Four Fours/Table of 4s

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Zero 4s

$1=\ln {e}$

$e^{\pi i}+e^{-\pi i}=-2$

One 4

Believe it or not, the rules allow you to do quite a bit with a single 4. The rules say you may combine 4s with any mathematical symbol except numbers. Thus, in addition to 4 alone, we also have:

$1=i^{4}$

$2={\sqrt {4}}$

$24=4!$

The following fractions are also useful:

${\dfrac {1}{4}}$

${\dfrac {1}{2}}={\dfrac {1}{\sqrt {4}}}$

${\dfrac {1}{24}}={\dfrac {1}{4!}}$

but these can't appear with 1 in the denominator.

Could possibly add constants (harmonic number), possibly special functions.

Once you allow variables like x into the mix, it's lights out.

One 4 with Variables

$5={\dfrac {\ln {\left({\dfrac {\ln {\left({\sqrt {\sqrt {\sqrt {\sqrt {\sqrt {x}}}}}}\right)}}{\ln {x}}}\right)}}{\ln {\sqrt {4}}}}$

Two 4s

Note: we aren't using fourth roots or one-quarter powers very much, e.g., Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\sqrt {2}}={\sqrt[{4}]{4}}=4^{\frac {1}{4}}} . Adding this would greatly expand the possibilities.

$1={\dfrac {4}{4}}$

$1=\log _{4}(4)$

$2=4-{\sqrt {4}}$

$2={\dfrac {4}{\sqrt {4}}}$

$3={\sqrt {4}}+i^{4}$

$3=4-i^{4}$

$4={\sqrt {4}}\times {\sqrt {4}}$

$4={\sqrt {4}}+{\sqrt {4}}$

$4={\dfrac {4}{i^{4}}}$

$5=4+i^{4}$

$5={\sqrt {4!+i^{4}}}$

$6=4+{\sqrt {4}}$

$6=(4-i^{4})!$

$8=4+4$

$8=4{\sqrt {4}}$

$12={\dfrac {4!}{\sqrt {4}}}$

$16=4\times 4$

$16=4^{\sqrt {4}}=({\sqrt {4}})^{4}$

$16={\sqrt {4^{4}}}$

$20=4!-4$

$22=4!-{\sqrt {4}}$

$23=4!-i^{4}$

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 24={\sqrt {(4!)^{\sqrt {4}}}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 25=4!+i^{4}}

$26=4!+{\sqrt {4}}$

$28=4!+4$

$44$

$48=4!\times {\sqrt {4}}$

$96=4!\times 4$

$120=(4+i^{4})!$

$256=4^{4}$

$576=(4!)^{\sqrt {4}}$

$720=(4+{\sqrt {4}})!$

$331,776=(4!)^{4}$

$16,777,216={\sqrt {4}}^{4!}$

$281,474,976,710,656=4^{4!}$

$1,333,735,776,850,284,124,449,081,472,843,776={4!}^{4!}$

(For those keeping score at home, that's 1 decillion 333 nonillion 735 octillion 776 septillion 850 sextillion 284 quintillion 124 quadrillion 449 trillion 81 billion 472 million 843 thousand 776)

Three 4s

These lists blow up pretty fast... as you can see, focusing on using a smaller number of 4s can force you to be creative. This makes it possible to combine 4 4's beyond the integers from 1 to 20, and keep on going.

One useful template for representing powers of 2 is:

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2^{n}=({\sqrt {4}})^{n}}

where $n$ is any number expressible with two 4's.

Once we can add three 4's, we can start to write expressions like

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\dfrac {\ln {P}}{\ln {Q}}}}

where P and Q are any expressions involving 1 or 2 fours. If we choose carefully, P and Q will have the same base and different exponents, so we can start to combine integer powers to obtain new integers. This trick will get us out of at least a few jams when constructing the integers from 1 to 100 using only four fours.

$2={\dfrac {4+4}{4}}$

$2={\sqrt {4}}\times \left({\dfrac {4}{4}}\right)$

$3={\dfrac {\ln {(4+4)}}{\ln {\sqrt {4}}}}$

$4=4\times \left({\dfrac {4}{4}}\right)$

$4=4+4-4$

$5=4+{\dfrac {4}{4}}$

$7=4+4-i^{4}$

$7=4+{\sqrt {4}}+i^{4}$

$8=4\times \left({\dfrac {4}{\sqrt {4}}}\right)$

$9=4+4+i^{4}$

$10=4+4+{\sqrt {4}}$

$11={\dfrac {4!-{\sqrt {4}}}{\sqrt {4}}}$

$12=4+4+4$

$18=4\times 4+{\sqrt {4}}$

$20=4\times 4+4$

$26=4!+4-{\sqrt {4}}$

$28=4!+{\sqrt {4}}+{\sqrt {4}}$

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 30=4!+4+{\sqrt {4}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 30={\dfrac {(4+{\sqrt {4}})!}{4!}}}

$30={\dfrac {(4+i^{4})!}{4}}$

$32=4(4+4)$

$32=4^{\sqrt {4}}{\sqrt {4}}$

$36=(4+{\sqrt {4}})^{\sqrt {4}}$

$46=4!+4!-{\sqrt {4}}$

$47=(4!)({\sqrt {4}})-i^{4}$

$49=(4!)({\sqrt {4}})+i^{4}$

$50=4!+4!+{\sqrt {4}}$

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 60={\dfrac {(4+i^{4})!}{\sqrt {4}}}}

$64={\dfrac {4^{4}}{4}}$

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 64=({\sqrt {4}})^{4+{\sqrt {4}}}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 74=(4A)_{4\cdot 4}}

(That is, 74 is 4A in hexidecimal, or base 16 = 4 * 4)

$75=(4B)_{4\cdot 4}$

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 76=(4C)_{4\cdot 4}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 77=(4D)_{4\cdot 4}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 78=(4E)_{4\cdot 4}}

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 79=(4F)_{4\cdot 4}}

$80=4(4!-4)$

$116=((4+i^{4})!-4$

$118=((4+i^{4})!-{\sqrt {4}}$

$119=((4+i^{4})!-i^{4})$

$120={\dfrac {(4+i^{4})!}{i^{4}}}$

$121=((4+i^{4})!+i^{4})$

$122=((4+i^{4})!+{\sqrt {4}})$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 124 = ( (4 + i^4)! + 4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 180 = \dfrac{(4+\sqrt{4})!}{4} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 252 = 4^4 - 4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 254 = 4^4 - \sqrt{4} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 258 = 4^4 + \sqrt{4} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 260 = 4^4 + 4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1,024 = 4^4 \times 4 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4,096 = (4+4)^{4} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 65,536 = (4 \times 4)^{4} }