--- In

primenumbers@yahoogroups.com, "David Broadhurst

<d.broadhurst@o...>" <d.broadhurst@o...> wrote:

> Let f(p)=phi(p^2-1)/(p^2-1).

> Say a prime p is "lowest yet" if there is

> no prime q<p with f(q)<f(p).

> The "lowest yet" sequence begins

<snip>

> 61577671, 117048931, ...

I believe it continues as follows

181333151

267190769

331413809

376754951

636510601

1737265531

3019962791

One can obtain fairly low values of f(p) realtively easily. Consider

for example

p=58531393985146662592474024598667898081212671 prime

p-1=2.5.7.13.19.43.53.67.71.73.43520821168673.98287085283258329

p+1=2^8.3^3.11.17.23.29.31.37.41.47.59.61.79.83.89.97.101.103.107.109.

113.127.131.661

So the first 33 primes are factors of p^2-1 and I believe this gives

an f(p) around 0.113

This is significantly smaller than the f(p) around 0.148 for the best

of the minimal examples listed.

To break the 0.10 barrier you need primes up to 257.

To break the 0.05 barrier you need primes up to 75029

By this stage, the numbers are getting rather large.

Richard Heylen