For discriminant d=4m with class number 8 and semiprime m:
If m=5p, then distinguish between p≡1mod8 and p≡5mod8.
If m=7p, then distinguish between p≡3mod8 and p≡7mod8.
The semiprime m=5p was in the previous entry. For m=7p, there are only four and Ramanujan found the radicals below.
For the 1st case, p=11,43, thus m=7p=77,301
G77=(8+3√7)1/8(√7+√112)1/8(√2+√114+√6+√114)1/2G301=(8+3√7)1/8(57√7+23√432)1/8(√42+7√434+√46+7√434)1/2
For the 2nd case, p=31,79, thus m=7p=217,553
G217=(√x1−12+√x1+12)1/2(√y1−12+√y1+12)1/2G553=(√x2−12+√x2+12)1/2(√y2−12+√y2+12)1/2 where x1=10+4√72,y1=14+5√74 x2=142+16√792,y2=98+11√794
But it seems not noticed that a fundamental unit is imbedded in these radicals as
x1+2y1=32(8+3√7)=32(3+√7√2)2=32U7
x2+2y2=32(80+9√79)=32(9+√79√2)2=32U79
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