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A general case and discussion

  I succeeded to obtain the general expression for arbitrary tex2html_wrap_inline425 . The expression is given by


We can obtain any tex2html_wrap_inline425 with arranging tex2html_wrap_inline425 pieces of tex2html_wrap_inline741 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, tex2html_wrap_inline441 , i and etc.) in one "1" problem, I know answers for N "N"s problem (N-piece-of-"N" problem) in the case tex2html_wrap_inline775 as follows. In the case N = 5 + 2m, tex2html_wrap_inline425 is given by


In the case N = 4 + 2m, tex2html_wrap_inline425 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 tex2html_wrap_inline425 with non-alphabetical symbols in the general case (N "N"s problem), please inform me.

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Up: Four "4"s Previous: Examples