Let S be the cube of a prime number, such that S is greater than 504.
Prove that (S-125)(S+125) is evenly divisible by 504.
Any prime number greater than 504 can be represented
in the form 6n ± 1 where n > 84 .
Let H = S^ 1/3 = 6n ± 1 where n = 85,86,87 ...
(H^3 + 5^3) (H^3 - 5^3)
= (H+5)(H^2 - 5H + 25)(H-5)(H^2 + 5H + 25)
Note (H-5) and (H+5) terms in the product expression
if H = 6n-1 type prime than
H-5 = 6n - 6 = 6(n-1) which is divisible by 504 for all n >84
if H = 6n +1 type prime than
H+5 = 6n+6 = 6(n+1) which is divisible by 504 for all n >84
Q.E.D
Edited on May 10, 2011, 4:10 pm
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Posted by phi
on 2011-05-10 16:06:39 |
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