Three numbers in A.P. whose sum is a 6-th power

Posted on August 31, 2016 by benvitalis

 
 
 
Find three numbers in arithmetical progression whose sum is a 6-th power.
 
 

 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

About benvitalis

math grad - Interest: Number theory
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1 Response to Three numbers in A.P. whose sum is a 6-th power

  1. Paul says:
    August 31, 2016 at 9:40 am

    Lets assume the numbers {a, b, c} > 0

    For all n when b = 243*n^6 and {a and c} vary accordingly to form the AP. So when n = 1, b = 243 and a can be any number from 1 to 242 and c is adjusted accordingly. If {a, b, c} are in AP, the sum of a, b, and c will always be 3b.
    The first few values of b are :-
    {243, 15552, 177147, 995328, 3796875, 11337408, 28588707, 63700992,129140163, 243000000}.

    The 6th root of 3 times those numbers are {3, 6, 9, 12, 15, 18, 21, 24, 27, 30}

    So there will be an infinite set of them where the total number in each set is 1 less than 243*n^6 for each n in the set.

    Paul.

    Reply

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