4つの4(ver.2.1)


なるしす さんの4つの4で1~100を作ろう(以下URL)が流行っている.よって便乗した.の6回目.
http://www.nicovideo.jp/watch/sm27096518

・4を4つ使ってできる自然数はどんなものがあるか?を調べた.

・使える2項演算子は次のようにした.
\(x+y\)
\(x-y\)
\(x\times y\)
\(\frac{x}{y}\)
\(x^y\)

・使える1項演算子は次のようにした.
\(x!\)(ただし、\(x=3,4\)の場合に限る)
\(\sqrt{x}\)(ただし、\(x>0\)の場合に限る)

また、1項演算子の連続した使用は3度までとする.

・4を使用した定数として次のものが使える.
\(44,444,4444\) (4を並べて使う)
\(.4\)(小数点を省略)
\(.\dot{4}\)(循環小数)\((=\frac{4}{9})\)

さらに、計算機の性能の制限が次のようにあるとする.
計算過程において,\(100000\)を超えた場合、これを捨てる.

以上の条件で可能な\(100000\)以下の正の整数は次の通りである.
計算アルゴリズムを少し変更して、簡単な式がだいたい出てくるようにしました(見やすい).

\[ 0 = 44-44 \]
\[ 1 = \frac{44}{44} \]
\[ 2 = \frac{4}{\left(4+4\right)}\times4 \]
\[ 3 = \frac{\left(4+\left(4+4\right)\right)}{4} \]
\[ 4 = 4+\left(4\times\left(4-4\right)\right) \]
\[ 5 = \frac{\left(4+\left(4\times4\right)\right)}{4} \]
\[ 6 = \sqrt{\left(\left(44-4\right)-4\right)} \]
\[ 7 = \frac{44}{4}-4 \]
\[ 8 = \left(4-\left(4-4\right)\right)+4 \]
\[ 9 = \left(4+4\right)+\frac{4}{4} \]
\[ 10 = \frac{\left(44-4\right)}{4} \]
\[ 11 = \frac{44}{\sqrt{\left(4\times4\right)}} \]
\[ 12 = \frac{\left(4+44\right)}{4} \]
\[ 13 = \frac{44}{4}+\sqrt{4} \]
\[ 14 = \frac{\left(4+4\right)}{.\dot{4}}-4 \]
\[ 15 = 4+\frac{44}{4} \]
\[ 16 = \left(44-4\right)\times.4 \]
\[ 17 = \left(4\times4\right)+\frac{4}{4} \]
\[ 18 = .4+\left(.4\times44\right) \]
\[ 19 = \frac{\left(4-\left(.4-4\right)\right)}{.4} \]
\[ 20 = .\dot{4}+\left(.\dot{4}\times44\right) \]
\[ 21 = \frac{\left(4+\left(.4+4\right)\right)}{.4} \]
\[ 22 = \frac{\left(4+4\right)}{.\dot{4}}+4 \]
\[ 23 = \frac{\left(\left(4!\times4\right)-4\right)}{4} \]
\[ 24 = \left(4+\left(4\times4\right)\right)+4 \]
\[ 25 = \frac{4}{.\dot{4}}+\left(4\times4\right) \]
\[ 26 = \left(4\times4\right)+\frac{4}{.4} \]
\[ 27 = \frac{\left(4+\left(4+4\right)\right)}{.\dot{4}} \]
\[ 28 = 44-\left(4\times4\right) \]
\[ 29 = 4+\frac{4}{\left(.4\times.4\right)} \]
\[ 30 = \frac{\left(4+\left(4+4\right)\right)}{.4} \]
\[ 31 = \frac{4!}{\left(.\dot{4}+.\dot{4}\right)}+4 \]
\[ 32 = \left(4\times4\right)+\left(4\times4\right) \]
\[ 33 = \frac{44}{\sqrt{\left(.\dot{4}\times4\right)}} \]
\[ 34 = 44-\frac{4}{.4} \]
\[ 35 = 44-\frac{4}{.\dot{4}} \]
\[ 36 = \left(44-4\right)-4 \]
\[ 37 = \frac{\left(.\dot{4}+\left(4\times4\right)\right)}{.\dot{4}} \]
\[ 38 = \left(44-4\right)-\sqrt{4} \]
\[ 39 = \frac{\left(\left(4\times4\right)-.4\right)}{.4} \]
\[ 40 = 44-\sqrt{\left(4\times4\right)} \]
\[ 41 = \frac{\left(.4+\left(4\times4\right)\right)}{.4} \]
\[ 42 = \left(44-4\right)+\sqrt{4} \]
\[ 43 = 44-\frac{4}{4} \]
\[ 44 = \left(4-4\right)+44 \]
\[ 45 = \frac{4}{4}+44 \]
\[ 46 = \left(4+44\right)-\sqrt{4} \]
\[ 47 = 44+\sqrt{\frac{4}{.\dot{4}}} \]
\[ 48 = \left(4+\left(4+4\right)\right)\times4 \]
\[ 49 = 44+\frac{\sqrt{4}}{.4} \]
\[ 50 = 44+\left(4+\sqrt{4}\right) \]
\[ 51 = \frac{\left(\left(4!-4\right)+.4\right)}{.4} \]
\[ 52 = \left(4+44\right)+4 \]
\[ 53 = \frac{4}{.\dot{4}}+44 \]
\[ 54 = \frac{4}{.4}+44 \]
\[ 55 = \frac{44}{\left(.4+.4\right)} \]
\[ 56 = \left(\frac{4}{.4}+4\right)\times4 \]
\[ 57 = \frac{\left(.4+4!\right)}{.4}-4 \]
\[ 58 = \frac{\left(4!+\left(.\dot{4}\times4\right)\right)}{.\dot{4}} \]
\[ 59 = \frac{\left(4+4!\right)}{.\dot{4}}-4 \]
\[ 60 = 44+\left(4\times4\right) \]
\[ 61 = \frac{4!}{.4}+\frac{4}{4} \]
\[ 62 = \frac{44}{\sqrt{.\dot{4}}}-4 \]
\[ 63 = \frac{\left({4}^{4}-4\right)}{4} \]
\[ 64 = \left(4+4\right)\times\left(4+4\right) \]
\[ 65 = \frac{\left(4+{4}^{4}\right)}{4} \]
\[ 66 = 44\times\frac{\sqrt{.\dot{4}}}{.\dot{4}} \]
\[ 67 = 4+\frac{\left(4+4!\right)}{.\dot{4}} \]
\[ 68 = 4+\left(\left(4\times4\right)\times4\right) \]
\[ 69 = \frac{\left(\sqrt{4}+44\right)}{\sqrt{.\dot{4}}} \]
\[ 70 = 4+\frac{44}{\sqrt{.\dot{4}}} \]
\[ 71 = \frac{\left(4!+\left(.4+4\right)\right)}{.4} \]
\[ 72 = \left(4+44\right)+4! \]
\[ 73 = \frac{\left(\left(4!\times\sqrt{4}\right)+\sqrt{.\dot{4}}\right)}{\sqrt{.\dot{4}}} \]
\[ 74 = 4+\frac{\left(4+4!\right)}{.4} \]
\[ 75 = \frac{44}{.\dot{4}}-4! \]
\[ 76 = \left(\left(4!-4\right)\times4\right)-4 \]
\[ 77 = \sqrt{{\frac{4}{.\dot{4}}}^{4}}-4 \]
\[ 78 = \left(\left(4!-4\right)\times4\right)-\sqrt{4} \]
\[ 79 = 4!+\frac{\left(.\dot{4}+4!\right)}{.\dot{4}} \]
\[ 80 = \left(4+\left(4\times4\right)\right)\times4 \]
\[ 81 = \frac{4}{\left(.\dot{4}\times.\dot{4}\right)}\times4 \]
\[ 82 = \left(\left(4!-4\right)\times4\right)+\sqrt{4} \]
\[ 83 = 4!-\frac{\left(.4-4!\right)}{.4} \]
\[ 84 = \left(\sqrt{4}\times44\right)-4 \]
\[ 85 = \frac{\left(4!+\frac{4}{.4}\right)}{.4} \]
\[ 86 = \frac{44}{.4}-4! \]
\[ 87 = \left(4!\times4\right)-\frac{4}{.\dot{4}} \]
\[ 88 = 44+44 \]
\[ 89 = \frac{\left(\frac{4!}{\sqrt{.\dot{4}}}-.4\right)}{.4} \]
\[ 90 = \frac{\left(44-4\right)}{.\dot{4}} \]
\[ 91 = \left(4!\times4\right)-\frac{\sqrt{4}}{.4} \]
\[ 92 = 4+\left(\sqrt{4}\times44\right) \]
\[ 93 = \left(4!\times4\right)-\sqrt{\frac{4}{.\dot{4}}} \]
\[ 94 = \frac{4}{\left(.\dot{4}-.4\right)}+4 \]
\[ 95 = \frac{44}{.\dot{4}}-4 \]
\[ 96 = \sqrt{4}\times\left(4+44\right) \]
\[ 97 = \frac{44}{.\dot{4}}-\sqrt{4} \]
