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May 4th, 2012, 05:40 PM   #1
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Integer fun...

p = product of 2 consecutive integers n-1 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 ?

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May 4th, 2012, 07:42 PM   #2
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Re: Integer fun...

With a bit of algebra, we find:



and so:

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May 4th, 2012, 08:16 PM   #3
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Re: Integer fun...

Yepper!!
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May 4th, 2012, 10:23 PM   #4
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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 trial-and-error, I found this sequence can be generated with the recursion:



where:



and hence, the closed form is:

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May 5th, 2012, 05:36 AM   #5
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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
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May 5th, 2012, 07:42 AM   #6
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Re: Integer fun...

I considered the negative solution, but decided to discard it, but I see no problem with it.
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May 5th, 2012, 08:48 AM   #7
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Re: Integer fun...

Ah, this is my kind of math! Thanks for a nice thread!
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May 5th, 2012, 10:34 AM   #8
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Re: Integer fun...

Quote:
Originally Posted by MarkFL
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 trial-and-error, I found this sequence can be generated with the recursion:

Interesting ... and are both perfect squares.
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May 5th, 2012, 11:04 AM   #9
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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
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May 6th, 2012, 12:08 AM   #10
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Re: Integer fun...

Hello MarkFL:

I think it should be:

[attachment=0:19nksj9y]n(m).jpg[/attachment:19nksj9y]

How do you think ?
Attached Images
File Type: jpg n(m).jpg (6.1 KB, 166 views)
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