Summarizing, we find tentative patterns for I. Levels4n+1=(5,13)II. Levels4n+2=(6,10)III. Levels4n+3=(3,7)
The definitions for these functions are in previous entries, but it may be good to have a few selected values together to get an overall picture. Note that Un are fundamental units, important to Pell equations.
I. Levels 4n+3=(3,7)
j3A(1+√−5/32)=−(√3)6j3A(1+√−17/32)=−(2√3)6j3A(1+√−41/32)=−(4√3)6j3B(1+√−5/32)=−33U25=−33(1+√52)2j3B(1+√−17/32)=−33U217=−33(4+√17)2j3B(1+√−41/32)=−33U241=−33(32+5√41)2
and one can observe its similarity to Level 7 j7A(1+√−5/72)=−(√7)2j7A(1+√−13/72)=−(3√7)2j7A(1+√−61/72)=−(39√7)2j7B(1+√−5/72)=−7U25=−7(1+√52)2j7B(1+√−13/72)=−7U213=−7(3+√132)2j7B(1+√−61/72)=−7U261=−7(39+5√612)2
II. Levels 4n+1=(5,13)
j5A(1+√−23/52)=−(6√23)2j5A(1+√−47/52)=−(18√47)2j5B(1+√−23/52)=−(√5)3(1+√52)9j5B(1+√−47/52)=−(√5)3(1+√52)15
where we see the golden ratio above and the bronze ratio below j13A(1+√−7/132)=−(√7)2j13A(1+√−31/132)=−(2√31)2j13B(1+√−7/132)=−√13(3+√132)j13B(1+√−31/132)=−√13(3+√132)3
III. Levels 4n+2=(6,10)
j6A(1+√−11/32)=−202j6A(1+√−59/32)=−10602j6B(1+√−11/32)=−U211=−(10+3√11)2j6B(1+√−59/32)=−U259=−(530+69√59)2
compared to j10A(12√−6/5)=62j10A(12√−38/5)=762j10D(12√−6/5)=U210=(3+√10)2j10D(12√−38/5)=U185=(1+√52)18
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