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

Posted by Ady TZIDON on 2008-03-07 07:41:28