Cosmological Theorem
There exists an integer N such that every string in the look and say sequence "decays" in at most N days to a compound of "common" and "transuranic elements."
The table below gives the periodic table of atoms associated with the look and say sequence as named by Conway (1987). The "abundance" is the average number of occurrences for long strings out of every million atoms. The asymptotic abundances are zero for transuranic elements, and 27.246... for arsenic (As), the next rarest element. The most common element is hydrogen (H), having an abundance of 91970.383.... The starting element is U, represented by the string "3," and subsequent terms are those giving a description of the current term: one three (13); one one, one three (1113); three ones, one three (3113), etc.
abundance n E_n E_n is the derivate of E_(n+1)
102.56285249 92 U 3
9883.5986392 91 Pa 13
7581.9047125 90 Th 1113
6926.9352045 89 Ac 3113
5313.7894999 88 Ra 132113
4076.3134078 87 Fr 1113122113
3127.0209328 86 Rn 311311222113
2398.7998311 85 At Ho.1322113
1840.1669683 84 Po 1113222113
1411.6286100 83 Bi 3113322113
1082.8883285 82 Pb Pm.123222113
830.70513293 81 Tl 111213322113
637.25039755 80 Hg 31121123222113
488.84742982 79 Au 132112211213322113
375.00456738 78 Pt 111312212221121123222113
287.67344775 77 Ir 3113112211322112211213322113
220.68001229 76 Os 1321132122211322212221121123222113
169.28801808 75 Re 111312211312113221133211322112211213322113
315.56655252 74 W Ge.Ca.312211322212221121123222113
242.07736666 73 Ta 13112221133211322112211213322113
2669.0970363 72 Hf 11132.Pa.H.Ca.W
2047.5173200 71 Lu 311312
1570.6911808 70 Yb 1321131112
1204.9083841 69 Tm 11131221133112
1098.5955997 68 Er 311311222.Ca.Co
47987.529438 67 Ho 1321132.Pm
36812.186418 66 Dy 111312211312
28239.358949 65 Tb 3113112221131112
21662.972821 64 Gd Ho.13221133112
20085.668709 63 Eu 1113222.Ca.Co
15408.115182 62 Sm 311332
29820.456167 61 Pm 132.Ca.Zn
22875.863883 60 Nd 111312
17548.529287 59 Pr 31131112
13461.825166 58 Ce 1321133112
10326.833312 57 La 11131.H.Ca.Co
7921.9188284 56 Ba 311311
6077.0611889 55 Cs 13211321
4661.8342720 54 Xe 11131221131211
3576.1856107 53 I 311311222113111221
2743.3629718 52 Te Ho.1322113312211
2104.4881933 51 Sb Eu.Ca.3112221
1614.3946687 50 Sn Pm.13211
1238.4341972 49 In 11131221
950.02745646 48 Cd 3113112211
728.78492056 47 Ag 132113212221
559.06537946 46 Pd 111312211312113211
428.87015041 45 Rh 311311222113111221131221
328.99480576 44 Ru Ho.132211331222113112211
386.07704943 43 Tc Eu.Ca.311322113212221
296.16736852 42 Mo 13211322211312113211
227.19586752 41 Nb 1113122113322113111221131221
174.28645997 40 Zr Er.12322211331222113112211
133.69860315 39 Y 1112133.H.Ca.Tc
102.56285249 38 Sr 3112112.U
78.678000089 37 Rb 1321122112
60.355455682 36 Kr 11131221222112
46.299868152 35 Br 3113112211322112
35.517547944 34 Se 13211321222113222112
27.246216076 33 As 11131221131211322113322112
1887.4372276 32 Ge 31131122211311122113222.Na
1447.8905642 31 Ga Ho.13221133122211332
23571.391336 30 Zn Eu.Ca.Ac.H.Ca.312
18082.082203 29 Cu 131112
13871.123200 28 Ni 11133112
45645.877256 27 Co Zn.32112
35015.858546 26 Fe 13122112
26861.360180 25 Mn 111311222112
20605.882611 24 Cr 31132.Si
15807.181592 23 V 13211312
12126.002783 22 Ti 11131221131112
9302.0974443 21 Sc 3113112221133112
56072.543129 20 Ca Ho.Pa.H.12.Co
43014.360913 19 K 1112
32997.170122 18 Ar 3112
25312.784218 17 Cl 132112
19417.939250 16 S 1113122112
14895.886658 15 P 311311222112
32032.812960 14 Si Ho.1322112
24573.006696 13 Al 1113222112
18850.441228 12 Mg 3113322112
14481.448773 11 Na Pm.123222112
11109.006696 10 Ne 111213322112
8521.9396539 9 F 31121123222112
6537.3490750 8 O 132112211213322112
5014.9302464 7 N 111312212221121123222112
3847.0525419 6 C 3113112211322112211213322112
2951.1503716 5 B 1321132122211322212221121123222112
2263.8860325 4 Be 111312211312113221133211322112211213322112
4220.0665982 3 Li Ge.Ca.312211322212221121123222112
3237.2968588 2 He 13112221133211322112211213322112
91790.383216 1 H Hf.Pa.22.Ca.Li
See also
Conway's Constant, Look and Say SequenceExplore with Wolfram|Alpha
WolframAlpha
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References
Conway, J. H. "The Weird and Wonderful Chemistry of Audioactive Decay." §5.11 in Open Problems in Communication and Computation (Ed. T. M. Cover and B. Gopinath). New York: Springer-Verlag, pp. 173-188, 1987.Conway, J. H. "The Weird and Wonderful Chemistry of Audioactive Decay." Eureka, 5-18, 1985.Ekhad, S. B. and Zeilberger, D. "Proof of Conway's Lost Cosmological Theorem." Electronic Research Announcement of the Amer. Math. Soc. 3, 78-82, 1997. http://www.math.temple.edu/~zeilberg/mamarim/mamarimhtml/horton.html.Hilgemeier, M. "Die Gleichniszahlen-Reihe." Bild der Wissensch. 12, 194-196, Dec. 1986.Hilgemeier, M. "'One Metaphor Fits All': A Fractal Voyage with Conway's Audioactive Decay." Ch. 7 in Pickover, C. A. (Ed.). Fractal Horizons: The Future Use of Fractals. New York: St. Martin's Press, 1996.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 905, 2002.Referenced on Wolfram|Alpha
Cosmological TheoremCite this as:
Weisstein, Eric W. "Cosmological Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CosmologicalTheorem.html