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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.

  Submitted by Praneeth    
Rating: 3.3333 (3 votes)
Solution: (Hide)
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.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re official solution -- IS NOT PERFECTAdy TZIDON2008-03-08 19:45:10
Hints/Tipshint- no spoilerAdy TZIDON2008-03-07 07:41:28
Some Thoughtsanother humble step forwardAdy TZIDON2008-03-06 12:42:29
A humble startAdy TZIDON2008-03-03 16:31:20
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