Eighth Order PHI – Fermat's Third Prime: 17

Fermat devised a series of integers based on this formula: 22n + 1, wherein n = {0, 1, 2, 3, 4}. We'll ignore his premise of these being an infinite series of primes, because I'm only interested in their associated series of numbers: 2(2n − 1), wherein n = {0, 1, 2, 3, 4, ...?}. And for the purposes of this webpage, I'll only focus on the third 'n' of n = 2, thereby: 2(22 − 1) = 2(4 − 1) = 23 = 8. The eighth order Golden Series of Numbers and Golden Integers and its associated Golden Polynomial are all embedded within Fermat's third prime for this series, namely: the 17-sided regular polygon, or heptadecagon...


PHI series of numbers, 8th order:


a b c d e f g h
1 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1
8 7 6 5 4 3 2 1
36 35 33 30 26 21 15 8
204 196 181 160 134 104 71 36
1086 1050 979 875 741 581 400 204
5916 5712 5312 4731 3990 3115 2136 1086
31998 30912 28776 25661 21671 16940 11628 5916
173502 167586 155958 139018 117347 91686 62910 31998
940005 908007 845097 753411 636064 497046 341088 173502
5094220 4920718 4579630 4082584 3446520 2693109 1848012 940005
27604798 26664793 24816781 22123672 18677152 14594568 10014938 5094220
149590922 144496702 134481764 119887196 101210044 79086372 54269591 27604798
810627389 783022591 728753000 649666628 548456584 428569388 294087624 149590922
4392774126 4243183204 3949095580 3520526192 2972069608 2322402980 1593649980 810627389
23804329059 22993701670 21400051690 19077648710 16105579102 12585052910 8635957330 4392774126
128995094597 124602320471 115966363141 103381310231 87275731129 68198082419 46798030729 23804329059
699021261776 675216932717 628418901988 560220819569 472945088440 369563778209 253597415068 128995094597
3787979292364 3658984197767 3405386782699 3035823004490 2562877916050 2002657096481 1374238194493 699021261776
20526967746120 19827946484344 18453708289851 16451051193370 13888173277320 10852350272830 7446963490131 3787979292364
111235140046330 107447160753966 100000197263835 89147846991005 75259673713685 58808622520315 40354914230464 20526967746120
602780523265720 582253555519600 541898641289136 483090018768821 407830345055136 318682498064131 218682300800296 111235140046330
3266453022809170 3155217882762840 2936535581962544 2617853083898413 2210022738843277 1726932720074456 1185034078785320 602780523265720
17700829632401740 17098049109136020 15913015030350700 14186082310276244 11976059571432967 9358206487534554 6421670905572010 3266453022809170
95920366069513405 92653913046704235 86232242141132225 76874035653597671 64897976082164704 50711893771888460 34798878741537760 17700829632401740
519790135138940200 502089305506538460 467290426765000700 416578532993112240 351680556910947536 274806521257349865 188574279116217640 95920366069513405
2816730123757620046 2720809757688106641 2532235478571889001 2257428957314539136 1905748400403591600 1489169867410479360 1021879440645478660 519790135138940200

After 26 iterations, the approximation of the eight roots of the 8th order of PHI accurate to eleven decimal places are...

X1 = a/h = 2816730123757620046 ÷ 519790135138940200 = 5.4189757237403
X2 = b/f = 2720809757688106641 ÷ 1489169867410479360 = 1.8270647407198
X3 = c/d = 2532235478571889001 ÷ 2257428957314539136 = 1.1217342943914
X4 = d/b = 2257428957314539136 ÷ 2720809757688106641 = 0.82969011373757
X5 = e/a = 1905748400403591600 ÷ 2816730123757620046 = 0.67658182242225
X6 = f/c = 1489169867410479360 ÷ 2532235478571889001 = 0.58808506555256
X7 = g/e = 1021879440645478660 ÷ 1905748400403591600 = 0.53620899822299
X8 = h/g = 519790135138940200 ÷ 1021879440645478660 = 0.50866091875829


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sTaTs