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 6 of 7: Verify the sum
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∑ai=sk2=(sk)k\sum a_i=s^{k^2}=(s^k)^k
Detailed analysis

We have ∑ai=sk2−1∑bi=sk2=(sk)k\sum a_i=s^{k^2-1}\sum b_i=s^{k^2}=(s^k)^k, a pure kk-th, hence 20092009-th, power.