Cai writes down the list of positive integers, excluding squares and cubes. His sequence starts

2, 3, 4, 6, 7, 10, 11, ...

What is the 1000th term in Cai's list?

 Jun 19, 2024

The sequence 2, 3, 5, 6, 7, 10, 11, ... consists of all positive integers that are neither perfect squares nor perfect cubes.


One way to find the 1000th term is to simply count the number of perfect squares and perfect cubes less than a very large number, then subtract that number from the very large number itself, assuming it is not a perfect square or cube itself.


The 33rd perfect square is 332=1089


The 12th perfect cube is 123=1728


Since 1089<1728, there are more perfect squares than perfect cubes less than 1728.


We know 1 is a perfect square and perfect cube, so we subtract 2 from the count.


Therefore, there are 33−2=31 perfect squares and cubes less than 1728.


Since 1728 is not a perfect square or cube, the 1728th term in Cai's list is 1728.


Following this logic, to find the 1000th term, we can look for the smallest perfect square greater than 1000.


The 32nd perfect square is 322=1024


There are 32−2=30 perfect squares and cubes less than 1024.


Since 1024 is a perfect square, the 1000th term in Cai's list is not 1024. However, we can see that the difference between the 1000th term and the 1024th term is less than or equal to the difference between the 999th term and the 1000th term (which is 1).


Therefore, the 1000th term must be 1 less than the 32nd perfect square, which is 322−1=1023​.

 Jun 19, 2024

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