04/11/2009, 08:11 PM
Almost forgot to mentions this, but most of those integer points are NOT on the main branch (where x=1, a=1) of infinite pentation.
So far we only know 3 exact points on this branch:
\( 1 = 1 \uparrow\uparrow\uparrow \infty \)
\( 2 = 1.559610469462 \uparrow\uparrow\uparrow \infty \)
\( 3 = 1.635078474636 \uparrow\uparrow\uparrow \infty \)
and so far we only know a closed form for \( \frac{\ln 2}{W(\ln 2)} = 1.559610469462 \)
aside from these 3 points, the rest of main branch of infinite pentation is all approximations.
Andrew Robbins
So far we only know 3 exact points on this branch:
\( 1 = 1 \uparrow\uparrow\uparrow \infty \)
\( 2 = 1.559610469462 \uparrow\uparrow\uparrow \infty \)
\( 3 = 1.635078474636 \uparrow\uparrow\uparrow \infty \)
and so far we only know a closed form for \( \frac{\ln 2}{W(\ln 2)} = 1.559610469462 \)
aside from these 3 points, the rest of main branch of infinite pentation is all approximations.
Andrew Robbins