Sunday, July 28, 2024

[xfelzoax] roots of 60

we subdivide the Babylonian range 1 to 60 by a geometric progression, i.e., we give values of 60^(a/b).

b=6 is pleasingly near integers: 1, 2, 4, 8 (or 7.7), 15, 30, 60.  (because 60 is close to 64.)

not sure what this might be useful for.  what applications have 1 and 60 as important endpoints?  maybe time expansion or reduction ratios.

previously, roots of 10 and 2.

2: 1 7.74596669241483 60

3: 1 3.91486764116886 15.3261886478711 60

4: 1 2.78315768371374 7.74596669241483 21.5582467177851 60

5: 1 2.26793315526605 5.14352079675504 11.6651613497612 26.4558061866516 60

6: 1 1.97860244646793 3.91486764116886 7.74596669241483 15.3261886478711 30.3244343537066 60

8: 1 1.66827985773183 2.78315768371374 4.64308590463121 7.74596669241483 12.9224402116173 21.5582467177851 35.9651887672942 60

10: 1 1.50596585461492 2.26793315526605 3.41542989237976 5.14352079675504 7.74596669241483 11.6651613497612 17.5673346813141 26.4558061866516 39.8415407734076 60

12: 1 1.40662804126319 1.97860244646793 2.78315768371374 3.91486764116886 5.50676260190201 7.74596669241483 10.8956939562414 15.3261886478711 21.5582467177851 30.3244343537066 42.6551996973686 60

15: 1 1.3138428340886 1.72618299268596 2.26793315526605 2.97970772423825 3.91486764116886 5.14352079675504 6.75777794080228 8.87865812188508 11.6651613497612 15.3261886478711 20.1362031288954 26.4558061866516 34.758771378369 45.667562697194 60

16: 1 1.29161908383696 1.66827985773183 2.15478210142724 2.78315768371374 3.59477957761214 4.64308590463121 5.99709836231608 7.74596669241483 10.0048384026885 12.9224402116173 16.6908703870671 21.5582467177851 27.8450428747567 35.9651887672942 46.4533241656359 60

18: 1 1.25541172535118 1.57605860014924 1.97860244646793 2.48396071110437 3.1183934020321 3.91486764116886 4.91477073992132 6.17006081431015 7.74596669241483 9.72437740983731 12.2080974220499 15.3261886478711 19.2406769334815 24.1549714259868 30.3244343537066 38.0696504522856 47.7930855578203 60

20: 1 1.22717800445368 1.50596585461492 1.84808817224173 2.26793315526605 2.78315768371374 3.41542989237976 4.19134043968205 5.14352079675504 6.31201558722787 7.74596669241483 9.50567994816233 11.6651613497612 14.3152294268302 17.5673346813141 21.5582467177851 26.4558061866516 32.4659834423485 39.8415407734076 48.8926625006703 60

24: 1 1.18601350804415 1.40662804126319 1.66827985773183 1.97860244646793 2.34664922856016 2.78315768371374 3.30086260790137 3.91486764116886 4.64308590463121 5.50676260190201 6.53109483144814 7.74596669241483 9.18682113006406 10.8956939562414 12.9224402116173 15.3261886478711 18.177066763208 21.5582467177851 25.5683718170415 30.3244343537066 35.9651887672942 42.6551996973686 50.5896430294 60

25: 1 1.17794781252771 1.38756104903882 1.63447450246393 1.92532566480971 2.26793315526605 2.67150689920472 3.14689570807088 3.70687891557494 4.36650990990659 5.14352079675504 6.05879907122839 7.1369491124984 8.40695359518908 9.90295259747496 11.6651613497612 13.740951294734 16.1861235196818 19.0664087933125 22.4592345308415 26.4558061866516 31.1635590262233 36.709046185518 43.2413406542096 50.9360426343918 60

30: 1 1.14622983475767 1.3138428340886 1.50596585461492 1.72618299268596 1.97860244646793 2.26793315526605 2.59957264580205 2.97970772423825 3.41542989237976 3.91486764116886 4.48733808943514 5.14352079675504 5.89565699293717 6.75777794080228 7.74596669241483 8.87865812188508 10.1769828319182 11.6651613497612 13.3709559663584 15.3261886478711 17.5673346813141 20.1362031288954 23.0807167850807 26.4558061866516 30.3244343537066 34.758771378369 39.8415407734076 45.667562697194 52.3455228441902 60

solving 60^x = N, where N is a divisor of 60:

60 ^ 0 = 1
60 ^ 0.169293807598781 = 2
60 ^ 0.268324336648371 = 3
60 ^ 0.338587615197563 = 4
60 ^ 0.393088048154066 = 5
60 ^ 0.437618144247152 = 6
60 ^ 0.562381855752848 = 10
60 ^ 0.606911951845934 = 12
60 ^ 0.661412384802437 = 15
60 ^ 0.731675663351629 = 20
60 ^ 0.830706192401219 = 30
60 ^ 1 = 60

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