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.

Note: a,d are positive integers.

THE SUM OF CUBES ( Si)^3 of AP is always divisible by the sum of correspondng members of that AP - therefore it cannot be prime.

E.G.

given 2,5,8,11,14 etc

2^3 +5^3 is divisible by 2+5

2^3 +5^3 + 8^3 is divisible by 2+5+8

2^3 +5^3 + 8^3+ 11^3+14^3 is divisible by 2+5+8+11+14

ETC

Now- go and prove it **formally**

*Edited on ***March 7, 2008, 7:43 am**