\[ 98 = \frac{\left(44-.\dot{4}\right)}{.\dot{4}} \]
\[ 99 = \frac{44}{\sqrt{\left(.\dot{4}\times.\dot{4}\right)}} \]
\[ 100 = \frac{\left(44-4\right)}{.4} \]
\[ 101 = \sqrt{4}+\frac{44}{.\dot{4}} \]
\[ 102 = \left({4}^{4}\times.4\right)-.4 \]
\[ 103 = \frac{44}{.\dot{4}}+4 \]
\[ 104 = 44+\frac{4!}{.4} \]
\[ 105 = \frac{\left(44-\sqrt{4}\right)}{.4} \]
\[ 106 = \frac{44}{.4}-4 \]
\[ 107 = \frac{\left(4!+\left(4!-.\dot{4}\right)\right)}{.\dot{4}} \]
\[ 108 = \frac{\left(4+44\right)}{.\dot{4}} \]
\[ 109 = \frac{\left(44-.4\right)}{.4} \]
\[ 110 = \frac{44}{\sqrt{\left(.4\times.4\right)}} \]
\[ 111 = \frac{444}{4} \]
\[ 112 = \frac{44}{.4}+\sqrt{4} \]
\[ \]
\[ 114 = 4+\frac{44}{.4} \]
\[ 115 = \frac{\left(\sqrt{4}+44\right)}{.4} \]
\[ 116 = \left(4\times\left(4+4!\right)\right)+4 \]
\[ 117 = \frac{\left(\left(4!\times\sqrt{4}\right)+4\right)}{.\dot{4}} \]
\[ 118 = \frac{\left(\left(4!-.4\right)\times\sqrt{4}\right)}{.4} \]
\[ 119 = \frac{\left(\left(4!\times\sqrt{4}\right)-.4\right)}{.4} \]
\[ 120 = \frac{\left(4+44\right)}{.4} \]
\[ 121 = {\frac{44}{4}}^{\sqrt{4}} \]
\[ 122 = \frac{\left(.4+4!\right)}{.4}\times\sqrt{4} \]
\[ 123 = 4!+\frac{44}{.\dot{4}} \]
\[ 124 = \frac{{4}^{4}}{\sqrt{4}}-4 \]
\[ 125 = \frac{\left(4!-4\right)}{\left(.4\times.4\right)} \]
\[ 126 = \frac{\left({4}^{4}-4\right)}{\sqrt{4}} \]
\[ 127 = \frac{\left({4}^{4}-\sqrt{4}\right)}{\sqrt{4}} \]
\[ 128 = \left(4+4\right)\times\left(4\times4\right) \]
\[ 129 = \frac{\left(\sqrt{4}+{4}^{4}\right)}{\sqrt{4}} \]
\[ 130 = \frac{\left(4+{4}^{4}\right)}{\sqrt{4}} \]
\[ 131 = \frac{4!}{\left(.4\times.\dot{4}\right)}-4 \]
\[ 132 = 44\times\sqrt{\frac{4}{.\dot{4}}} \]
\[ 133 = \frac{4!}{\left(.4\times.\dot{4}\right)}-\sqrt{4} \]
\[ 134 = \frac{44}{.4}+4! \]
\[ 135 = \frac{4!}{\left(\left(.\dot{4}-.4\right)\times4\right)} \]
\[ 136 = \sqrt{4}\times\left(4!+44\right) \]
\[ 137 = \frac{4!}{\left(.4\times.\dot{4}\right)}+\sqrt{4} \]
\[ 138 = \frac{\left(\left(4!\times4\right)-4\right)}{\sqrt{.\dot{4}}} \]
\[ 139 = 4+\frac{4!}{\left(.4\times.\dot{4}\right)} \]
\[ 140 = 44+\left(4!\times4\right) \]
\[ 141 = \frac{\left(\left(4!\times4\right)-\sqrt{4}\right)}{\sqrt{.\dot{4}}} \]
\[ 142 = \left(\left(4+\sqrt{4}\right)\times4!\right)-\sqrt{4} \]
\[ 143 = \frac{\left(\left(4!\times4!\right)-4\right)}{4} \]
\[ 144 = 4\times\left(\frac{4}{.\dot{4}}\times4\right) \]
\[ 145 = \frac{\left(4+\frac{4!}{.\dot{4}}\right)}{.4} \]
\[ 146 = \frac{4!}{\left(.4\times.4\right)}-4 \]
\[ 147 = \frac{\left(\sqrt{4}+\left(4!\times4\right)\right)}{\sqrt{.\dot{4}}} \]
\[ 148 = 4+\left(\left(4+\sqrt{4}\right)\times4!\right) \]
\[ 149 = \frac{\left(\frac{4!}{.4}-.4\right)}{.4} \]
\[ 150 = \frac{\left(4+\left(4!\times4\right)\right)}{\sqrt{.\dot{4}}} \]
\[ 151 = \frac{\left(.4+\frac{4!}{.4}\right)}{.4} \]
\[ 152 = \left(4\times44\right)-4! \]
\[ 153 = \frac{\left(4!+44\right)}{.\dot{4}} \]
\[ 154 = \frac{4!}{\left(.4\times.4\right)}+4 \]
\[ 155 = \frac{\left(\sqrt{4}+\frac{4!}{.4}\right)}{.4} \]
\[ 156 = \frac{4!}{.4}+\left(4!\times4\right) \]
\[ 158 = \frac{{\sqrt{\sqrt{\sqrt{4}}}}^{4!}}{.4}-\sqrt{4} \]
\[ 159 = \frac{4!}{\left(.4\times.\dot{4}\right)}+4! \]
\[ 160 = \left(44-4\right)\times4 \]
\[ 161 = \frac{\left(.4+{\sqrt{\sqrt{\sqrt{4}}}}^{4!}\right)}{.4} \]
\[ 162 = \sqrt{\left(4\times{\frac{4}{.\dot{4}}}^{4}\right)} \]
\[ 164 = \frac{{\sqrt{\sqrt{\sqrt{4}}}}^{4!}}{.4}+4 \]
\[ 165 = \frac{44}{\left(\sqrt{.\dot{4}}-.4\right)} \]
\[ 166 = \]
\[ 168 = \left(44-\sqrt{4}\right)\times4 \]
\[ 169 = \sqrt{{\left(4+\frac{4}{.\dot{4}}\right)}^{4}} \]
\[ 170 = \frac{\left(4!+44\right)}{.4} \]
\[ 172 = \left(4\times44\right)-4 \]
\[ 174 = \left(4\times44\right)-\sqrt{4} \]
\[ 175 = \frac{\left(4+4!\right)}{\left(.4\times.4\right)} \]
\[ 176 = 44\times\sqrt{\left(4\times4\right)} \]
\[ 178 = \left(4\times44\right)+\sqrt{4} \]
\[ 180 = \left(4\times44\right)+4 \]
\[ 181 = \]
\[ 184 = 4\times\left(\sqrt{4}+44\right) \]
\[ 188 = \left(4!\times\left(4+4\right)\right)-4 \]
\[ 189 = \frac{\left(4!+\frac{4!}{.4}\right)}{.\dot{4}} \]
\[ 190 = \left(4!\times\left(4+4\right)\right)-\sqrt{4} \]
\[ 192 = 4\times\left(4+44\right) \]
\[ 194 = \left(4!\times\left(4+4\right)\right)+\sqrt{4} \]
\[ 195 = \frac{\left(\frac{4!}{.\dot{4}}+4!\right)}{.4} \]
\[ 196 = 4+\left(4!\times\left(4+4\right)\right) \]
\[ 198 = \frac{44}{.\dot{4}}\times\sqrt{4} \]
\[ 200 = \left(4\times44\right)+4! \]
\[ 202 = {4}^{4}-\frac{4!}{.\dot{4}} \]
\[ 204 = 4!\times\left(4+\frac{\sqrt{4}}{.\dot{4}}\right) \]
\[ 207 = \frac{\left(\left(4!\times4\right)-4\right)}{.\dot{4}} \]
\[ 208 = \left(\left(4!\times\sqrt{4}\right)+4\right)\times4 \]
\[ 210 = \frac{\left(4!-\sqrt{.\dot{4}}\right)}{.\dot{4}}\times4 \]
\[ 212 = {4}^{4}-44 \]
\[ 214 = \left(4\times\frac{4!}{.\dot{4}}\right)-\sqrt{4} \]
\[ 215 = \frac{\left(\left(4!\times4\right)-.\dot{4}\right)}{.\dot{4}} \]
\[ 216 = \frac{4!}{.4}\times\left(4-.4\right) \]
\[ 217 = \frac{\left(\left(4!\times4\right)+.\dot{4}\right)}{.\dot{4}} \]
\[ 218 = \sqrt{4}+\left(4\times\frac{4!}{.\dot{4}}\right) \]
\[ 220 = \frac{\left(\sqrt{4}\times44\right)}{.4} \]
