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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 4510 or 361 
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  
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Powers expressed as sum of consecutive numbers Quote:
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  
Senior Member Joined: May 2013 From: España Posts: 151 Thanks: 4  Re: Powers expressed as sum of consecutive numbers Quote:
Demonstration: Example: Regards.  
June 1st, 2013, 02:11 AM  #7  
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  Re: Powers expressed as sum of consecutive numbers Quote:
 
June 1st, 2013, 02:48 AM  #8  
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Powers expressed as sum of consecutive numbers Quote:
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!  

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