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May 5th, 2013, 10:09 AM  #1 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Little LCM lemma for the first n pronics
Pronics are of course numbers of the form (x)(x+1). The first pronic can for different purposes be considered 0*1 or 1*2. In this case, I am going with 1*2 as the first pronic. First n pronics will therefore be 1*2, 2*3, ..., n(n+1) ALL pronics are even, as one of n and n+1 is always even. But there is no other common factor to all n of the first n pronics, even when you consider only the first 2, ie 1*2 and 2*3. So the LCM of the of the first n pronics will always be (n+1)!/2 eg, for the first 4, ie 1*2, 2*3, 3*4, 4*5, it is 5!/2 or 60. Did I screw anything up? Pretty sure I didn't, but would appreciate a check off! (This will I hope be part of a larger if not exactly profound proof concerning pronics and harmonic means.) 
May 5th, 2013, 10:32 AM  #2  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Little LCM lemma for the first n pronics Quote:
 
May 5th, 2013, 10:34 AM  #3  
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Little LCM lemma for the first n pronics Quote:
 

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