Useful fact: Near powers of 3 are 8X
Because of the 3x role in Collatz rise steps, it's useful to look at the relationship between powers of 3 and powers of 2.
Specifically, even powers of 3 (e) and odd powers of 3 (o) are always near multiples of 8 (and higher powers of 2):
3^e-1 = 8x = x*2^3 (and x*2^4, x*2^5, x*2^6, etc. for higher even powers of 3)
3^o-3 = 8x = x*2^3 (and x*2^4, x*2^5, x*2^6, etc. for higher odd powers of 3)
(Examples and proof below)
I think this may be useful when thinking about how upper terms of Collatz values change after N rise steps.
For instance, we know that a Collatz sequence starting at 2^H*C+L (where L is odd) rises to 3^H*C+Cz(L) after H steps (where Cz(L) is Collatz steps applied to lower term L).
So, any sequence that rises an even number of steps will have an upper term 3^E which by the fact given above will be (8x+1)*C. Sometimes that will be 16x, 32x, 128x.
That's more clearly interesting in binary. If you start with a number:
CCCC00000LLLL
It will rise to:
XXXCCCMMMM where MMM is Collatz expansion of L
That means the lower three binary digits (maybe more) of C are untouched after H rises.
So, even though it seems like multiple 3x+1 steps completely scramble the upper bits of the starting number, some bits are preserved.
| 3^(o)-3 | 3^(o)-3 | Binary | Multiple of 2^(N) |
|---|---|---|---|
| 3^(3)-3 | 24 | ...0000000000011000 | 8x |
| 3^(5)-3 | 240 | ...0000000011110000 | 16x |
| 3^(7)-3 | 2184 | ...0000100010001000 | 8x |
| 3^(9)-3 | 19680 | ...0100110011100000 | 32x |
| 3^(11)-3 | 177144 | ...0011001111111000 | 8x |
| ... | ... | ... | ... |
| 3^(17)-3 | 129140160 | ...0000010111000000 | 64x |
| 3^(33)-3 | 5.55906E+15 | ...0011101110000000 | 128x |
| 3^(e)-1 | 3^(e)-1 | Binary | Multiple of 2^(N) |
|---|---|---|---|
| 3^(2)-1 | 8 | ...0000000000001000 | 8x |
| 3^(4)-1 | 80 | ...0000000001010000 | 16x |
| 3^(6)-1 | 728 | ...0000001011011000 | 8x |
| 3^(8)-1 | 6560 | ...0001100110100000 | 32x |
| 3^(10)-1 | 59048 | ...0110011010101000 | 8x |
| ... | ... | ... | ... |
| 3^(16-1) | 43046720 | ...0101011101000000 | 64x |
| 3^(32-1) | 1.85302E+15 | ...0011111010000000 | 128x |
Its easy enough to prove that these near powers of 3 are always 8x.
If 3^o-3 = 8y for any odd o (which is true for 3^3 - 3 = 24)
then: 9*(8y+3)-3 = 72y + 27 - 3 = 72y + 24 = 8*(9y+3)
so: 3^(o+2)-3 is also a multiple of 8 and by induction all 3^o-3 = 8x
If 3^e-1 = 8y for any even e (which is true for 3^2 - 1 = 8)
then: 9*(8y+1)-1 = 72y + 9 - 1 = 72y + 8 = 8*(9y+1)
so: 3^(e+2)-1 is also a multiple of 8 and by induction all 3^e-1 = 8x