Mathematician Proves Huge Result on ‘Dangerous’ Problem

Quanta Magazine ^ | December 11, 2019 | Kevin Hartnett

[C19fan]

Take a number, any number. If it’s even, halve it. If it’s odd, multiply by 3 and add 1. Repeat. Do all starting numbers lead to 1?

Experienced mathematicians warn up-and-comers to stay away from the Collatz conjecture. It’s a siren song, they say: Fall under its trance and you may never do meaningful work again.

The Collatz conjecture is quite possibly the simplest unsolved problem in mathematics — which is exactly what makes it so treacherously alluring.

“This is a really dangerous problem. People become obsessed with it and it really is impossible,” said Jeffrey Lagarias, a mathematician at the University of Michigan and an expert on the Collatz conjecture.

Earlier this year one of the top mathematicians in the world dared to confront the problem — and came away with one of the most significant results on the Collatz conjecture in decades.

[Bloody Sam Roberts]

Same answer as the "How many days and how many men does it take to dig a half of a hole?" question.

Ain't no such thing as half of a hole...and no such thing as a half of a chicken. And even still...if there were...it couldn't lay eggs. Of any kind.

[Bloody Sam Roberts]

Is 0 a number?

No. It is a numeral. Just like black is not a color. It is the absence of all color. Zero is the absence of all numbers.

[The Chid]

Purple. Ice Cream has no bones.

Duh.

[Robert DeLong]

Your first one is wrong 8>):

It's:

1 4 2 1 4 2 1 4 2 1 (and continues repeating forever)

If you were starting with 4, not 1 (1 was example #):

4 2 1 4 2 1 4 2 1

[SubMareener]

I know, but it was a fun mistake!

[Scrambler Bob]

Take any number.

Add 1.

Subtract original number.

Resolves to 1, always.

Scrambler’s conjecture.

Don’t spend the rest of your life trying to solve this.

[Scrambler Bob]

If it is 0 degrees and it gets twice as cold, how cold is it?
= = =

Well, if it gets half as cold, how cold is That?

[Lonesome in Massachussets]

0 is a number.

The conjecture only involves positive integers. 0 is not only a number, it is also a integer, but not (strictly) a positive integer.

The conjecture is affirmative, you can disprove it with one counter-example. If the conjecture were that no Collatz sequence ever reached 1 then two obvious counter examples would be 1, with sequence length one: {1} or 2 with sequence length two: {2,1}.

Some other examples: Three goes {3, 5, 8, 4, 2, 1}. Ten goes {10,5,8,4,2,1}. YMMV.

1001 has sequence length 92: {1001, 1502, 751, 1127, 1691, 2537, 3806, 1903, 2855, 4283, 6425, 9638, 4819, 7229, 10844, 5422, 2711, 4067, 6101, 9152, 4576, 2288, 1144, 572, 286, 143, 215, 323, 485, 728, 364, 182, 91, 137, 206, 103, 155, 233, 350, 175, 263, 395, 593, 890, 445, 668, 334, 167, 251, 377, 566, 283, 425, 638, 319, 479, 719, 1079, 1619, 2429, 3644, 1822, 911, 1367, 2051, 3077, 4616, 2308, 1154, 577, 866, 433, 650, 325, 488, 244, 122, 61, 92, 46, 23, 35, 53, 80, 40, 20, 10, 5, 8, 4, 2, 1}

[CharlesWayneCT]

It’s the proof that eludes us.

The problem is fascinating, and simple to express, and really comes down to two questions:

First, are there any numbers other than 1 which, if multiplied by 3 and then adding 1, and then dividing by 2, yield the same number.

Second, are their any repeating sequences of numbers where, having done those same calculations, you end up in a loop.

The 3rd, more esoteric question, is whether there is a sequence that never repeats, but also never leads to 1 — but given an infinite number of tries, this must necessarily either be proven false, or fall into the 1st or 2nd questions.

[shotgun]

I thought for a minute to demonstrate “1/2” by using the example of the their peanut butter sandwich that their mom made them for lunch and that she cut in half, corner to corner, but then these chitlin’s most likely don’t bring their own lunches to school..

[sanjuanbob]

If it is 0 degrees and it gets twice as cold, how cold is it?

Wouldn’t it be 32 degrees below 0?

[elbook]

Seems so simple and obviously true but the devil is in that ‘take any number’ (assumed to be a positive ‘whole’ number - if that’s not to be assumed THEN all bets are off immediately anyway)’

I suspect that there possibly there are whole numbers that when that process is applied form a loop with numbers going back up to the loop start number OR even there are patterns of an infinite upward ramp — AND PROVING THAT THAT DOES NOT HAPPEN (for ANY number upto Infinity ) IS THE HARD PART... (so to not always lead to the final ‘1’)

Any number ...

Consider a possible convolution — consider the ‘mult by 3 and add one’ creates an even (a bigger number) which subsequently being even gets ‘divided by 2’ (a less small number than originally) but still is an even - numbers like 10 but when divided by 2 become odd again.