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 May 31st, 2013, 01:28 AM #1 Senior Member   Joined: May 2013 From: España Posts: 151 Thanks: 4 Powers expressed as sum of consecutive numbers . Option 1) If "n"=couple Demonstration: Option 2) If "n"=odd Demonstration: Examples: 1) 2) Regards. May 31st, 2013, 02:52 AM #2 Math Team   Joined: Apr 2012 Posts: 1,579 Thanks: 22 Re: Powers expressed as sum of consecutive numbers Indeed, not just powers, but any composite number can be expressed as the sum of consecutive numbers, eg 35 = 5*7 = 5+6+7+8+9 = 2+3+4+5+6+7+8 Therefore, any composite can be expressed as the difference between triangular numbers, eg 35 45-10 or 36-1 May 31st, 2013, 03:46 AM #3 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Re: Powers expressed as sum of consecutive numbers Primes too. for example: 5 = 15 - 10 and 17 = 153 - 136. May 31st, 2013, 05:36 AM #4 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: Powers expressed as sum of consecutive numbers Nice problem! See A001227 in the OEIS: number of ways to write n as difference of two triangular numbers; number of odd divisors of n. May 31st, 2013, 02:44 PM   #5
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Re: Powers expressed as sum of consecutive numbers

Quote:
 Originally Posted by Hoempa Primes too. for example: 5 = 15 - 10 and 17 = 153 - 136.
Indeed. Primes can be the difference either of consecutive triangles, since any number at all can obviously be so expressed:

66 - 55 = 11, since 1+2+3+4+5+6+7+8+9+10+11 - 1+2+3+4+5+6+7+8+9+10 = 11

Primes can also be the difference between triangles spaced two apart, as every odd number can be so expressed:

1+5+3+4+5+6 - 1+5+3+4 = 5+6 = 11

But that's it.

Composites have other possibilities. May 31st, 2013, 05:46 PM   #6
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Re: Powers expressed as sum of consecutive numbers

Quote:
 Originally Posted by mente oscura Option 2) If "n"=odd
Curious application:

Demonstration:

Example:

Regards. June 1st, 2013, 02:11 AM   #7
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Re: Powers expressed as sum of consecutive numbers

Quote:
 Originally Posted by johnr Composites have other possibilities
Not all of them. What about n = 4? See the sequence CRGreathouse gave. June 1st, 2013, 02:48 AM   #8
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Re: Powers expressed as sum of consecutive numbers

Quote:
Originally Posted by Hoempa
Quote:
 Originally Posted by johnr Composites have other possibilities
Not all of them. What about n = 4? See the sequence CRGreathouse gave.
Oh, sorry. Yes, evens are a different story. I should have specified that I was, as usual, talking about odd composites.

Yes, powers of 2 are even more restricted than odd primes in how they can be expressed as the difference between triangular numbers.

I never thought about this stuff simply in terms of how many odd factors a number has and find CRG's link quite fascinating! Tags consecutive, expressed, numbers, powers, sum Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post johnr Number Theory 5 March 5th, 2014 11:03 AM Dacu Number Theory 3 May 31st, 2013 09:06 PM daigo Algebra 1 May 18th, 2012 02:59 PM coax Number Theory 1 July 24th, 2009 05:36 AM natus zeri Math Events 2 December 29th, 2007 01:52 PM

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