[D66] Moeilijk, deel 3! (On the coincidence between the Collatz conjecture and the 37 leap seconds between TAI-UTC)

[René Oudeweg]

...uit mijn maelstroom.pdf:



On the coincidence between the Collatz conjecture and the 37 leap 
seconds between TAI-UTC

August 21, 2023


Since the inception of UTC time in 1972 (bound to the rotation of the 
Earth axis) there have been 37 leap seconds introduced with respect to 
TAI (International Atomic Time)

1961 JAN 1 =JD 2437300.5 TAI-UTC= 1.4228180 S + (MJD - 37300.) X 0.001296 S
1961 AUG 1 =JD 2437512.5 TAI-UTC= 1.3728180 S + (MJD - 37300.) X 0.001296 S
1962 JAN 1 =JD 2437665.5 TAI-UTC= 1.8458580 S + (MJD - 37665.) X 0.0011232S
1963 NOV 1 =JD 2438334.5 TAI-UTC= 1.9458580 S + (MJD - 37665.) X 0.0011232S
1964 JAN 1 =JD 2438395.5 TAI-UTC= 3.2401300 S + (MJD - 38761.) X 0.001296 S
1964 APR 1 =JD 2438486.5 TAI-UTC= 3.3401300 S + (MJD - 38761.) X 0.001296 S
1964 SEP 1 =JD 2438639.5 TAI-UTC= 3.4401300 S + (MJD - 38761.) X 0.001296 S
1965 JAN 1 =JD 2438761.5 TAI-UTC= 3.5401300 S + (MJD - 38761.) X 0.001296 S
1965 MAR 1 =JD 2438820.5 TAI-UTC= 3.6401300 S + (MJD - 38761.) X 0.001296 S
1965 JUL 1 =JD 2438942.5 TAI-UTC= 3.7401300 S + (MJD - 38761.) X 0.001296 S
1965 SEP 1 =JD 2439004.5 TAI-UTC= 3.8401300 S + (MJD - 38761.) X 0.001296 S
1966 JAN 1 =JD 2439126.5 TAI-UTC= 4.3131700 S + (MJD - 39126.) X 0.002592 S
1968 FEB 1 =JD 2439887.5 TAI-UTC= 4.2131700 S + (MJD - 39126.) X 0.002592 S
1972 JAN 1 =JD 2441317.5 TAI-UTC= 10.0 S + (MJD - 41317.) X 0.0 S
1972 JUL 1 =JD 2441499.5 TAI-UTC= 11.0 S + (MJD - 41317.) X 0.0 S
1973 JAN 1 =JD 2441683.5 TAI-UTC= 12.0 S + (MJD - 41317.) X 0.0 S
1974 JAN 1 =JD 2442048.5 TAI-UTC= 13.0 S + (MJD - 41317.) X 0.0 S
1975 JAN 1 =JD 2442413.5 TAI-UTC= 14.0 S + (MJD - 41317.) X 0.0 S
1976 JAN 1 =JD 2442778.5 TAI-UTC= 15.0 S + (MJD - 41317.) X 0.0 S
1977 JAN 1 =JD 2443144.5 TAI-UTC= 16.0 S + (MJD - 41317.) X 0.0 S
1978 JAN 1 =JD 2443509.5 TAI-UTC= 17.0 S + (MJD - 41317.) X 0.0 S
1979 JAN 1 =JD 2443874.5 TAI-UTC= 18.0 S + (MJD - 41317.) X 0.0 S
1980 JAN 1 =JD 2444239.5 TAI-UTC= 19.0 S + (MJD - 41317.) X 0.0 S
1981 JUL 1 =JD 2444786.5 TAI-UTC= 20.0 S + (MJD - 41317.) X 0.0 S
1982 JUL 1 =JD 2445151.5 TAI-UTC= 21.0 S + (MJD - 41317.) X 0.0 S
1983 JUL 1 =JD 2445516.5 TAI-UTC= 22.0 S + (MJD - 41317.) X 0.0 S
1985 JUL 1 =JD 2446247.5 TAI-UTC= 23.0 S + (MJD - 41317.) X 0.0 S
1988 JAN 1 =JD 2447161.5 TAI-UTC= 24.0 S + (MJD - 41317.) X 0.0 S
1990 JAN 1 =JD 2447892.5 TAI-UTC= 25.0 S + (MJD - 41317.) X 0.0 S
1991 JAN 1 =JD 2448257.5 TAI-UTC= 26.0 S + (MJD - 41317.) X 0.0 S
1992 JUL 1 =JD 2448804.5 TAI-UTC= 27.0 S + (MJD - 41317.) X 0.0 S
1993 JUL 1 =JD 2449169.5 TAI-UTC= 28.0 S + (MJD - 41317.) X 0.0 S
1994 JUL 1 =JD 2449534.5 TAI-UTC= 29.0 S + (MJD - 41317.) X 0.0 S
1996 JAN 1 =JD 2450083.5 TAI-UTC= 30.0 S + (MJD - 41317.) X 0.0 S
1997 JUL 1 =JD 2450630.5 TAI-UTC= 31.0 S + (MJD - 41317.) X 0.0 S
1999 JAN 1 =JD 2451179.5 TAI-UTC= 32.0 S + (MJD - 41317.) X 0.0 S
2006 JAN 1 =JD 2453736.5 TAI-UTC= 33.0 S + (MJD - 41317.) X 0.0 S
2009 JAN 1 =JD 2454832.5 TAI-UTC= 34.0 S + (MJD - 41317.) X 0.0 S
2012 JUL 1 =JD 2456109.5 TAI-UTC= 35.0 S + (MJD - 41317.) X 0.0 S
2015 JUL 1 =JD 2457204.5 TAI-UTC= 36.0 S + (MJD - 41317.) X 0.0 S
2017 JAN 1 =JD 2457754.5 TAI-UTC= 37.0 S + (MJD - 41317.) X 0.0 S
63

The current difference between UTC and TAI is thus 37 seconds.

