Infinite Range of Golden Ratios
Infinite PHI – ∞ φ


PHI series of numbers, 1st order: 1

Since this is a 1st order PHI, it has no further iterations. Once is all unity needs to authenticate itself.

It also has no relationship with any other value, so there is no need to approximate its solution by dividing some other number into unity!

Hence, all we get is... X = 1

Subtracting 1 from both sides, we get...
A linear polynomial in one unknown: X − 1 = 0




PHI series of numbers, 2nd order:


a b
1 ÷ 0 = NULL
1 ÷ 1 = 1
2 ÷ 1 = 2
3 ÷ 2 = 1.5
5 ÷ 3 = 1.6666666666667
8 ÷ 5 = 1.6
13 ÷ 8 = 1.625
21 ÷ 13 = 1.6153846153846
34 ÷ 21 = 1.6190476190476
55 ÷ 34 = 1.6176470588235
89 ÷ 55 = 1.6181818181818
144 ÷ 89 = 1.6179775280899
233 ÷ 144 = 1.6180555555556
377 ÷ 233 = 1.618025751073
610 ÷ 377 = 1.6180371352785
987 ÷ 610 = 1.6180327868852
1597 ÷ 987 = 1.6180344478217
2584 ÷ 1597 = 1.6180338134001
4181 ÷ 2584 = 1.6180340557276
6765 ÷ 4181 = 1.6180339631667
10946 ÷ 6765 = 1.6180339985218
17711 ÷ 10946 = 1.6180339850174
28657 ÷ 17711 = 1.6180339901756
46368 ÷ 28657 = 1.6180339882053
75025 ÷ 46368 = 1.6180339889579
121393 ÷ 75025 = 1.6180339886704
196418 ÷ 121393 = 1.6180339887802
317811 ÷ 196418 = 1.6180339887383
514229 ÷ 317811 = 1.6180339887543
832040 ÷ 514229 = 1.6180339887482
1346269 ÷ 832040 = 1.6180339887505
2178309 ÷ 1346269 = 1.6180339887496
3524578 ÷ 2178309 = 1.61803398875
5702887 ÷ 3524578 = 1.6180339887499
9227465 ÷ 5702887 = 1.6180339887499
14930352 ÷ 9227465 = 1.6180339887499


After 35 iterations, it's pretty obvious that the approximation of the 2nd order of PHI accurate to eleven decimal places is... 1.6180339887499

And it's reciprocal is... 0.6180339887499

To be able to make a 2nd order polynomial in one unknown out of these two numbers will require that one of them is given a negative sign value... − 0.6180339887499

Now we can form this polynomial by multiplying these two values together. But first, we have to turn them into linear expressions in one unknown...

X1 = 1.6180339887499

Subtract 1.6180339887499 from both sides of the equal sign...

X1 − 1.6180339887499 = 1.6180339887499 − 1.6180339887499

Yields...
(X − 1.6180339887499) = 0

X2 = − 0.6180339887499

Add 0.6180339887499 to both sides of the equal sign...

X2 + 0.6180339887499 = − 0.6180339887499 + 0.6180339887499

Yields...
(X + 0.6180339887499) = 0

Multiplying these two roots together...
(X − 1.6180339887499) × (X + 0.6180339887499) = 0

Yields...
X2 + 0.6180339887499X1 − 1.6180339887499X1 − 1 = 0

Simplifying further, yields a quadratic polynomial in one unknown...
X2 − X1 − 1 = 0




