The Barge Problem: Coming Soon: The collection of positive integers

A state of the art collection is coming to the Skillman Library this spring! Due to a recent breakthrough in library storage technology, our library is able to house, on site, the infinite collection of all positive integers! (Due to storage limits only positive integers are available at this time, the collection of all real numbers is WAY TOO large to be housed on-site.) In fact, a special thanks is in order to our Director of Special Collections Diane Shaw, as Lafayette is the first college in the nation to offer this service!

What this means is that you can now check out any number you wish. Need the number 23 for a project? Check it out! (If you need to check out a larger number, such as the largest known prime with over 20 million digits, a special suitcase will be available for transport.)

In order to check out any number a special system has been devised. For very technical reasons, you must express the number you want in the following form:

±12 ± 22 ± 32 · · · ± n2

where you can choose last square n2 and whether you add or subtract a given square. For example, if you want to check out the number 23 you could write

+1 + 4 + 9 − 16 + 25.

(Note: For any n chosen all squares between 1 and n2 must appear.) The question is: using this system can every positive integer be requested? If so, how?