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Old 2021-08-10, 12:58   #16
ThomasK
 
Aug 2021

10002 Posts
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To give you a feeling how the distribution of the numbers is developing, I give you the function values for all powers of ten from 1 to 12.


W0(10): 3
W1(10): 6
W2(10): 0
W3(10): 0
W4(10): 0


W0(100): 17
W1(100): 71
W2(100): 11
W3(100): 0
W4(100): 0


W0(103): 108
W1(103): 686
W2(103): 201
W3(103): 4
W4(103): 0


W0(104): 755
W1(104): 6598
W2(104): 2592
W3(104): 54
W4(104): 0


W0(105): 5936
W1(105): 63449
W2(105): 29916
W3(105): 698
W4(105): 0


W0(106): 48474
W1(106): 614400
W2(106): 328988
W3(106): 8137
W4(106): 0


W0(107): 406270
W1(107): 5952657
W2(107): 3550745
W3(107): 90324
W4(107): 3


W0(108): 3532031
W1(108): 58088295
W2(108): 37432690
W3(108): 946964
W4(108): 19


W0(109): 31295358
W1(109): 568932663
W2(109): 390065916
W3(109): 9705879
W4(109): 183


W0(1010): 279591668
W1(1010): 5588087493
W2(1010): 4034529147
W3(1010): 97790090
W4(1010): 1601


W0(1011): 2521429242
W1(1011): 54968844332
W2(1011): 41532029309
W3(1011): 977682518
W4(1011): 14598


W0(1012): 22996137423
W1(1012): 541664112990
W2(1012): 425608837164
W3(1012): 9730782305
W4(1012): 130117


Note: Always the sum is not 10n but 10n - 1, because W (1) is not defined.


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When we calculated a total analysis up to 1010 for the first time in the early years, we thought that, contrary to our theoretical estimates, which say that the asymptotic density for the Zweiwertzahlen is 1 and the asymptotic density for the Dreiwertzahlen is 0, there was a calculation error, because after the total analysis by the computer up to 1010 the proportion of Dreiwertzahlen increases.

That is a contradiction. To solve the problem we decided to do the total analysis up to 1012.

The relief and the cheering was very great when we read the intermediate results on the screen in the middle of the night during the calculation and the proportion of Dreiwertzahlen - as had long been expected - finally fell. If this hadn't been the case, then we could have thrown away all of our theoretical estimates.

The reason this takes so long is that the function ln ln n increases extremely slowly and it takes a very long time before the function ln ln n can assert itself against multiplicative constants.

We expect that around 1024 the Zweiwertzahlen will overtake the Einswertzahlen, i.e. W2(n) > W1(n) for all n > around 1024.

However, the Einswertzahlen offer very strong resistance against the Zweiwertzahlen. According to our extrapolations, it takes an extremely long time before the Zweiwertzahlen goes to the asyptotic density 1 and the Einswertzahlen goes to the asyptotic density 0 approach.


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In the next posts I will tell you something about our algorithm and prove a very important theorem from the Minimum theory.
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