PHI series of numbers, 3rd order:











































a b c
1 0 0
1 1 1
3 2 1
6 5 3
14 11 6
31 25 14
70 56 31
157 126 70
353 283 157
793 636 353
1782 1429 793
4004 3211 1782
8997 7215 4004
20216 16212 8997
45425 36428 20216
102069 81853 45425
229347 183922 102069
515338 413269 229347
1157954 928607 515338
2601899 2086561 1157954
5846414 4688460 2601899
13136773 10534874 5846414
29518061 23671647 13136773
66326481 53189708 29518061
149034250 119516189 66326481
334876920 268550439 149034250
752461609 603427359 334876920
1690765888 1355888968 752461609
3799116465 3046654856 1690765888
8536537209 6845771321 3799116465
19181424995 15382308530 8536537209
43100270734 34563733525 19181424995
96845429254 77664004259 43100270734
217609704247 174509433513 96845429254
488964567014 392119137760 217609704247
1098693409021 881083704774 488964567014
2468741680809 1979777113795 1098693409021
5547212203625 4448518794604 2468741680809
12464472679038 9995730998229 5547212203625
28007415880892 22460203677267 12464472679038
62932092237197 50467619558159 28007415880892
141407127676248 113399711795356 62932092237197
317738931708801 254806839471604 141407127676248
713952898856653 572545771180405 317738931708801
1604237601745859 1286498670037058 713952898856653
3604689170639570 2890736271782917 1604237601745859
8099663044168346 6495425442422487 3604689170639570
18199777657230403 14595088486590833 8099663044168346
40894529187989582 32794866143821236 18199777657230403
91889172989041221 73689395331810818 40894529187989582
206473097508841621 165578568320852039 91889172989041221
463940838818734881 372051665829693660 206473097508841621
1042465602157270162 835992504648428541 463940838818734881
2342398945624433584 1878458106805698703 1042465602157270162
5263322654587402449 4220857052430132287 2342398945624433584




After 54 iterations, the approximation of the three roots of the 2nd order of PHI accurate to eleven decimal places are...

X1 = a/c = 5263322654587402449 ÷ 2342398945624433584 = 2.2469796037175
X2 = b/a = 4220857052430132287 ÷ 5263322654587402449 = 0.80193773580484
X3 = c/b = 2342398945624433584 ÷ 4220857052430132287 = 0.55495813208737

To be able to make a 3rd order polynomial in one unknown out of these three numbers will require that one of them is given a negative sign value... − 0.80193773580484

Now we can form this polynomial by multiplying these three values together. But first, we have to turn them into linear expressions in one unknown...

X1 = 2.2469796037175

Subtract 2.2469796037175 from both sides of the equal sign...

X1 − 2.2469796037175 = 2.2469796037175 − 2.2469796037175

Yields...
(X − 2.2469796037175) = 0

X2 = − 0.80193773580484

Add 0.80193773580484 to both sides of the equal sign...

X2 + 0.80193773580484 = − 0.80193773580484 + 0.80193773580484

Yields...
(X + 0.80193773580484) = 0

X3 = 0.55495813208737

Subtract 0.55495813208737 from both sides of the equal sign...

X3 − 0.55495813208737 = 0.55495813208737 − 0.55495813208737

Yields...
(X − 0.55495813208737) = 0

Multiplying these three roots together...
(X − 2.2469796037175) × (X + 0.80193773580484) × (X − 0.55495813208737) = 0

Yields...
X3 − 2.2469796037175X2 + 0.80193773580484X2 − 0.55495813208737X2 − 1.8019377358048X1 + 1.2469796037175X1 − 0.44504186791263X1 + 1 = 0

Simplifying further, yields a 3rd order polynomial in one unknown...
X3 -2X2 -1X1 + 1 = 0