\[ 222 = \frac{444}{\sqrt{4}} \]
\[ 224 = \left(4+4!\right)\times\left(4+4\right) \]
\[ 225 = \frac{4}{\left(.4\times\left(.\dot{4}-.4\right)\right)} \]
\[ 228 = \left({4}^{4}-4!\right)-4 \]
\[ 230 = \frac{\left(\left(4!\times4\right)-4\right)}{.4} \]
\[ 232 = 4\times\left(4+\frac{4!}{.\dot{4}}\right) \]
\[ 234 = \frac{\left(\sqrt{4}+4!\right)}{.\dot{4}}\times4 \]
\[ 235 = \frac{\left(\left(4!\times4\right)-\sqrt{4}\right)}{.4} \]
\[ 236 = \frac{\left(4!\times4\right)}{.4}-4 \]
\[ 238 = \frac{\left(4!\times4\right)}{.4}-\sqrt{4} \]
\[ 239 = \frac{\left(\left(4!\times4\right)-.4\right)}{.4} \]
\[ 240 = {4}^{4}-\left(4\times4\right) \]
\[ 241 = \frac{\left(\left(4!\times4\right)+.4\right)}{.4} \]
\[ 242 = \frac{\left(4!\times4\right)}{.4}+\sqrt{4} \]
\[ 243 = \frac{4!}{\left(.\dot{4}\times.\dot{4}\right)}\times\sqrt{4} \]
\[ 244 = \frac{\left(4!\times4\right)}{.4}+4 \]
\[ 245 = \frac{\left(\sqrt{4}+\left(4!\times4\right)\right)}{.4} \]
\[ 246 = {4}^{4}-\frac{4}{.4} \]
\[ 247 = {4}^{4}-\frac{4}{.\dot{4}} \]
\[ 248 = {4}^{4}-\left(4+4\right) \]
\[ 250 = \frac{\left(4+\left(4!\times4\right)\right)}{.4} \]
\[ 251 = {4}^{4}-\frac{\sqrt{4}}{.4} \]
\[ 252 = \frac{\left(4\times\left(4+4!\right)\right)}{.\dot{4}} \]
\[ 253 = {4}^{4}-\sqrt{\frac{4}{.\dot{4}}} \]
\[ 254 = \left({4}^{4}-4\right)+\sqrt{4} \]
\[ 255 = {4}^{4}-\frac{4}{4} \]
\[ 256 = \left(\left(4\times4\right)\times4\right)\times4 \]
\[ 257 = {4}^{4}+\frac{4}{4} \]
\[ 258 = \left(4+{4}^{4}\right)-\sqrt{4} \]
\[ 259 = {4}^{4}+\sqrt{\frac{4}{.\dot{4}}} \]
\[ 260 = \sqrt{{\left(4\times4\right)}^{4}}+4 \]
\[ 261 = {4}^{4}+\frac{\sqrt{4}}{.4} \]
\[ 262 = \left(4+\sqrt{4}\right)+{4}^{4} \]
\[ 264 = 44\times\left(4+\sqrt{4}\right) \]
\[ 265 = \frac{4}{.\dot{4}}+{4}^{4} \]
\[ 266 = \frac{4}{.4}+{4}^{4} \]
\[ 268 = \frac{4!}{\sqrt{4}}+{4}^{4} \]
\[ 270 = \frac{\left(\left(4!\times4\right)+4!\right)}{.\dot{4}} \]
\[ 272 = 4\times\left(4!+44\right) \]
\[ 275 = \frac{44}{\left(.4\times.4\right)} \]
\[ 276 = {4}^{4}+\left(4!-4\right) \]
\[ 278 = \left({4}^{4}-\sqrt{4}\right)+4! \]
\[ 280 = \left(4+4!\right)\times\frac{4}{.4} \]
\[ 282 = 4!+\left(\sqrt{4}+{4}^{4}\right) \]
\[ 284 = \left(4+{4}^{4}\right)+4! \]
\[ 286 = \frac{\left(\left(4!\times4!\right)-4\right)}{\sqrt{4}} \]
\[ 287 = \frac{\left(\left(4!\times4!\right)-\sqrt{4}\right)}{\sqrt{4}} \]
\[ 288 = 4!\times\left(4+\left(4+4\right)\right) \]
\[ 289 = \frac{\left(\sqrt{4}+\left(4!\times4!\right)\right)}{\sqrt{4}} \]
\[ 290 = \frac{\left(\left(4!\times4!\right)+4\right)}{\sqrt{4}} \]
\[ 292 = 4+\sqrt{\frac{{4!}^{4}}{4}} \]
\[ 294 = {\sqrt{\left(\frac{\sqrt{4!}}{.4}+\sqrt{4!}\right)}}^{4} \]
\[ 296 = 444\times\sqrt{.\dot{4}} \]
\[ 300 = {4}^{4}+44 \]
\[ 304 = 4!+\left({4}^{4}+4!\right) \]
\[ 310 = \frac{4!}{.\dot{4}}+{4}^{4} \]
\[ 312 = \left(4+\frac{4}{.\dot{4}}\right)\times4! \]
\[ 316 = {4}^{4}+\frac{4!}{.4} \]
\[ 320 = \left(4\times4\right)\times\left(4!-4\right) \]
\[ 324 = \frac{{\left(4+\sqrt{4}\right)}^{4}}{4} \]
\[ 336 = 4!\times\left(\frac{4}{.4}+4\right) \]
\[ 338 = \sqrt{\frac{{\left(\sqrt{4}+4!\right)}^{4}}{4}} \]
\[ 343 = {\sqrt{\sqrt{\sqrt{\frac{\left(4+4!\right)}{4}}}}}^{4!} \]
\[ 348 = \frac{\left({4}^{4}-4!\right)}{\sqrt{.\dot{4}}} \]
\[ 351 = \frac{4!}{\sqrt{\sqrt{{\sqrt{.4}}^{4!}}}}-4! \]
\[ 352 = 44\times\left(4+4\right) \]
\[ 360 = \frac{4}{\left(.\dot{4}-.4\right)}\times4 \]
\[ 361 = {\sqrt{\left(4!-\frac{\sqrt{4}}{.4}\right)}}^{4} \]
\[ 368 = \left(\left(4!\times4\right)-4\right)\times4 \]
\[ 371 = \frac{4!}{\sqrt{\sqrt{{\sqrt{.4}}^{4!}}}}-4 \]
\[ 373 = \frac{4!}{\sqrt{\sqrt{{\sqrt{.4}}^{4!}}}}-\sqrt{4} \]
\[ 375 = \frac{4!}{\left(.4\times\left(.4\times.4\right)\right)} \]
\[ 376 = \left(\left(4!\times4\right)-\sqrt{4}\right)\times4 \]
\[ 377 = \frac{4!}{\sqrt{\sqrt{{\sqrt{.4}}^{4!}}}}+\sqrt{4} \]
\[ 378 = \frac{\left({4}^{4}-4\right)}{\sqrt{.\dot{4}}} \]
\[ 379 = 4+\frac{4!}{\sqrt{\sqrt{{\sqrt{.4}}^{4!}}}} \]
\[ 380 = \left(\left(4!\times4\right)\times4\right)-4 \]
\[ 381 = \frac{\left({4}^{4}-\sqrt{4}\right)}{\sqrt{.\dot{4}}} \]
\[ 382 = \left(\left(4!\times4\right)\times4\right)-\sqrt{4} \]
\[ 383 = \frac{\left({4}^{4}-\sqrt{.\dot{4}}\right)}{\sqrt{.\dot{4}}} \]
\[ 384 = \left(\sqrt{\left(4\times4\right)}\times4\right)\times4! \]
\[ 385 = \frac{\left({4}^{4}+\sqrt{.\dot{4}}\right)}{\sqrt{.\dot{4}}} \]
\[ 386 = \sqrt{4}+\left(\left(4!\times4\right)\times4\right) \]
\[ 387 = \frac{\left(\sqrt{4}+{4}^{4}\right)}{\sqrt{.\dot{4}}} \]
\[ 388 = \left(\left(4!\times4\right)\times4\right)+4 \]
\[ 390 = \frac{\left(4+{4}^{4}\right)}{\sqrt{.\dot{4}}} \]
\[ 392 = 4\times\left(\sqrt{4}+\left(4!\times4\right)\right) \]
\[ 396 = 44\times\frac{4}{.\dot{4}} \]
\[ 398 = {\left(4!-4\right)}^{\sqrt{4}}-\sqrt{4} \]
\[ 399 = 4!+\frac{4!}{\sqrt{\sqrt{{\sqrt{.4}}^{4!}}}} \]
\[ 400 = 4\times\left(4+\left(4!\times4\right)\right) \]
\[ 402 = \sqrt{4}+{\left(4!-4\right)}^{\sqrt{4}} \]
\[ 404 = \sqrt{{\left(4!-4\right)}^{4}}+4 \]
\[ 408 = \left(\left(4!\times4\right)\times4\right)+4! \]
