Number Pattern Game

[BMAD]

Pick any positive integer.

If you pick an odd positive integer then your next picked number is 3n+1

If you pick an even number then your next picked number is n/2

Repeat with your picked numbers until you notice something interesting.

1. What interesting event happens?

2. Why is it happening?

3. Can you prove your hypothesis in number 2?

[TimeSpaceLightForce]

i picked : then picked ..... interesting
1:4 3:10 5:16 7:22 9:28 +6 every next odd
2:1 4:2 6:3 8:4 10:5 +1 every next even

[Barcallica]

we will end up in 1, 4, 2, 1 cycle whatever n we choose?

[BMAD]

we will end up in 1, 4, 2, 1 cycle whatever n we choose?

yes. now we need to answer the last two questions

[BMAD]

i picked : then picked ..... interesting

1:4 3:10 5:16 7:22 9:28 +6 every next odd

2:1 4:2 6:3 8:4 10:5 +1 every next even

If you keep repeating (more than a single recurssion) you should come to the same conclusion as Barcallica.

[Rainman]

This is a "famous" unsolved problem. The conjecture that every positive integer eventually leads back to 1 is called the Collatz conjecture. We won't be able to prove or disprove it here, sorry

[bonanova]

It has not been proved, but most mathematicians believe this sequence eventually reaches the value of unity.
Thereafter it repeats 4 2 1 ...

A proof would rule out two alternatives.

1. A cycle other than 4 2 1 ... exists
2. The sequence is unbounded.

Four other cycles are known if the starting integer can be negative.

Strong heuristic arguments suggest the sequence is always bounded.

No other cycle has been found for starting numbers up to ~ 5x10¹⁸.

An interesting way to create the sequence in binary is to append a trailing "1" and add, then remove all trailing zeros.

For a starting point of 7(10) = 1 1 1(2), this gives

111
1111
10110
10111
100010
100011
110100
11011
101000
1011
10000

[BMAD]

This is a "famous" unsolved problem. The conjecture that every positive integer eventually leads back to 1 is called the Collatz conjecture. We won't be able to prove or disprove it here, sorry

I was hoping the fabulous minds at Brainden could pull together. Oh well.