Up:** A better method for
** Previous:** Simple algebraic relations

In general, if we want to handle several levels of nested
exponentials, we will have to precompute several primes
*n*_{1}, *n*_{2}, ... so that
*n*_{i+1} = 4*k*_{i}*n*_{i}+1.
This will make it possible to represent *i* at all levels.
The values of *k*_{i} should be chosen as small as possible,
as we want to keep the ratio between the largest and smallest
primes small.
The following Maple code generates a sequence of 5 primes
with this property.
It should be noticed that the ratio between
*n*_{5}/*n*_{1} = 1536.
The minimum ratio, for all *k*_{i}=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;
lprint(n1,n2,n3,n4,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.

*Gaston Gonnet*

*1999-07-04*