\[ 416 = \left(4\times\left(\sqrt{4}+4!\right)\right)\times4 \]
\[ 420 = 444-4! \]
\[ 424 = \sqrt{{\left(4!-4\right)}^{4}}+4! \]
\[ 432 = \frac{4!}{.\dot{4}}\times\left(4+4\right) \]
\[ 440 = 444-4 \]
\[ 441 = \sqrt{{\left(4!-\sqrt{\frac{4}{.\dot{4}}}\right)}^{4}} \]
\[ 442 = 444-\sqrt{4} \]
\[ 444 = {\sqrt{\sqrt{444}}}^{4} \]
\[ 446 = 444+\sqrt{4} \]
\[ 448 = 444+4 \]
\[ 450 = \frac{\left(4-4!\right)}{\left(.4-.\dot{4}\right)} \]
\[ 456 = \left(4!\times\left(4!-4\right)\right)-4! \]
\[ 460 = \sqrt{{\left(4!-\sqrt{4}\right)}^{4}}-4! \]
\[ 464 = \sqrt{4}\times\left({4}^{4}-4!\right) \]
\[ 468 = 4!+444 \]
\[ 472 = \left(4-4!\right)\times\left(.4-4!\right) \]
\[ 476 = \left(4!\times\left(4!-4\right)\right)-4 \]
\[ 478 = \left(4!\times\left(4!-4\right)\right)-\sqrt{4} \]
\[ 480 = 4!\times\left(4+\left(4\times4\right)\right) \]
\[ 482 = \left(4!\times\left(4!-4\right)\right)+\sqrt{4} \]
\[ 484 = \frac{{\sqrt{44}}^{4}}{4} \]
\[ 486 = \frac{\left(4!\times4\right)}{\left(.\dot{4}\times.\dot{4}\right)} \]
\[ 488 = \left(.4+4!\right)\times\left(4!-4\right) \]
\[ 495 = \frac{\left(\sqrt{4}-4!\right)}{\left(.4-.\dot{4}\right)} \]
\[ 496 = 4!\times\left(\left(\sqrt{.\dot{4}}-4\right)+4!\right) \]
\[ 500 = \sqrt{\frac{{\sqrt{\sqrt{\frac{4}{.4}}}}^{4!}}{4}} \]
\[ 504 = \left({4}^{4}-4\right)\times\sqrt{4} \]
\[ 508 = \left(\sqrt{4}\times{4}^{4}\right)-4 \]
\[ 510 = \left(\sqrt{4}\times{4}^{4}\right)-\sqrt{4} \]
\[ 512 = {4}^{4}+{4}^{4} \]
\[ 514 = \sqrt{4}+\left(\sqrt{4}\times{4}^{4}\right) \]
\[ 516 = \left(\sqrt{4}\times{4}^{4}\right)+4 \]
\[ 520 = \sqrt{4}\times\left(4+{4}^{4}\right) \]
\[ 522 = \frac{\left({4}^{4}-4!\right)}{.\dot{4}} \]
\[ 524 = \left(\left(4!-\sqrt{4}\right)\times4!\right)-4 \]
\[ 525 = \frac{\left(4!-\sqrt{.\dot{4}}\right)}{\left(.\dot{4}-.4\right)} \]
\[ 526 = \left(\left(4!-\sqrt{4}\right)\times4!\right)-\sqrt{4} \]
\[ 528 = \frac{44}{\sqrt{4}}\times4! \]
\[ 529 = \sqrt{{\left(4!-\frac{4}{4}\right)}^{4}} \]
\[ 530 = \frac{\left(.\dot{4}-4!\right)}{\left(.4-.\dot{4}\right)} \]
\[ 531 = \frac{\left(4!-.4\right)}{\left(.\dot{4}-.4\right)} \]
\[ 532 = \left(4!\times4!\right)-44 \]
\[ 536 = \frac{4!}{\left(.\dot{4}-.4\right)}-4 \]
\[ 538 = \frac{4!}{\left(.\dot{4}-.4\right)}-\sqrt{4} \]
\[ 540 = \frac{\left(4!\times4\right)}{\left(.4\times.\dot{4}\right)} \]
\[ 542 = \sqrt{4}-\frac{4!}{\left(.4-.\dot{4}\right)} \]
\[ 544 = 4-\frac{4!}{\left(.4-.\dot{4}\right)} \]
\[ 548 = \left(\left(4!\times4!\right)-4!\right)-4 \]
\[ 549 = \frac{\left(.4+4!\right)}{\left(.\dot{4}-.4\right)} \]
\[ 550 = \frac{\left(.\dot{4}+4!\right)}{\left(.\dot{4}-.4\right)} \]
\[ 552 = \left(4!-\frac{4}{4}\right)\times4! \]
\[ 554 = \left(\left(4!\times4!\right)-4!\right)+\sqrt{4} \]
\[ 555 = \frac{\left(\sqrt{.\dot{4}}+4!\right)}{\left(.\dot{4}-.4\right)} \]
\[ 556 = \left(\left(4!\times4!\right)-4!\right)+4 \]
\[ 558 = \left(\left(4!-\sqrt{.\dot{4}}\right)\times4!\right)-\sqrt{4} \]
\[ 560 = \left(4!\times4!\right)-\left(4\times4\right) \]
\[ 562 = \left(\left(4!-\sqrt{.\dot{4}}\right)\times4!\right)+\sqrt{4} \]
\[ 564 = 4!+\frac{4!}{\left(.\dot{4}-.4\right)} \]
\[ 566 = \left(4!\times4!\right)-\frac{4}{.4} \]
\[ 567 = \frac{\left({4}^{4}-4\right)}{.\dot{4}} \]
\[ 568 = \left(4!\times4!\right)-\left(4+4\right) \]
\[ 570 = \left(\left(4!\times4!\right)-\sqrt{4}\right)-4 \]
\[ 571 = \left(4!\times4!\right)-\frac{\sqrt{4}}{.4} \]
\[ 572 = \frac{{4}^{4}}{.\dot{4}}-4 \]
\[ 573 = \left(4!\times4!\right)-\sqrt{\frac{4}{.\dot{4}}} \]
\[ 574 = \frac{{4}^{4}}{.\dot{4}}-\sqrt{4} \]
\[ 575 = \frac{\left({4}^{4}-.\dot{4}\right)}{.\dot{4}} \]
\[ 576 = \left(\left(4!\times4!\right)+4\right)-4 \]
\[ 577 = \frac{\left({4}^{4}+.\dot{4}\right)}{.\dot{4}} \]
\[ 578 = \sqrt{4}+\frac{{4}^{4}}{.\dot{4}} \]
\[ 579 = \sqrt{\frac{4}{.\dot{4}}}+\left(4!\times4!\right) \]
\[ 580 = 4+\frac{{4}^{4}}{.\dot{4}} \]
\[ 581 = \frac{\sqrt{4}}{.4}+\left(4!\times4!\right) \]
\[ 582 = \left(\left(4!\times4!\right)+4\right)+\sqrt{4} \]
\[ 584 = \left(4!\times4!\right)+\left(4+4\right) \]
\[ 585 = \frac{\left(4+{4}^{4}\right)}{.\dot{4}} \]
\[ 586 = \frac{4}{.4}+\left(4!\times4!\right) \]
\[ 588 = 4!\times\left(\frac{\sqrt{4}}{4}+4!\right) \]
\[ 590 = \left(4!\times\left(\sqrt{.\dot{4}}+4!\right)\right)-\sqrt{4} \]
\[ 592 = \left(4!\times4!\right)+\left(4\times4\right) \]
\[ 594 = \left(4!\times\left(\sqrt{.\dot{4}}+4!\right)\right)+\sqrt{4} \]
\[ 596 = \left(\left(4!\times4!\right)+4!\right)-4 \]
\[ 598 = 4!-\left(\sqrt{4}-\left(4!\times4!\right)\right) \]
\[ 600 = \frac{4}{\left(.4\times.4\right)}\times4! \]
\[ 601 = {\frac{\sqrt{4}}{.4}}^{4}-4! \]
\[ 602 = \left(\left(4!\times4!\right)+4!\right)+\sqrt{4} \]
\[ 604 = \left(\left(4!\times4!\right)+4\right)+4! \]
\[ 608 = \left(4!+\sqrt{\left(.\dot{4}\times4\right)}\right)\times4! \]
\[ 612 = 4!\times\left(\frac{\sqrt{.\dot{4}}}{.\dot{4}}+4!\right) \]
\[ 616 = \frac{{4}^{4}}{.4}-4! \]
\[ 620 = 44+\left(4!\times4!\right) \]
\[ 621 = {\frac{\sqrt{4}}{.4}}^{4}-4 \]
\[ 622 = \left(\left(\sqrt{4}+4!\right)\times4!\right)-\sqrt{4} \]
\[ 623 = {\frac{\sqrt{4}}{.4}}^{4}-\sqrt{4} \]
