Super Happy Numbers

GrimReaper · 2012-02-08 10:26:32

Do you Make up Names for People! MUG.

bobbym · 2012-02-08 10:27:39

Mug? I have not heard that in a long time.

GrimReaper · 2012-02-08 10:29:48

Well Suited!

bobbym · 2012-02-08 10:30:34

Sorry, GiB you are barred!

Stop trying to sneak back in under different names or I will have to report you to your isp.

anonimnystefy · 2012-02-08 10:47:54

is there a way to ban him by his IP?

bobbym · 2012-02-08 10:49:32

He is using a proxy to scramble his ip address. But there are other ways to detect him.

wintersolstice · 2012-03-02 07:59:54

if you can exhaust all numbers less than a million you'll find all possible loops this is because

when put into the procedure:

a 1 interval number will become a 1 or 2 interval number

a 2 interval number will become a 1,2 or 3 interval number

a 3 interval number will become a 1,2 or 3 interval number

a n interval number will become a "less than n" interval number (where n is 4 or more)

so any number will fall below 4 intervals and then stay there

these rules are universal

meaning they work regardless of what the interval size is!

in other words they work for

happy numbers

super happy numbers

hyper-n happy numbers (regardless of what n is!)

when I can I'll post a complete proof of all of this:D

Last edited by wintersolstice (2012-03-02 08:03:10)

anonimnystefy · 2012-03-02 08:17:51

Hi ws

Sorry about the off topic posts there!

phrontister · 2012-03-03 05:08:22

Hi wintersolstice,

You don't have to check numbers >9999 because that is the largest number that produces a number greater than itself in its first iteration...which means that numbers higher than that only lead to re-checking numbers already checked.

I discovered that property when first working on these cycles, but forgot to tell you.

A test I did shows that any number will eventually iterate to one of the twelve cycles that I mentioned in post #6, so I reckon we won't find any others.

wintersolstice · 2012-03-03 08:16:04

phrontister wrote:

Hi wintersolstice,
You don't have to check numbers >9999 because that is the largest number that produces a number greater than itself in its first iteration...which means that numbers higher than that only lead to re-checking numbers already checked.

well what I've done is to demonstrate that if you search between one and "the largest number with 3 intervals" you can always find the entire list of cycles for any type of Hyper-n happy numbers (although as you've demonstrated it might not be necessary to search them all:D)

phrontister wrote:

I discovered that property when first working on these cycles, but forgot to tell you.

no worries:D I had a lot of fun with all that complicated proving (I'm now going to take it to other bases and other powers and to all Hyper-n cases:D)

do you think it's worth still posting the proofs? (because it's strange how those rules work with all interval sizes)

phrontister wrote:

A test I did shows that any number will eventually iterate to one of the twelve cycles that I mentioned in post #6, so I reckon we won't find any others.

well it's looks like for super happy numbers the problem is solved:D

phrontister · 2012-03-04 05:49:39

Hi wintersolstice,

"the largest number with 3 intervals"

For SHNs, that probably only needs to be "the largest number with 2 intervals" - depending on your proof wording (ie, if "largest number" refers to the test number only and not any of its iterations: eg, 9999 iterates to 19602, which has 3 intervals).

Yes...I'd like to see those proofs. Always interested to see how others do things...might be handy for something later.

the BASIC program I used to prove there were no more cycles. I simply adapted the one from my post #11.

The program checks each number from 1 up to your 'input' number to see if any of its iterations match the first number of one of the twelve already-known cycles...which all numbers do.

If you run this program and don't see anything happening for a while in the output window, that is because it only prints the final result. My computer takes about 40 seconds to test 9999 numbers in LibertyBASIC and 4 seconds with the QB64 compiler.

The lines that commence with a single quotation mark are just my explanatory notes for various stages, and aren't required for the program to run.

Last edited by phrontister (2012-03-04 12:03:58)

wintersolstice · 2012-03-04 08:49:03

phrontister wrote:

Hi wintersolstice,
"the largest number with 3 intervals"
For SHNs, that probably only needs to be "the largest number with 2 intervals" - depending on your proof wording (ie, if "largest number" refers to the test number only and not any of its iterations: eg, 9999 iterates to 19602, which has 3 intervals).
Yes...I'd like to see those proofs. Always interested to see how others do things...might be handy for something later.

well actually my proof proves the following statement

when

implying that a number with 4 or more intervals will always have less intervals after an iteration and that a number with 3 intervals can have up to 3 intervals (so it may have an equal value after an iteration)

this means your proof has a slight hole in it:

there could be a number between 9999 and 999999 (i.e 3 intervals) that is "equal" after an iteration

you haven't proven/disproven that yet

such a number could form a cycle on it's own!

so you still need to check up to 999999 just in case (though I doubt such a number exists, it's best to be on the safe side )

Last edited by wintersolstice (2012-03-04 08:53:23)

bobbym · 2012-03-04 09:10:21

Hi wintersolstice;

All fixed! Have a good day!

phrontister · 2012-03-04 12:03:13

Hi wintersolstice,

I'll change my axiom to:

No first iteration of n is >=n, where n>9999.

99999's first iteration is 19683 and 999999's first iteration is 29403, which are both waaaaay short of their parent numbers. The higher the parent number, the greater the disparity with their first iteration.

1, 1233 and 8833 are the only single-iteration numbers.

Thanks for posting that proof. I'll have to look at it later...off to work now.

so you still need to check up to 999999 just in case (though I doubt such a number exists, it's best to be on the safe side )

Just for fun I checked up to 9999999, and to my great surprise no other cycles turned up!

117 iterations are the most required to find all cycles (unique & duplicates).

Last edited by phrontister (2012-03-05 23:51:28)

wintersolstice · 2012-03-05 01:08:52

that new axiom sounds better:D

btw why are you thanking for the proof I haven't posted it yet?:D

I only posted the statement that I'd proved:m

when I post the proof I'll also prove that the rules are the same for "Happy numbers, Super happy numbers, Hyper 3 happy numbers etc

I'm also looking at higher powers, other bases (binary, ternary etc) and like I've already said other intrval sizes:D

It's become an obsession atm!

phrontister · 2012-03-05 23:58:03

wintersolstice wrote:

It's become an obsession atm!

I enjoy trotting out LibertyBasic and wiping the dust off it to deal with these exercises. And that also led to me installing the QB64 compiler and learning something about that.

So, thanks for giving me this opportunity.

Math Is Fun Forum

#26 2012-02-08 10:26:32

Re: Super Happy Numbers

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Re: Super Happy Numbers

#28 2012-02-08 10:29:48

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#29 2012-02-08 10:30:34

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#30 2012-02-08 10:47:54

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#32 2012-03-02 07:59:54

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#33 2012-03-02 08:17:51

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