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Prime as sum of Cubes of terms of AP? (Posted on 2008-03-02) Difficulty: 2 of 5
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.

Note: a,d are positive integers.

See The Solution Submitted by Praneeth    
Rating: 3.3333 (3 votes)

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Hints/Tips hint- no spoiler | Comment 3 of 4 |

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

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