\[ 624 = 4!\times\left(4-\left(\sqrt{4}-4!\right)\right) \]
\[ 625 = {\left(\frac{4}{4}+4\right)}^{4} \]
\[ 626 = \sqrt{4}+\left(\left(\sqrt{4}+4!\right)\times4!\right) \]
\[ 627 = \sqrt{4}+{\frac{\sqrt{4}}{.4}}^{4} \]
\[ 628 = \left(\left(\sqrt{4}+4!\right)\times4!\right)+4 \]
\[ 629 = 4+{\frac{\sqrt{4}}{.4}}^{4} \]
\[ 630 = \frac{\left({4}^{4}-4\right)}{.4} \]
\[ 635 = \frac{\left({4}^{4}-\sqrt{4}\right)}{.4} \]
\[ 636 = \frac{{4}^{4}}{.4}-4 \]
\[ 638 = \frac{{4}^{4}}{.4}-\sqrt{4} \]
\[ 639 = \frac{\left({4}^{4}-.4\right)}{.4} \]
\[ 640 = \frac{{\left(4\times4\right)}^{\sqrt{4}}}{.4} \]
\[ 641 = \frac{\left(.4+{4}^{4}\right)}{.4} \]
\[ 642 = \frac{{4}^{4}}{.4}+\sqrt{4} \]
\[ 644 = 4+\frac{{4}^{4}}{.4} \]
\[ 645 = \frac{\left(\sqrt{4}+{4}^{4}\right)}{.4} \]
\[ 648 = \frac{4!}{\left(.\dot{4}+.\dot{4}\right)}\times4! \]
\[ 649 = {\frac{\sqrt{4}}{.4}}^{4}+4! \]
\[ 650 = \frac{\left(4+{4}^{4}\right)}{.4} \]
\[ 652 = {\sqrt{\left(\sqrt{4}+4!\right)}}^{4}-4! \]
\[ 656 = \left(\left(4-\sqrt{.\dot{4}}\right)+4!\right)\times4! \]
\[ 664 = 4!+\frac{{4}^{4}}{.4} \]
\[ 666 = \frac{444}{\sqrt{.\dot{4}}} \]
\[ 668 = \left(\left(4+4!\right)\times4!\right)-4 \]
\[ 670 = \left(\left(4+4!\right)\times4!\right)-\sqrt{4} \]
\[ 672 = \sqrt{\left(4\times4\right)}!\times\left(4+4!\right) \]
\[ 674 = \left(\left(4+4!\right)\times4!\right)+\sqrt{4} \]
\[ 676 = \left(\left(4+4!\right)\times4!\right)+4 \]
\[ 678 = \sqrt{{\left(\sqrt{4}+4!\right)}^{4}}+\sqrt{4} \]
\[ 680 = \sqrt{{\left(\sqrt{4}+4!\right)}^{4}}+4 \]
\[ 684 = \left(\frac{\sqrt{4}}{.\dot{4}}+4!\right)\times4! \]
\[ 688 = 4!\times\left(\left(4+\sqrt{.\dot{4}}\right)+4!\right) \]
\[ 696 = \left(\left(4+4!\right)\times4!\right)+4! \]
\[ 700 = \frac{\left({4}^{4}+4!\right)}{.4} \]
\[ 704 = 4\times\left(4\times44\right) \]
\[ 705 = \sqrt{{\sqrt{\sqrt{\frac{4}{.\dot{4}}}}}^{4!}}-4! \]
\[ 720 = \frac{\left(4!\times4!\right)}{\left(.4+.4\right)} \]
\[ 725 = \sqrt{{\sqrt{\sqrt{\frac{4}{.\dot{4}}}}}^{4!}}-4 \]
\[ 726 = \sqrt{\frac{{\left(4!-\sqrt{4}\right)}^{4}}{.\dot{4}}} \]
\[ 727 = \sqrt{{\sqrt{\sqrt{\frac{4}{.\dot{4}}}}}^{4!}}-\sqrt{4} \]
\[ 728 = \left(4+4!\right)\times\left(\sqrt{4}+4!\right) \]
\[ 729 = {\frac{4}{.\dot{4}}}^{\sqrt{\frac{4}{.\dot{4}}}} \]
\[ 731 = \sqrt{4}+{\sqrt{\sqrt{\sqrt{\frac{4}{.\dot{4}}}}}}^{4!} \]
\[ 733 = 4+\sqrt{{\sqrt{\sqrt{\frac{4}{.\dot{4}}}}}^{4!}} \]
\[ 750 = 4!\times\sqrt{\frac{4}{\sqrt{{\sqrt{.4}}^{4!}}}} \]
\[ 753 = \sqrt{{\sqrt{\sqrt{\frac{4}{.\dot{4}}}}}^{4!}}+4! \]
\[ 760 = \sqrt{{\left(4+4!\right)}^{4}}-4! \]
\[ 768 = 4\times\left(4!\times\left(4+4\right)\right) \]
\[ 780 = {\left(4+4!\right)}^{\sqrt{4}}-4 \]
\[ 782 = {\left(4+4!\right)}^{\sqrt{4}}-\sqrt{4} \]
\[ 784 = \left(4+4!\right)\times\left(4+4!\right) \]
\[ 786 = \sqrt{4}+\sqrt{{\left(4+4!\right)}^{4}} \]
\[ 788 = {\left(4+4!\right)}^{\sqrt{4}}+4 \]
\[ 792 = 4!\times\left(\frac{4}{.\dot{4}}+4!\right) \]
\[ 800 = \sqrt{\left({\left(4!-4\right)}^{4}\times4\right)} \]
\[ 808 = {\left(4+4!\right)}^{\sqrt{4}}+4! \]
\[ 810 = \frac{4!}{\left(\left(.\dot{4}-.4\right)\times\sqrt{.\dot{4}}\right)} \]
\[ 816 = \left(4!+\frac{4}{.4}\right)\times4! \]
\[ 828 = \frac{\left(\left(4!\times4!\right)-4!\right)}{\sqrt{.\dot{4}}} \]
\[ 832 = {4}^{4}+\left(4!\times4!\right) \]
\[ 840 = \sqrt{\frac{{4!}^{4}}{.\dot{4}}}-4! \]
\[ 841 = {\sqrt{\left(\frac{\sqrt{4}}{.4}+4!\right)}}^{4} \]
\[ 848 = 4!\times\frac{\left(4!-.\dot{4}\right)}{\sqrt{.\dot{4}}} \]
\[ 858 = \frac{\left(\left(4!\times4!\right)-4\right)}{\sqrt{.\dot{4}}} \]
\[ 860 = \sqrt{\frac{{4!}^{4}}{.\dot{4}}}-4 \]
\[ 861 = \frac{\left(\left(4!\times4!\right)-\sqrt{4}\right)}{\sqrt{.\dot{4}}} \]
\[ 862 = \sqrt{\frac{{4!}^{4}}{.\dot{4}}}-\sqrt{4} \]
\[ 863 = \frac{\left(\left(4!\times4!\right)-\sqrt{.\dot{4}}\right)}{\sqrt{.\dot{4}}} \]
\[ 864 = \frac{\left(\left(4!\times4\right)\times4\right)}{.\dot{4}} \]
\[ 865 = \frac{\left(\left(4!\times4!\right)+\sqrt{.\dot{4}}\right)}{\sqrt{.\dot{4}}} \]
\[ 866 = \sqrt{\frac{{4!}^{4}}{.\dot{4}}}+\sqrt{4} \]
\[ 867 = \frac{\left(\sqrt{4}+\left(4!\times4!\right)\right)}{\sqrt{.\dot{4}}} \]
\[ 868 = \sqrt{\frac{{4!}^{4}}{.\dot{4}}}+4 \]
\[ 870 = \frac{\left(\left(4!\times4!\right)+4\right)}{\sqrt{.\dot{4}}} \]
\[ 880 = \left(4!-4\right)\times44 \]
\[ 887 = \left(\sqrt{\sqrt{.\dot{4}}}-4\right)+{4}^{\sqrt{4!}} \]
\[ 888 = 444\times\sqrt{4} \]
\[ 889 = \left(\sqrt{\sqrt{.\dot{4}}}+{4}^{\sqrt{4!}}\right)-\sqrt{4} \]
\[ 891 = {4}^{\sqrt{4!}}+\frac{4}{\sqrt{4!}} \]
\[ 893 = {4}^{\sqrt{4!}}+\left(\sqrt{4}+\sqrt{\sqrt{.\dot{4}}}\right) \]
\[ 895 = \left(\sqrt{\sqrt{.\dot{4}}}+4\right)+{4}^{\sqrt{4!}} \]
\[ 896 = \left(\left(\sqrt{4}-.\dot{4}\right)\times4!\right)\times4! \]
\[ 900 = \frac{\sqrt{{\left(4!-4\right)}^{4}}}{.\dot{4}} \]
\[ 912 = \left(\sqrt{4}+\frac{4!}{\sqrt{.\dot{4}}}\right)\times4! \]
\[ 915 = \left(4!+{4}^{\sqrt{4!}}\right)+\sqrt{\sqrt{.\dot{4}}} \]
\[ 928 = 4\times\left({4}^{4}-4!\right) \]
\[ 936 = \frac{\left(\sqrt{4}+4!\right)}{\sqrt{.\dot{4}}}\times4! \]
\[ 960 = \left(44-4\right)\times4! \]
