MathLabs

Problem 2

For a positive integer kk, call an integer a pure kk-th power if it is mkm^k for some integer mm. Show that for every positive integer nn there exist nn distinct positive integers whose sum is a pure 20092009-th power and whose product is a pure 20102010-th power.
Step 2 of 7: Choose bases
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bi=ik+1 (1≤i≤n)b_i=i^{k+1}\ (1\le i\le n)
Detailed analysis

The integers bi=ik+1b_i=i^{k+1} are positive and distinct, and ∏bi=(n!)k+1\prod b_i=(n!)^{k+1}.