Problems & Puzzles: Puzzles

Puzzle 532. Puzzle 529 re-loaded: sizing the prime nebulas.

Related Puzzle 529 I have a few additional questions.

For these questions I consider the 3 basic graphs (O, L & Γ) as a kind of nebulas in expansion. I think that the graphs provided by Mr X & by WW suggest & justify the nebula term very naturally.

For these three nebula each prime will be considered as particles of unitary mass.

So, the mass center of the nebula up to the prime-k will be defined as (Xi, Yi) where:

{Xi =(Σ(xi))/k & Yi =(Σ(yi))/k}, i=1, 2, ...k; x1=y1=0, i≠3

Q1. For each nebula type, the coordinates (Xi,Yi) of the mass center converges to some fixed point as i goes to infinite?

Q2. Can you plot the route of the mass center for a few millions of primes, just to have a bit of empirical/visual data related to Q1?

The following question is related to the size of the nebulas.

At every prime-k considered, a rectangular box XY oriented will get inside all the primes from 2 to pk. Up to this point (prime) at the most we will have at the most four primes defining four maximal & minimal coordinates reached: Xmin, Xmax, Ymin, Ymax.

Accordingly, the size of this rectangular box will be defined as:

LxW = (Ymax-Ymin)x(Xmax-Xmin)

The size of the nebula will be defined as the diagonal of the rectangular box (as is used to define the size of the TV screens):

S=(L^2+W^2)^(1/2)

Q3. Do you devise a law relating the size of the nebula up to the prime-k against the prime-k value: S(pk)=f(pk)?

Q4. In order to share all of us some hard-data, can you send your values for these two tables below, only for the "O" type-nebula?:

 Mass center of the "O" nebula k pk Xk Yk 10^6 10^8 10^10 10^12 More?

 Size of the "O" nebula k pk Xmax Xmin Ymax Ymin S 10^6 10^8 10^10 10^12 More?

Andreas Höglund wrote (Feb 2011)

I collected data on the O, L and G nebula up to k=5*10^11, pk=14*10^12. I added an extra variable “Max Distance” which is maximum distance from the origin the graph has been, that is maximum of sqrt(x^2+y^2). See tables below.

The data for the O and L nebula is very similar. The mass centers follow the same direction and almost same speed, and the size S and Max Distance is also remarkably close, except the last datapoint k=5*10^11 where the size of L gets abit larger than the size of O.
The G nebular is different. The mass center stays much closer to the origin and doesn’t not move in any particular direction and the Max Distance is 50-70% lower than for O and L, but interestingly the size S is still almost the same as for O and L nebula.

Q1:
For the O and L graph the mass center (Xi,Yi) roughly follows the line y = –x, so it seems Xi->∞ and Yi-> -∞ as pk->∞ .
For the G graph the mass center seems to be “randomly” moving around the origin though further and further away, but not very far from the origin compared to O and L graph.

Q3:
I tried using Excel regression analysis to find a function for S(pk) but none fit very well except a second order polynomial y=ax^2+bx+c with a negative constant a which would imply a maximum size S(pk) at around pk=11*10^12 which can’t be correct.
Later I found that S(pk) “very” rougly follows:  S(pk) ~ 2*sqrt(pk)/ln (pk)   where ln is the natural logarithm.

 Mass center of the "O" nebula Size of the "O" nebula k Pk Xk Yk Xmin Xmax Ymin Ymax S Max Distance 10^3 7,919 6 -4 1 13 -10 1 16 15 10^4 104,729 13 -12 -1 31 -37 4 52 37 10^5 1,299,709 29 -33 -7 70 -96 19 138 105 10^6 15,485,863 114 -87 -40 368 -275 19 503 386 10^7 179,424,673 238 -290 -99 877 -695 202 1326 979 10^8 2,038,074,743 811 -823 -563 2262 -2226 517 3938 2971 10^9 22,801,763,489 2735 -2183 -566 8203 -6493 1158 11638 8767 10^10 252,097,800,623 5788 -7209 -7231 17252 -24244 7986 40475 25919 10^11 2,760,727,302,517 23656 -19993 -7644 70839 -65732 15902 113242 81582 5*10^11 14,638,944,639,703 48105 -43605 -37817 133249 -109022 21169 214973 156156

 Mass center of the "L" nebula Size of the "L" nebula k Pk Xk Yk Xmin Xmax Ymin Ymax S Max Distance 10^3 7,919 5 -4 -2 14 -10 1 19 15 10^4 104,729 11 -13 -6 31 -35 2 52 36 10^5 1,299,709 34 -28 -18 87 -77 12 138 99 10^6 15,485,863 98 -103 -62 281 -257 19 440 347 10^7 179,424,673 238 -289 -62 703 -764 171 1208 934 10^8 2,038,074,743 921 -712 -379 2719 -1837 466 3860 2986 10^9 22,801,763,489 2168 -2750 -2002 6163 -7858 713 11838 8844 10^10 252,097,800,623 7012 -5984 -2612 20770 -17481 2073 30481 25086 10^11 2,760,727,302,517 21871 -21778 -8524 69489 -55595 19309 108151 81855 5*10^11 14,638,944,639,703 51744 -39967 -19921 151094 -130223 56476 253185 161310

 Mass center of the "G" nebula Size of the "G" nebula k Pk Xk Yk Xmin Xmax Ymin Ymax S Max Distance 10^3 7,919 0 1 -6 5 -3 9 16 9 10^4 104,729 1 0 -22 23 -18 14 55 25 10^5 1,299,709 -5 0 -59 50 -66 49 158 67 10^6 15,485,863 16 11 -99 149 -134 193 410 200 10^7 179,424,673 0 -51 -319 346 -478 424 1121 486 10^8 2,038,074,743 -110 98 -1232 1235 -1200 1477 3640 1672 10^9 22,801,763,489 566 -15 -3710 4063 -4177 3673 11047 4823 10^10 252,097,800,623 -1224 -196 -15501 11738 -10509 10368 34319 15892 10^11 2,760,727,302,517 1785 1878 -40746 49777 -24366 41635 112029 55863 5*10^11 14,638,944,639,703 -3638 8138 -80724 104281 -50181 90577 232464 114746

 Records   |  Conjectures  |  Problems  |  Puzzles