\[ 968 = \left(4!-\sqrt{4}\right)\times44 \]
\[ 976 = \sqrt{{\sqrt{\sqrt{\frac{4}{.4}}}}^{4!}}-4! \]
\[ 990 = \frac{44}{\left(.\dot{4}-.4\right)} \]
\[ 996 = {\sqrt{\sqrt{\sqrt{\frac{4}{.4}}}}}^{4!}-4 \]
\[ 998 = \sqrt{{\sqrt{\sqrt{\frac{4}{.4}}}}^{4!}}-\sqrt{4} \]
\[ 999 = \frac{444}{.\dot{4}} \]
\[ 1000 = \left(4\times{4}^{4}\right)-4! \]
\[ 1002 = \sqrt{4}+{\sqrt{\sqrt{\sqrt{\frac{4}{.4}}}}}^{4!} \]
\[ 1004 = {\sqrt{\sqrt{\sqrt{\frac{4}{.4}}}}}^{4!}+4 \]
\[ 1008 = 4\times\left({4}^{4}-4\right) \]
\[ 1014 = \frac{\sqrt{{\left(\sqrt{4}+4!\right)}^{4}}}{\sqrt{.\dot{4}}} \]
\[ 1016 = \left({4}^{4}-\sqrt{4}\right)\times4 \]
\[ 1018 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-4!\right)}{4} \]
\[ 1020 = \left(4\times{4}^{4}\right)-4 \]
\[ 1022 = \left(4\times{4}^{4}\right)-\sqrt{4} \]
\[ 1023 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-4\right)}{4} \]
\[ 1024 = \frac{{\left(4+4\right)}^{4}}{4} \]
\[ 1025 = \frac{\left(4+{\sqrt{\sqrt{4}}}^{4!}\right)}{4} \]
\[ 1026 = \left(4\times{4}^{4}\right)+\sqrt{4} \]
\[ 1028 = 4+\left(4\times{4}^{4}\right) \]
\[ 1030 = \frac{\left(4!+{\sqrt{\sqrt{4}}}^{4!}\right)}{4} \]
\[ 1032 = \left(4!\times44\right)-4! \]
\[ 1040 = 4\times\left(4+{4}^{4}\right) \]
\[ 1048 = \left(4\times{4}^{4}\right)+4! \]
\[ 1052 = \left(4!\times44\right)-4 \]
\[ 1054 = \left(4!\times44\right)-\sqrt{4} \]
\[ 1056 = \sqrt{\left(4\times4\right)}!\times44 \]
\[ 1058 = \left(4!\times44\right)+\sqrt{4} \]
\[ 1060 = \left(4!\times44\right)+4 \]
\[ 1072 = \left(44+\sqrt{.\dot{4}}\right)\times4! \]
\[ 1080 = 4!+\left(4!\times44\right) \]
\[ 1089 = {\sqrt{\left(\frac{4}{.\dot{4}}+4!\right)}}^{4} \]
\[ 1104 = 4!\times\left(\sqrt{4}+44\right) \]
\[ 1110 = \frac{444}{.4} \]
\[ 1120 = \left({4}^{4}+4!\right)\times4 \]
\[ 1128 = \left(\left(4!\times4!\right)\times\sqrt{4}\right)-4! \]
\[ 1136 = 4!\times\left(4!+\left(4!-\sqrt{.\dot{4}}\right)\right) \]
\[ 1144 = \left(\sqrt{4}+4!\right)\times44 \]
\[ 1148 = \left(\left(4!\times4!\right)\times\sqrt{4}\right)-4 \]
\[ 1150 = \left(\left(4!\times4!\right)\times\sqrt{4}\right)-\sqrt{4} \]
\[ 1152 = \left(4+44\right)\times4! \]
\[ 1154 = \left(\left(4!\times4!\right)\times\sqrt{4}\right)+\sqrt{4} \]
\[ 1156 = 4+\left(\left(4!\times4!\right)\times\sqrt{4}\right) \]
\[ 1160 = \left(\left(4!\times4!\right)+4\right)\times\sqrt{4} \]
\[ 1168 = 4!\times\left(\left(4!\times\sqrt{4}\right)+\sqrt{.\dot{4}}\right) \]
\[ 1176 = \left(\left(4!\times4!\right)\times\sqrt{4}\right)+4! \]
\[ 1184 = \left(\sqrt{.\dot{4}}+4!\right)\times\left(4!\times\sqrt{4}\right) \]
\[ 1188 = \frac{\left(\left(4!-\sqrt{4}\right)\times4!\right)}{.\dot{4}} \]
\[ 1200 = \frac{\left(4!\times\left(4!-4\right)\right)}{.4} \]
\[ 1210 = \frac{{\left(4!-\sqrt{4}\right)}^{\sqrt{4}}}{.4} \]
\[ 1215 = \frac{4!}{\left(.\dot{4}\times\left(.\dot{4}-.4\right)\right)} \]
\[ 1225 = \frac{{\left(4!-\sqrt{.\dot{4}}\right)}^{\sqrt{4}}}{.\dot{4}} \]
\[ 1232 = \left(4+4!\right)\times44 \]
\[ 1242 = \frac{\left(\left(4!\times4!\right)-4!\right)}{.\dot{4}} \]
\[ 1248 = \left(\left(4!\times\sqrt{4}\right)+4\right)\times4! \]
\[ 1250 = \sqrt{4}\times{\frac{\sqrt{4}}{.4}}^{4} \]
\[ 1260 = \frac{\left(\left(4!-\sqrt{.\dot{4}}\right)\times4!\right)}{.\dot{4}} \]
\[ 1272 = {\left(4+\sqrt{4}\right)}^{4}-4! \]
\[ 1280 = \sqrt{4}\times\frac{{4}^{4}}{.4} \]
\[ 1287 = \frac{\left(\left(4!\times4!\right)-4\right)}{.\dot{4}} \]
\[ 1292 = {\left(4+\sqrt{4}\right)}^{4}-4 \]
\[ 1294 = {\left(4+\sqrt{4}\right)}^{4}-\sqrt{4} \]
\[ 1295 = \frac{\left(\left(4!\times4!\right)-.\dot{4}\right)}{.\dot{4}} \]
\[ 1296 = {\left(\frac{4}{.4}-4\right)}^{4} \]
\[ 1297 = \frac{\left(\left(4!\times4!\right)+.\dot{4}\right)}{.\dot{4}} \]
\[ 1298 = {\left(4+\sqrt{4}\right)}^{4}+\sqrt{4} \]
\[ 1300 = {\left(4+\sqrt{4}\right)}^{4}+4 \]
\[ 1305 = \frac{\left(\left(4!\times4!\right)+4\right)}{.\dot{4}} \]
\[ 1312 = \left(\frac{4!}{.\dot{4}}+\sqrt{.\dot{4}}\right)\times4! \]
\[ 1320 = {\left(4+\sqrt{4}\right)}^{4}+4! \]
\[ 1331 = {\sqrt{\sqrt{\sqrt{\frac{44}{4}}}}}^{4!} \]
\[ 1332 = \frac{\left(4!\times\left(\sqrt{.\dot{4}}+4!\right)\right)}{.\dot{4}} \]
\[ 1344 = 4!\times\left(\frac{4!}{.4}-4\right) \]
\[ 1350 = \frac{4!}{\left(.4\times\left(.\dot{4}-.4\right)\right)} \]
\[ 1352 = {\sqrt{\left(\sqrt{4}+4!\right)}}^{4}\times\sqrt{4} \]
\[ 1369 = \frac{\sqrt{{\left(\sqrt{.\dot{4}}+4!\right)}^{4}}}{.\dot{4}} \]
\[ 1380 = \frac{\left(\left(4!\times4!\right)-4!\right)}{.4} \]
\[ 1392 = \left(4+\frac{4!}{.\dot{4}}\right)\times4! \]
\[ 1400 = \frac{\left(4!-\sqrt{.\dot{4}}\right)}{.4}\times4! \]
\[ 1404 = 4!\times\frac{\left(\sqrt{4}+4!\right)}{.\dot{4}} \]
\[ 1408 = \left(\sqrt{4}+.\dot{4}\right)\times\left(4!\times4!\right) \]
\[ 1416 = \left(4!-.4\right)\times\frac{4!}{.4} \]
\[ 1424 = \left(\frac{4!}{.4}-\sqrt{.\dot{4}}\right)\times4! \]
\[ 1430 = \frac{\left(\left(4!\times4!\right)-4\right)}{.4} \]
\[ 1435 = \frac{\left(\left(4!\times4!\right)-\sqrt{4}\right)}{.4} \]
\[ 1436 = \left(4!\times\frac{4!}{.4}\right)-4 \]
\[ 1438 = \left(4!\times\frac{4!}{.4}\right)-\sqrt{4} \]
