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# A general case and discussion

I succeeded to obtain the general expression for arbitrary . The expression is given by

We can obtain any with arranging pieces of signs. The solution without alphabetical functions still is unknown.

The next possible generalization of the problem must be in terms of the different base number from "4" (e.g. three "3"s, five "5"s, ... N "N"s). It is considered that one "1" problem seems to be impossible to be solved. But we know the answer that it's possible. See the following equation

This equation uses only 1. Though I do not know if we have a solution without symbols considered as a number (as e, , i and etc.) in one "1" problem, I know answers for N "N"s problem (N-piece-of-"N" problem) in the case as follows. In the case N = 5 + 2m, is given by

In the case N = 4 + 2m, is given by

In each equation, there are m pieces of (N-N)=0 added with no effect (underlined parts). If you could have an expression for with non-alphabetical symbols in the general case (N "N"s problem), please inform me.

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