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...
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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