\[ 1439 = \frac{\left(\left(4!\times4!\right)-.4\right)}{.4} \]
\[ 1440 = \frac{{4}^{4}}{\left(.4\times.\dot{4}\right)} \]
\[ 1441 = \frac{\left(.4+\left(4!\times4!\right)\right)}{.4} \]
\[ 1442 = \left(4!\times\frac{4!}{.4}\right)+\sqrt{4} \]
\[ 1444 = 4+\left(4!\times\frac{4!}{.4}\right) \]
\[ 1445 = \frac{\left(\sqrt{4}+\left(4!\times4!\right)\right)}{.4} \]
\[ 1450 = \frac{\left(\left(4!\times4!\right)+4\right)}{.4} \]
\[ 1456 = 4!\times\left(\frac{4!}{.4}+\sqrt{.\dot{4}}\right) \]
\[ 1458 = \frac{{\frac{4!}{.\dot{4}}}^{\sqrt{4}}}{\sqrt{4}} \]
\[ 1464 = \frac{\left(.4+4!\right)}{.4}\times4! \]
\[ 1480 = 4!\times\frac{\left(\sqrt{.\dot{4}}+4!\right)}{.4} \]
\[ 1488 = 4!\times\left(\sqrt{4}+\frac{4!}{.4}\right) \]
\[ 1500 = \frac{\left(\left(4!\times4!\right)+4!\right)}{.4} \]
\[ 1512 = \frac{\left(4+4!\right)}{.\dot{4}}\times4! \]
\[ 1520 = \left({\sqrt{\sqrt{\sqrt{4}}}}^{4!}-\sqrt{.\dot{4}}\right)\times4! \]
\[ 1521 = \frac{\sqrt{{\left(\sqrt{4}+4!\right)}^{4}}}{.\dot{4}} \]
\[ 1532 = \left({\sqrt{\sqrt{\sqrt{4}}}}^{4!}\times4!\right)-4 \]
\[ 1534 = \left({\sqrt{\sqrt{\sqrt{4}}}}^{4!}\times4!\right)-\sqrt{4} \]
\[ 1536 = \left(\left(4!\times4\right)\times4\right)\times4 \]
\[ 1538 = \sqrt{4}+\left({\sqrt{\sqrt{\sqrt{4}}}}^{4!}\times4!\right) \]
\[ 1540 = \left({\sqrt{\sqrt{\sqrt{4}}}}^{4!}\times4!\right)+4 \]
\[ 1552 = 4!\times\left(\sqrt{.\dot{4}}+{\sqrt{\sqrt{\sqrt{4}}}}^{4!}\right) \]
\[ 1560 = \left(\sqrt{4}+4!\right)\times\frac{4!}{.4} \]
\[ 1568 = \sqrt{4}\times{\left(4+4!\right)}^{\sqrt{4}} \]
\[ 1576 = {\frac{4}{\sqrt{.4}}}^{4}-4! \]
\[ 1584 = \frac{44}{\sqrt{.\dot{4}}}\times4! \]
\[ 1596 = {\frac{4}{\sqrt{.4}}}^{4}-4 \]
\[ 1598 = {\frac{4}{\sqrt{.4}}}^{4}-\sqrt{4} \]
\[ 1600 = {\sqrt{\left(44-4\right)}}^{4} \]
\[ 1602 = {\frac{4}{\sqrt{.4}}}^{4}+\sqrt{4} \]
\[ 1604 = {\frac{4}{\sqrt{.4}}}^{4}+4 \]
\[ 1624 = {\frac{4}{\sqrt{.4}}}^{4}+4! \]
\[ 1632 = 4!\times\left(4!+44\right) \]
\[ 1638 = \left(.4\times{\sqrt{\sqrt{4}}}^{4!}\right)-.4 \]
\[ 1640 = \left(4+{\sqrt{\sqrt{4}}}^{4!}\right)\times.4 \]
\[ 1648 = .4\times\left(4!+{\sqrt{\sqrt{4}}}^{4!}\right) \]
\[ 1664 = \left(\sqrt{4}+4!\right)\times\sqrt{{\sqrt{\sqrt{4}}}^{4!}} \]
\[ 1680 = \frac{\left(\left(4+4!\right)\times4!\right)}{.4} \]
\[ 1690 = \frac{\sqrt{{\left(\sqrt{4}+4!\right)}^{4}}}{.4} \]
\[ 1704 = {\sqrt{\sqrt{\sqrt{\frac{4!}{\sqrt{4}}}}}}^{4!}-4! \]
\[ 1724 = {\sqrt{\sqrt{\sqrt{\frac{4!}{\sqrt{4}}}}}}^{4!}-4 \]
\[ 1726 = {\sqrt{\sqrt{\sqrt{\frac{4!}{\sqrt{4}}}}}}^{4!}-\sqrt{4} \]
\[ 1728 = 4!\times\left(\left(4!\times4\right)-4!\right) \]
\[ 1730 = \sqrt{4}+{\sqrt{\sqrt{\sqrt{\frac{4!}{\sqrt{4}}}}}}^{4!} \]
\[ 1732 = {\sqrt{\sqrt{\sqrt{\frac{4!}{\sqrt{4}}}}}}^{4!}+4 \]
\[ 1752 = {\sqrt{\sqrt{\sqrt{\frac{4!}{\sqrt{4}}}}}}^{4!}+4! \]
\[ 1764 = {\sqrt{\left(44-\sqrt{4}\right)}}^{4} \]
\[ 1776 = 4\times444 \]
\[ 1792 = \left(4+4!\right)\times\sqrt{{\sqrt{\sqrt{4}}}^{4!}} \]
\[ 1800 = \frac{{\sqrt{\frac{4!}{.4}}}^{4}}{\sqrt{4}} \]
\[ 1820 = \left(.\dot{4}\times{\sqrt{\sqrt{4}}}^{4!}\right)-.\dot{4} \]
\[ 1872 = 4!\times\left(\frac{4!}{.\dot{4}}+4!\right) \]
\[ 1875 = \frac{\left(4!\times\sqrt{4}\right)}{{.4}^{4}} \]
\[ 1912 = {\sqrt{44}}^{4}-4! \]
\[ 1920 = \left(4!\times\left(4!-4\right)\right)\times4 \]
\[ 1932 = {\sqrt{44}}^{4}-4 \]
\[ 1934 = {\sqrt{44}}^{4}-\sqrt{4} \]
\[ 1936 = 44\times44 \]
\[ 1938 = {\sqrt{44}}^{4}+\sqrt{4} \]
\[ 1940 = {\sqrt{44}}^{4}+4 \]
\[ 1944 = \frac{{\left(4+\sqrt{4}\right)}^{4}}{\sqrt{.\dot{4}}} \]
\[ 1960 = {\sqrt{44}}^{4}+4! \]
\[ 2000 = \frac{{\sqrt{\sqrt{\sqrt{\left(4!-4\right)}}}}^{4!}}{4} \]
\[ 2016 = 4!\times\left(4!+\frac{4!}{.4}\right) \]
\[ 2024 = {\sqrt{\sqrt{\sqrt{4}}}}^{44}-4! \]
\[ 2025 = \frac{4}{{\left(.\dot{4}-.4\right)}^{\sqrt{4}}} \]
\[ 2036 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-4!\right)}{\sqrt{4}} \]
\[ 2039 = \sqrt{\sqrt{.\dot{4}}}+{\sqrt{4!}}^{\left(4+\sqrt{\sqrt{.4}}\right)} \]
\[ 2044 = {\sqrt{\sqrt{\sqrt{4}}}}^{44}-4 \]
\[ 2046 = {\sqrt{\sqrt{\sqrt{4}}}}^{44}-\sqrt{4} \]
\[ 2047 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-\sqrt{4}\right)}{\sqrt{4}} \]
\[ 2048 = \left(4+4\right)\times{4}^{4} \]
\[ 2049 = \frac{\left(\sqrt{4}+{\sqrt{\sqrt{4}}}^{4!}\right)}{\sqrt{4}} \]
\[ 2050 = \sqrt{4}+{\sqrt{\sqrt{\sqrt{4}}}}^{44} \]
\[ 2052 = 4+{\sqrt{\sqrt{\sqrt{4}}}}^{44} \]
\[ 2060 = \frac{\left(4!+{\sqrt{\sqrt{4}}}^{4!}\right)}{\sqrt{4}} \]
\[ 2072 = 4!+{\sqrt{\sqrt{\sqrt{4}}}}^{44} \]
\[ 2112 = \left(\sqrt{4}\times44\right)\times4! \]
\[ 2116 = {\sqrt{\left(\sqrt{4}+44\right)}}^{4} \]
\[ 2160 = \frac{4!}{\left(.\dot{4}-.4\right)}\times4 \]
\[ 2187 = {\sqrt{\sqrt{\sqrt{\frac{4}{.\dot{4}}}}}}^{\left(4+4!\right)} \]
\[ 2197 = {\sqrt{\sqrt{\sqrt{\left(4+\frac{4}{.\dot{4}}\right)}}}}^{4!} \]
\[ 2208 = 4!\times\left(\left(4!\times4\right)-4\right) \]
\[ 2240 = \left(\left(4!-\sqrt{.\dot{4}}\right)\times4!\right)\times4 \]
\[ 2250 = \frac{\sqrt{{\sqrt{\sqrt{\frac{4}{.4}}}}^{4!}}}{.\dot{4}} \]
\[ 2256 = \left(\left(4!\times4\right)-\sqrt{4}\right)\times4! \]
\[ 2280 = \left(\left(4!\times4\right)\times4!\right)-4! \]
