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Happy Numbers - Comments and DiscussionThis is the comment and discussion page for the article Mathematical Fun With Happy Numbers, published in Directory : Mathematics : Fun With Numbers.
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Humphrey Nowlin says:
Hi there, I just got done reading your "HAPPY NUMBERS" article, and just wanted to say, you DON'T have all the happy cubes.
Of course, zero works, but that's trivial.
There are also 370, 371 and 407!
Is that all of them?
Douglas Twitchell says:
Thanks for pointing that out. I must not have actually looked for other happy cubes, because I would have done it by writing a computer program, and I wouldn't have missed any.
Is there egg on my face? 
Quote There are also 370, 371 and 407!
Is that all of them?
If no one checks this out in the next couple weeks I'll crank out a proof.
*waits with bated breath for someone else to provide a proof* 
Humphrey Nowlin says:
Can you give me a suggestion on how to get started with this? If I had an idea how to begin, maybe I could figure it out myself.
Douglas Twitchell says:
Well, I would find a value X such that for all Y>X, F(Y)<X, where F(Y) is the value you get by summing the cubes of the digits of Y.
Once you've done that, you just need to write a computer program to find all happy cubes less than X, and you're done!
Humphrey Nowlin says:
Thanks! 
MartynJohnston says:
I wrote something a while ago that told me that there are only 5 'Happy Cubes'.
These are 1, 153, 370, 371, and 407. It also told me I didn't need to look any higher than 2916 to determine that there weren't any more (but I can't remember why that was).
MartynJohnston says:
There are also several 'Happy (whatever you call something to the power of 4)'.
ie. when you sum the fourth power of the digits, you get the number you started with.
I reckon that 1 (of course), 1634, 8208, and 9474 are the only ones (I stopped looking beyond 33000).
MartynJohnston says:
There are also several 'Happy (whatever you call something to the power of 5)'.
ie. when you sum the fifth power of the digits, you get the number you started with.
I reckon that 1 (of course), 4150, 4151, 54748, 92727, 93084 and 194979 are the only ones (I stopped looking beyond 355,000).
MartynJohnston says:
There are also several 'Happy (whatever you call something to the power of 6)'.
ie. when you sum the sixth power of the digits, you get the number you started with.
I reckon that 1 (of course) and 548834 are the only ones (I stopped looking beyond 3,721,000).
MartynJohnston says:
I can't find any 'Happy (whatever you call something to the power of 7)'.
ie. when you sum the seventh power of the digits, you get the number you started with.
I've looked up to 4,000,000, and reckon I've got to check up to 38,263,752 to prove whether there are any or not.
MartynJohnston says:
I just found a Happy Seventh Power thing.
4,210,818 seems to work.
ie. when you sum the seventh power of the digits, you get the number you started with.
Are there any others ?
Michael Dehmlow says:
1,1741725,9800817,9926315,14459929
Douglas Twitchell says:
wow! You guys have been busy! Thanks for those posts. Martyn, there was a problem with the forum the day you posted those, and several of your posts got lost...but they're all back now.
Thanks again.
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