PHI series of numbers, 4th order:





















a b c d
1 0 0 0
1 1 1 1
4 3 2 1
10 9 7 4
30 26 19 10
85 75 56 30
246 216 160 85
707 622 462 246
2037 1791 1329 707
5864 5157 3828 2037
16886 14849 11021 5864
48620 42756 31735 16886
139997 123111 91376 48620
403104 354484 263108 139997
1160693 1020696 757588 403104
3342081 2938977 2181389 1160693
9623140 8462447 6281058 3342081
27708726 24366645 18085587 9623140
79784098 70160958 52075371 27708726
229729153 202020427 149945056 79784098
661478734 581694636 431749580 229729153
1904652103 1674922950 1243173370 661478734
5484227157 4822748423 3579575053 1904652103
15791202736 13886550633 10306975580 5484227157
45468956106 39984728949 29677753369 15791202736
130922641160 115131438424 85453685055 45468956106
376976720745 331507764639 246054079584 130922641160
1085461206128 954538564968 708484485384 376976720745
3125460977225 2748484256480 2039999771096 1085461206128
8999406210929 7913945004801 5873945233705 3125460977225
25912757426660 22787296449435 16913351215730 8999406210929
74612811302754 65613405091825 48700053876095 25912757426660
214839027697334 188926270270674 140226216394579 74612811302754
618604325665341 543991514362587 403765297968008 214839027697334
1781200165693270 1566361137995936 1162595840027928 618604325665341
5128761469382475 4510157143717134 3347561303689206 1781200165693270
14767680082482085 12986479916788815 9638918613099609 5128761469382475
42521840081752984 37393078612370509 27754159999270900 14767680082482085
122436758775876478 107669078693394393 79914918694123493 42521840081752984
352542596245147348 310020756163394364 230105837469270871 122436758775876478
1015105948653689061 892669189877812583 662563352408541712 352542596245147348
2922881087185190704 2570338490940043356 1907775138531501644 1015105948653689061
8416100665310424765 7400994716656735704 5493219578125234060 2922881087185190704




After 42 iterations, the approximation of the four roots of the 4th order of PHI accurate to eleven decimal places are...

X1 = a/d = 8416100665310424765 ÷ 2922881087185190704 = 2.8793852415718
X2 = b/b = 7400994716656735704 ÷ 7400994716656735704 = 1
X3 = c/a = 5493219578125234060 ÷ 8416100665310424765 = 0.65270364466614
X4 = d/c = 2922881087185190704 ÷ 5493219578125234060 = 0.53208888623796

To be able to make a 4th order polynomial in one unknown out of these four numbers will require that two of them are given a negative sign value... − 1 and − 0.53208888623796

Now we can form this polynomial by multiplying these four values together. But first, we have to turn them into linear expressions in one unknown...

X1 = 2.8793852415718

Subtract 2.8793852415718 from both sides of the equal sign...

X1 − 2.8793852415718 = 2.8793852415718 − 2.8793852415718

Yields...
(X − 2.8793852415718) = 0

X2 = − 1

Add 1 to both sides of the equal sign...

X2 + 1 = − 1 + 1

Yields...
(X + 1) = 0

X3 = 0.65270364466614

Subtract 0.65270364466614 from both sides of the equal sign...

X3 − 0.65270364466614 = 0.65270364466614 − 0.65270364466614

Yields...
(X − 0.65270364466614) = 0

X4 = − 0.53208888623796

Add 0.53208888623796 to both sides of the equal sign...

X4 + 0.53208888623796 = − 0.53208888623796 + 0.53208888623796

Yields...
(X + 0.53208888623796) = 0

Multiplying these four roots together...
(X − 2.8793852415718) × (X + 1) × (X − 0.65270364466614) × (X + 0.53208888623796) = 0

Yields...
X4 − 2.8793852415718X3 + 1X3 − 0.65270364466614X3 + 0.53208888623796X3 Ê
Ä − 2.8793852415718X2 + 1.8793852415718X2 − 1.532088886238X2 Ê
Ä − 0.65270364466614X2 + 0.53208888623796X2 − 0.34729635533386X2 Ê
Ä + 1.8793852415718X1 − 1.532088886238X1 + 1X1 − 0.34729635533386X1 + 1 = 0


Simplifying further, yields a 4th order polynomial in one unknown...
X4 -2X3 -3X2 − 1X1 + 1 = 0




The Expanded Euclidean Algorithm arises from the Infinite Range of Golden Ratios...


sTaTs