Prime as sum of Cubes of terms of AP? (Posted on 2008-03-02)
Let S_{i} be the i^{th} term of an Arithmetic Progression whose 1^{st} term is a and common difference d. Show that for any 2 positive integers m,n(>m), Σ(i:m to n)(S_{i})^{3} can't be a prime number.
The given sum is divisible by mean of S_{i} {i:m to n}
Explanation:
Sum = (S_{m}^3+S_{n}^3)+(S_{m+1}^3+S_{n-1}^3)+..
Use a^3+b^3=(a+b)(a^2+b^2-ab)
As all of them are in AP
S_{m}+S_{n}= S_{m+1}+S_{n-1} = S_{m+2}+S_{n-2} = ....
Sum={(S_{m}+S_{n})(S_{m}^2+S_{n}^2-S_{m}S_{n}+...)}
Hence the given sum can't be a prime.
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