Processing math: 100%

Sunday, June 15, 2025

Entry 173

There are many in the set d=4m with class number 8. When m is a prime or a semiprime (a product of two primes) like m=3p, then it may be well-behaved. For this set, there are only three, namely p=23,47,71, thus m=3p=69,141,213. Ramanujan found the radicals below and the G-function have a common form

G3p=U1/24pU1/163px1/2p

with fundamental unit Un and where x2p is a root of a unit quartic

G69=(5+232)1/12(33+232)1/8(2+334+6+334)1/2G141=(7+472)1/12(43+47)1/8(14+934+18+934)1/2G213=(59+7712)1/12(53+712)1/8(19+1232+21+1232)1/2

How Ramanujan found these is unknown as it is uncertain if he was aware of class field theory. Note also that without the factor 3, then d=23,47,71 are the smallest d with class number h(d)=3,5,7, respectively, while h(3d)=8.

P.S. Checking h(3d)=16, one finds the only primes are d=167,191,239,383,311 which are the smallest d with class number h(d)=11,13,15,17,19, respectively. Makes me wonder if their G3d would be analogous.

No comments:

Post a Comment