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 3 of 7: Name the sum
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s=∑i=1nbis=\sum_{i=1}^n b_i
Detailed analysis

Let s=∑bis=\sum b_i; it is a positive integer.