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Indian Team Selection Test 2010 ST2 P2

Source:

May 22, 2010

number theoryrelatively primenumber theory unsolved

Problem Statement

Given an integer

k>1

, show that there exist an integer an

n>1

and distinct positive integers

a_1,a_2,\cdots a_n

, all greater than

1

, such that the sums

\sum_{j=1}^n a_j

and

\sum_{j=1}^n \phi (a_j)

are both

k

-th powers of some integers. (Here

\phi (m)

denotes the number of positive integers less than

m

and relatively prime to

m

.)

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