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January 1st, 2013, 01:19 PM  #1 
Newbie Joined: Jan 2013 Posts: 1 Thanks: 0  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:
I thought this was intresting, so I posted it here. Did anyone find something like this before? 
January 1st, 2013, 04:38 PM  #2 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  Re: Some Numbers With Strange Qualityies
*deleted* oh shucks, i misread the procedure.. but i still recommend the book. http://www.amazon.com/RecreationsTheor ... 0486210960 
January 1st, 2013, 08:31 PM  #3 
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  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. 
January 1st, 2013, 10:01 PM  #4 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  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. 
January 2nd, 2013, 04:49 AM  #5 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  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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