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