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January 1st, 2013, 01:19 PM   #1
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Some Numbers With Strange Qualityies

Hello everyone!
I'm a 1st grade student in a slovenian high school. Recently, I've became very intrested in numbers and math itsself.
I was just playing around with a few numbers, when I came across a few that are unique. I was wondering wich numbers can loop back
to themselfes after this procedure; n is a digit, m is a digit, o is an integer, p is another integer
o = mn
p = m^2 + n^2
o = p
...repeat...

For instance, 16 is one of these numbers;
1^2 + 6^2 = 1+36 = 37 ==> 3^2 + 7^2 = 9+49 = 58 ==> 5^2 + 8^2 = 25+64 = 89 ==>
8^2 + 9^2 = 64+81 = 145 ==> 1^2 + 4^2 + 5^2 = 1+16+25 = 42 ==> 4^2 + 2^2 = 16+4 = 20 ==> 2^2 + 0^2 = 4+0 = 4 ==> 4^2 = 16

I then created a program to find every single number that has this property. It turns out, there are only a limited few:
  • 1
    4
    16
    20
    37
    42
    58
    89
    145

I thought this was intresting, so I posted it here.
Did anyone find something like this before?
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January 1st, 2013, 04:38 PM   #2
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Re: Some Numbers With Strange Qualityies

*deleted*

oh shucks, i misread the procedure.. but i still recommend the book.

http://www.amazon.com/Recreations-Theor ... 0486210960

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January 1st, 2013, 08:31 PM   #3
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Re: Some Numbers With Strange Qualityies

Hello, agentredlum!

One of my favorite books: Recreations in the Theory of Numbers, Albert H. Beiler.

I've recommended it at a number of math sites and got no feedback.
I'm gratified that there is at least one other person who likes the book.

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January 1st, 2013, 10:01 PM   #4
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Re: Some Numbers With Strange Qualityies

I'm glad too. That book has similar loop procedures to the OP but i'm not sure if it has the exact example since i no longer have my copy given to me by prof. rosenberg.

I wish there were more books on number theory like that book.

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January 2nd, 2013, 04:49 AM   #5
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Re: Some Numbers With Strange Qualityies

This is Sloane's A039943. It looks like it was studied by Arthur Porges in the 40s.
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