Let's suppose that \(n\), \(x\) and \(y\) are positive integers such that:$$n^{\, x+y}=x||y$$where \(x||y\) represents the concatenation of \(x\) and \(y\). What values of \(n\), \(x\) and \(y\) satisfy this equation? Well it seems that there is only one set of values, namely \(n=2\), \(x=3\) and \(y=2\) where$$2^{3+2}=32$$This curious fact struck me when looking at the periodicity of conjunctions of Mars and Venus, that occur in almost the same location in the tropical zodiac at intervals 32 years (+0.5 to 4 days). See my post Periodicity of Mars-Venus Conjunctions.
This got me thinking about what other interesting properties the number 32 might possess. Naturally I turned to Numbers Aplenty to find out.
32 can be written using four fours as in \( (4+4)^{\! ^{\frac{\sqrt{\overline{.4}}}{.4}}} \) where:$$\sqrt{\overline{.4}}=\sqrt{\frac{4}{9}}=\frac{2}{3}\\\ \frac{\frac{2}{3}}{.4}=\frac{\frac{2}{3}}{\frac{2}{5}}=\frac{5}{3}\\(4+4)^{\! ^{\frac{5}{3}}}=8^{\! ^{\frac{5}{3}}}=\left ( 2^3 \right )^{ \!\frac{5}{3}}=2^5=32$$Actually this is not a property unique to 32. In fact, all the numbers up 112 can be represented thus and 113 is the smallest natural number that cannot be obtained using four fours, the common arithmetic operations, factorial, roots, and the notations$$.4=0.4=\frac{2}{5} \text{ and }\overline{.4}=0.4444\dots=\frac{4}{9}$$The use of the vinculum or overline is potentially confusing because the symbol is also used for the rising factorial as in:$$4^{\overline{4}}=4 \times 5 \times 6 \times 7=840$$I've always used the overdot as in \( 0.\dot{4}=0.4444 \dots\) to represent the repeating decimal.
There are other ways to represent 32 using four fours:$$4!! \times 4 + 4 -4 = 32\\4^{\sqrt{4}}+4^{\sqrt{4}}=32\\4 \times 4+4 \times 4=32$$The use of the double factorial can be noted in the previous example, viz. 4!!= 4 x 2 = 8. When attempting the four fours representation, it needs to be made clear what's allowable. If the simple factorial is allowed (4!= 4 x 3 x 2 = 24), then it needs to be determined whether the double factorial (4!!=4 x2) and subfactorial (!4 = 9) are also permitted. I've written about the various types of factorials in my post Subfactorials, Semifactorials and Others. It's also in this post that I address the problem of representing 10 using three threes, which involves the use of the subfactorial (!3=2).
The key to solving these puzzles is the collect various building blocks. These might include (depending on what's allowed):
- \(\sqrt{4}=2\)
- \(4!=24\)
- \(4!!=12\)
- \(4^{\overline{4}}=840\)
- \(!4=9 \)
- \(.\dot{4}=0.4444 \dots =4/9 \)
Four fours is just one of a variety of mathematical puzzles that attempt to represent the counting numbers in terms of a fixed number of identical digits, according to certain prescribed rules. Here are links to a variety of representations:
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