The Collatz conjecture (3x + 1) states that for every positive integer 
the Collatz sequence will eventually end in the loop (4,2,1).


The Collatz Conjecture, also known as the 3n + 1 conjecture, is a famous 
unsolved problem in mathematics. The conjecture is named after the 
German mathematician Lothar Collatz, who first proposed it in 1937. The 
conjecture is simple to state, but its resolution remains an open question.

Here's the statement of the Collatz Conjecture:

     Start with any positive integer n.
     If n is even, divide it by 2 (n/2).
     If nn is odd, multiply it by 3 and add 1 (3n+1).
     Repeat the process indefinitely for the resulting values.

The conjecture suggests that, regardless of the initial value of nn, the 
sequence generated by this process will always eventually reach the 
cycle 4,2,14,2,1. Once the sequence reaches 1, it will continue in the 
cycle indefinitely: 1,4,2,1,4,2,…1,4,2,1,4,2,….




I calculated the number of steps for the numbers between 2 and 100:

x: 2 collatz steps: 1
x: 3 collatz steps: 7
x: 4 collatz steps: 2
x: 5 collatz steps: 5
x: 6 collatz steps: 8
x: 7 collatz steps: 16
x: 8 collatz steps: 3
x: 9 collatz steps: 19
x: 10 collatz steps: 6
x: 11 collatz steps: 14
x: 12 collatz steps: 9
x: 13 collatz steps: 9
x: 14 collatz steps: 17
x: 15 collatz steps: 17
x: 16 collatz steps: 4
x: 17 collatz steps: 12
x: 18 collatz steps: 20
x: 19 collatz steps: 20
x: 20 collatz steps: 7
x: 21 collatz steps: 7
x: 22 collatz steps: 15
x: 23 collatz steps: 15
x: 24 collatz steps: 10
x: 25 collatz steps: 23
x: 26 collatz steps: 10
x: 27 collatz steps: 111
x: 28 collatz steps: 18
x: 29 collatz steps: 18
x: 30 collatz steps: 18
x: 31 collatz steps: 106
x: 32 collatz steps: 5
x: 33 collatz steps: 26
x: 34 collatz steps: 13
x: 35 collatz steps: 13
x: 36 collatz steps: 21
x: 37 collatz steps: 21
x: 38 collatz steps: 21
x: 39 collatz steps: 34
x: 40 collatz steps: 8
x: 41 collatz steps: 109
x: 42 collatz steps: 8
x: 43 collatz steps: 29
x: 44 collatz steps: 16
x: 45 collatz steps: 16
x: 46 collatz steps: 16
x: 47 collatz steps: 104
x: 48 collatz steps: 11
x: 49 collatz steps: 24
x: 50 collatz steps: 24
x: 51 collatz steps: 24
x: 52 collatz steps: 11
x: 53 collatz steps: 11
x: 54 collatz steps: 112
x: 55 collatz steps: 112
x: 56 collatz steps: 19
x: 57 collatz steps: 32
x: 58 collatz steps: 19
x: 59 collatz steps: 32
x: 60 collatz steps: 19
x: 61 collatz steps: 19
x: 62 collatz steps: 107
x: 63 collatz steps: 107
x: 64 collatz steps: 6
x: 65 collatz steps: 27
x: 66 collatz steps: 27
x: 67 collatz steps: 27
x: 68 collatz steps: 14
x: 69 collatz steps: 14
x: 70 collatz steps: 14
x: 71 collatz steps: 102
x: 72 collatz steps: 22
x: 73 collatz steps: 115
x: 74 collatz steps: 22
x: 75 collatz steps: 14
x: 76 collatz steps: 22
x: 77 collatz steps: 22
x: 78 collatz steps: 35
x: 79 collatz steps: 35
x: 80 collatz steps: 9
x: 81 collatz steps: 22
x: 82 collatz steps: 110
x: 83 collatz steps: 110
x: 84 collatz steps: 9
x: 85 collatz steps: 9
x: 86 collatz steps: 30
x: 87 collatz steps: 30
x: 88 collatz steps: 17
x: 89 collatz steps: 30
x: 90 collatz steps: 17
x: 91 collatz steps: 92
x: 92 collatz steps: 17
x: 93 collatz steps: 17
x: 94 collatz steps: 105
x: 95 collatz steps: 105
x: 96 collatz steps: 12
x: 97 collatz steps: 118
x: 98 collatz steps: 25
x: 99 collatz steps: 25

We see a first spike at x=27 (111 steps)
What is more curious are the results at x=65,66,67:

27

Now 27 is exactly the number of leap seconds introduced by the BIPM and 
IERS (minus the 10 seconds introduced at the inception of UTC) until today!

Hypothesis:

The number of steps in a Collatz sequence is in a mysterious way related 
to TAI-UTC time calculation.

The second, symbol s, is the SI unit of time. It is defined by taking 
thefixed numerical value of the caesium frequency, , the unperturbed 
ground-state hyperfine transition frequency of the caesium 133 atom, to 
be 9192631770 when expressed in the unit Hz, which is equal to s−1.

The Collatz steps for x=9192631770 = 222
Thus:

Let N(x) be the number of Collatz steps
and Ct = 9192631770
For x=27 :
N(27)=111

For x=9192631770 :
N(9192631770)=222

N(Ct) = N(27) * 2