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Monday, June 16, 2025

Entry 175

For discriminant d=4m with class number 8 and semiprime m

If m=5p, then distinguish between p1mod8 and p5mod8

If m=7p, then distinguish between p3mod8 and p7mod8

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+37)1/8(7+112)1/8(2+114+6+114)1/2G301=(8+37)1/8(577+23432)1/8(42+7434+46+7434)1/2

For the 2nd case, p=31,79, thus m=7p=217,553 

G217=(x112+x1+12)1/2(y112+y1+12)1/2G553=(x212+x2+12)1/2(y212+y2+12)1/2 where x1=10+472,y1=14+574 x2=142+16792,y2=98+11794

But it seems not noticed that a fundamental unit is imbedded in these radicals as

x1+2y1=32(8+37)=32(3+72)2=32U7

x2+2y2=32(80+979)=32(9+792)2=32U79

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