May 4th, 2012, 05:40 PM  #1 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  Integer fun...
p = product of 2 consecutive integers n1 and n. s = sum of m consecutive integers, the first being n+1. s = p Example ( n = 12, m = 8 ): p = 11 * 12 = 132 s = 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 132 If m = 4,541,160 then what's n ? 
May 4th, 2012, 07:42 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 522 Math Focus: Calculus/ODEs  Re: Integer fun...
With a bit of algebra, we find: and so: 
May 4th, 2012, 08:16 PM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  Re: Integer fun...
Yepper!!

May 4th, 2012, 10:23 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 522 Math Focus: Calculus/ODEs  Re: Integer fun...
I was interested in which values of m lead to an integral value of n, so I wrote a short program on my calculator, and found the sequence: With some good old trialanderror, I found this sequence can be generated with the recursion: where: and hence, the closed form is: 
May 5th, 2012, 05:36 AM  #5 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  Re: Integer fun...
Nice! The negative solution also valid; right? If m = 4,541,160 then what's n ? n = 6,203,341 or 1,662,180 Using my m = 8 example: n = 12 or n = 3 p = 4 * 3 = 12 s = 2 + 1 + 0 + 1 + 2 + 3 + 5 + 5 = 12 
May 5th, 2012, 07:42 AM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 522 Math Focus: Calculus/ODEs  Re: Integer fun...
I considered the negative solution, but decided to discard it, but I see no problem with it. 
May 5th, 2012, 08:48 AM  #7 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Integer fun...
Ah, this is my kind of math! Thanks for a nice thread!

May 5th, 2012, 10:34 AM  #8  
Senior Member Joined: Feb 2010 Posts: 714 Thanks: 151  Re: Integer fun... Quote:
 
May 5th, 2012, 11:04 AM  #9 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  Re: Integer fun...
97512 [97513 + 97514 +...+ 235415] 235416 97512 * 235416 = 97513 + 97514 +...+ 235415 = 22,955,884,992 Did that in my head watching the New York : Washington hockey game 
May 6th, 2012, 12:08 AM  #10 
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  Re: Integer fun...
Hello MarkFL: I think it should be: [attachment=0:19nksj9y]n(m).jpg[/attachment:19nksj9y] How do you think ? 

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