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Von Neumann and the Collatz Conjecture

He famously said that: "Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin."

I wonder if he knew about the Collatz Conjecture when he said that. 

On his Wikipedia page, I read:

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At times he could be ignorant of the standard mathematical literature, finding it easier to rederive basic information he needed rather than chase references.

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John von Neumann - Wikipedia 

So it's possible, even likely, he did not know about it.

The Collatz Conjecture definitely involves a deterministic process, and I say it produces random numbers.

Here is my informal proof of that: given any number, there is no algorithm for determining how many iterations it will take the Collatz Conjecture process to reach 1. There is also no algorithm or equation you could use to find the maximum value of the iterations. Small changes in the initial value produce drastically different results, which is a hallmark of chaos theory. 

In this way, the process is akin to rolling dice. The overall and final results can be calculated beforehand, but the specific and intermediate results are unknowable. 

If something is deterministic but unpredictable, it is effectively random. 

icky history

I try to keep things family friendly here. Sometimes that is not possible. People often shit and piss themselves when they die, so various warrior cultures developed a tradition of fasting before battle. This has obvious downsides.

Battle of Fallen Timbers - Wikipedia

Cannibalism of slain enemies was common across Polynesia. If there was nothing left to eat, stealing shit from the guts of a dead warrior was proof you were in the battle.

During WW2 in the Pacific, US troops sometimes took ears, teeth, and other body parts from dead Japanese soldiers as grisly souvenirs.

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In the May 22, 1944 issue of LIFE magazine, a striking and controversial image was featured as the “Picture of the Week.” It showed 20-year-old Natalie Nickerson of Phoenix, Arizona, gazing at a human skull — reportedly of a Japanese soldier — that she had received from her boyfriend, a Navy lieutenant serving in the Pacific

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American mutilation of Japanese war dead - Wikipedia

The Japanese mutilated the dead too. 

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The Ear Mound, or Mimizuka, in Kyoto is a memorial burial mound for the severed noses of Korean and Chinese victims killed during Toyotomi Hideyoshi's invasions of Korea in the late 16th century.

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Mimizuka - Wikipedia

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A plaque, which was later removed, stood in front of the Ear Mound in the 1960s with the passage, "One cannot say that cutting off noses was so atrocious by the standard of the time." Most guidebooks do not mention the Ear Mound, and only a few Japanese or foreign tourists visit the site.[10] The majority of visiting tourists are Korean—Korean tour buses are often seen parked near the Ear Mound.[citation needed]

In 1982, not a single Japanese school textbook mentioned the Ear Mound. 

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But sure, let's all get extra pissed off about Confederate monuments...

conjecture: can all primes greater than 2 be written in the form 2n + 3?

All primes greater than 5 end in either 1, 3, 7, or 9. If n is an integer, 2n + 3 will always be odd, but not necessarily prime. If we further require that n not be a multiple of 3, we eliminate all odd multiples of 3 since 2n + 3 will always be a multiple of 3 if n is a multiple of 3. We can also require that n be an integer whose last digit is not 1, because in those cases, 2n + 3 will always have 5 as the last digit. Likewise, if the last digit of n is a 6, the last digit of 2n + 3 will be 5. 

If the conjecture is true, then it makes sense that 1 cannot be written form 2n + 3 since it is not prime. Also, it intuitively makes sense that all other primes would be some combination of the two smallest primes.

Let's make a table

n                            2n + 3                         prime?

0                            3                                  yes

1                            5                                  yes

2                            7                                  yes

4                            11                                yes

5                            13                                yes

7                            17                                yes                 

8                            19                                yes

10                          23                                yes

13                          29                                yes

14                          31                                yes

17                          37                                yes 

19                          41                                yes

20                          43                                yes

22                          47                                yes

23                          49                                no 

25                          53                                yes

28                          59                                yes

29                          61                                yes

32                          67                                yes

34                          71                                yes

35                          73                                yes

37                          77                                no

38                          79                                yes

40                          83                                yes

43                          89                                yes

44                          91                                yes  

For larger primes, we know that because of the possible last digits, the result of subtracting 3 will always be an even number, and so the result can always be written in the form of 2n.                             

I have a hunch I'm missing something but feel I'm on the right track. Could this be a new primality test? It may even be a way to find the next largest prime. It seems the conjecture occasionally returns a perfect square or a semiprime instead of a prime number. 

The largest known prime number is 2^136279841 − 1. That's not really relevant for the conjecture, but I thought I'd put it here. The prime numbers used for RSA and other modern cryptography are in the range of about 2^300, which has roughly 100 digits.

Let's try n = 100,000.

2n + 3 = 200,003

200,003 is prime. 

n = 1,000,000

2n + 3 = 2,000,003

2,000,003 is prime. 

n = 5,000,000,000

2n + 3 = 10,000,000,003

10,000,000,003 = 13 × 769231

Well, looks like I have to think about this some more. 

pipe dream

 

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