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Topic: Pairs of Power Numbers
Replies: 1   Last Post: Jun 29, 2013 8:15 AM

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Gerhard Lantzsch

Posts: 2
Registered: 6/28/13
Pairs of Power Numbers
Posted: Jun 28, 2013 9:02 PM
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There are many pairs of power numbers of the form  a^(2n) and
(a+1)^(2n) which have equal sums of digits. The simplest pair for  a=4
 and  n=1  which gives us
4^2 = 16  and  5^2 = 25. For both the sum of digits = 7. Two more
a = 3154,  n=7,  sum of digits = 214;  a = 4093, n = 30, sum of digits
= 1063.
If the form of the power numbers is a^(2n+1)  and  (a+1)^(2n+1) then
there are no pairs of power numbers which have equal sums of digits.
Has anyone such a pair? Or does anyone know a proof that it is
impossible to find such pairs?   

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