\[ 2288 = \left(\left(4!\times4!\right)-4\right)\times4 \]
\[ 2296 = \left(\left(4!\times4!\right)-\sqrt{4}\right)\times4 \]
\[ 2300 = \left(\left(4!\times4\right)\times4!\right)-4 \]
\[ 2302 = \left(\left(4!\times4\right)\times4!\right)-\sqrt{4} \]
\[ 2304 = {\sqrt{\left(4+44\right)}}^{4} \]
\[ 2306 = \sqrt{4}+\left(\left(4!\times4\right)\times4!\right) \]
\[ 2308 = 4+\left(\left(4!\times4\right)\times4!\right) \]
\[ 2312 = \left(\sqrt{4}+\left(4!\times4!\right)\right)\times4 \]
\[ 2320 = \left(\left(4!\times4!\right)+4\right)\times4 \]
\[ 2328 = \left(\left(4!\times4\right)\times4!\right)+4! \]
\[ 2352 = 4!\times\left(\sqrt{4}+\left(4!\times4\right)\right) \]
\[ 2368 = \left(\left(\sqrt{.\dot{4}}+4!\right)\times4\right)\times4! \]
\[ 2376 = \frac{\left(4!\times44\right)}{.\dot{4}} \]
\[ 2400 = \left(4+\left(4!\times4\right)\right)\times4! \]
\[ 2401 = {\frac{\left(4+4!\right)}{4}}^{4} \]
\[ 2496 = \left(4\times\left(\sqrt{4}+4!\right)\right)\times4! \]
\[ 2500 = \frac{{\frac{4}{.4}}^{4}}{4} \]
\[ 2560 = \frac{{4}^{4}}{.4}\times4 \]
\[ 2592 = \sqrt{4}\times{\left(4+\sqrt{4}\right)}^{4} \]
\[ 2640 = \frac{4!}{.4}\times44 \]
\[ 2662 = \frac{{\sqrt{\sqrt{\sqrt{\left(4!-\sqrt{4}\right)}}}}^{4!}}{4} \]
\[ 2688 = \left(\left(4+4!\right)\times4!\right)\times4 \]
\[ 2704 = 4\times{\sqrt{\left(\sqrt{4}+4!\right)}}^{4} \]
\[ 2728 = \sqrt{.\dot{4}}\times\left({\sqrt{\sqrt{4}}}^{4!}-4\right) \]
\[ 2730 = \left({\sqrt{\sqrt{4}}}^{4!}\times\sqrt{.\dot{4}}\right)-\sqrt{.\dot{4}} \]
\[ 2732 = \left(\sqrt{4}+{\sqrt{\sqrt{4}}}^{4!}\right)\times\sqrt{.\dot{4}} \]
\[ 2744 = {\sqrt{\sqrt{\sqrt{\left(\frac{4}{.4}+4\right)}}}}^{4!} \]
\[ 2809 = {\sqrt{\frac{\left(4!-.\dot{4}\right)}{.\dot{4}}}}^{4} \]
\[ 2816 = {\sqrt{\sqrt{\sqrt{4}}}}^{4!}\times44 \]
\[ 2880 = 4!\times\left(\left(4!\times4\right)+4!\right) \]
\[ 2892 = {\frac{4!}{.\dot{4}}}^{\sqrt{4}}-4! \]
\[ 2904 = \frac{{\sqrt{44}}^{4}}{\sqrt{.\dot{4}}} \]
\[ 2908 = {\left(\left(\sqrt{4!}+\sqrt{4}\right)+.\dot{4}\right)}^{4} \]
\[ 2912 = {\frac{4!}{.\dot{4}}}^{\sqrt{4}}-4 \]
\[ 2914 = {\frac{4!}{.\dot{4}}}^{\sqrt{4}}-\sqrt{4} \]
\[ 2916 = .\dot{4}\times{\frac{4}{.\dot{4}}}^{4} \]
\[ 2918 = \sqrt{4}+{\frac{4!}{.\dot{4}}}^{\sqrt{4}} \]
\[ 2920 = 4+{\frac{4!}{.\dot{4}}}^{\sqrt{4}} \]
\[ 2940 = {\frac{4!}{.\dot{4}}}^{\sqrt{4}}+4! \]
\[ 3000 = \sqrt{{\sqrt{\sqrt{\frac{\sqrt{4}}{.4}}}}^{4!}}\times4! \]
\[ 3025 = {\sqrt{\frac{\left(.\dot{4}+4!\right)}{.\dot{4}}}}^{4} \]
\[ 3072 = \frac{{4}^{4}}{\sqrt{4}}\times4! \]
\[ 3125 = {\frac{\sqrt{4}}{.4}}^{\frac{\sqrt{4}}{.4}} \]
\[ 3136 = {\left(4+4!\right)}^{\sqrt{4}}\times4 \]
\[ 3141 = {\sqrt{\sqrt{\sqrt{\frac{4!}{\sqrt{\left(\sqrt{4!}-\sqrt{\sqrt{4!}}\right)}}}}}}^{4!} \]
\[ 3200 = {\frac{4}{\sqrt{.4}}}^{4}\times\sqrt{4} \]
\[ 3240 = \frac{{\left(4+\sqrt{4}\right)}^{4}}{.4} \]
\[ 3364 = {\sqrt{\left(4+\frac{4!}{.\dot{4}}\right)}}^{4} \]
\[ 3375 = {\sqrt{\sqrt{\sqrt{\frac{4!}{\left(4\times.4\right)}}}}}^{4!} \]
\[ 3456 = \frac{{4!}^{4}}{\left(4!\times4\right)} \]
\[ 3481 = {\sqrt{\frac{\left(4!-.4\right)}{.4}}}^{4} \]
\[ 3520 = {\sqrt{\sqrt{4}}}^{4!}-\left(4!\times4!\right) \]
\[ 3576 = {\sqrt{\frac{4!}{.4}}}^{4}-4! \]
\[ 3596 = {\sqrt{\frac{4!}{.4}}}^{4}-4 \]
\[ 3598 = {\sqrt{\frac{4!}{.4}}}^{4}-\sqrt{4} \]
\[ 3600 = \frac{\left(4!\times4!\right)}{\left(.4\times.4\right)} \]
\[ 3602 = {\sqrt{\frac{4!}{.4}}}^{4}+\sqrt{4} \]
\[ 3604 = {\sqrt{\frac{4!}{.4}}}^{4}+4 \]
\[ 3624 = {\sqrt{\frac{4!}{.4}}}^{4}+4! \]
\[ 3721 = {\sqrt{\frac{\left(.4+4!\right)}{.4}}}^{4} \]
\[ 3750 = 4\times\frac{4!}{{.4}^{4}} \]
\[ 3840 = {\sqrt{\sqrt{4}}}^{4!}-{4}^{4} \]
\[ 3844 = {\sqrt{\left(\sqrt{4}+\frac{4!}{.4}\right)}}^{4} \]
\[ 3872 = {\sqrt{44}}^{4}\times\sqrt{4} \]
\[ 3888 = \frac{{\sqrt{\sqrt{\sqrt{\frac{4!}{\sqrt{4}}}}}}^{4!}}{.\dot{4}} \]
\[ 3969 = {\frac{\left(4+4!\right)}{.\dot{4}}}^{\sqrt{4}} \]
\[ 4000 = .4\times{\frac{4}{.4}}^{4} \]
\[ 4032 = {\sqrt{\sqrt{4}}}^{4!}-{\sqrt{\sqrt{\sqrt{4}}}}^{4!} \]
\[ 4036 = {\sqrt{\sqrt{4}}}^{4!}-\frac{4!}{.4} \]
\[ 4042 = {\sqrt{\sqrt{4}}}^{4!}-\frac{4!}{.\dot{4}} \]
\[ 4048 = \left({\sqrt{\sqrt{4}}}^{4!}-4!\right)-4! \]
\[ 4052 = {\sqrt{\sqrt{4}}}^{4!}-44 \]
\[ 4060 = {\sqrt{\sqrt{4}}}^{4!}-\frac{4!}{\sqrt{.\dot{4}}} \]
\[ 4068 = \left({\sqrt{\sqrt{4}}}^{4!}-4!\right)-4 \]
\[ 4070 = \left({\sqrt{\sqrt{4}}}^{4!}-\sqrt{4}\right)-4! \]
\[ 4072 = {\left(4+4\right)}^{4}-4! \]
\[ 4074 = \left({\sqrt{\sqrt{4}}}^{4!}-4!\right)+\sqrt{4} \]
\[ 4076 = \left(4+{\sqrt{\sqrt{4}}}^{4!}\right)-4! \]
\[ 4080 = {\sqrt{\sqrt{4}}}^{4!}-\left(4\times4\right) \]
\[ 4084 = {\sqrt{\sqrt{4}}}^{4!}-\frac{4!}{\sqrt{4}} \]
\[ 4086 = {\sqrt{\sqrt{4}}}^{4!}-\frac{4}{.4} \]
\[ 4087 = {\sqrt{\sqrt{4}}}^{4!}-\frac{4}{.\dot{4}} \]
\[ 4088 = {\sqrt{\sqrt{4}}}^{4!}-\left(4+4\right) \]
\[ 4090 = \left({\sqrt{\sqrt{4}}}^{4!}-4\right)-\sqrt{4} \]
\[ 4091 = {\sqrt{\sqrt{4}}}^{4!}-\frac{\sqrt{4}}{.4} \]
\[ 4092 = {\left(4+4\right)}^{4}-4 \]
\[ 4093 = {\sqrt{\sqrt{4}}}^{4!}-\sqrt{\frac{4}{.\dot{4}}} \]
\[ 4094 = {\left(4+4\right)}^{4}-\sqrt{4} \]
\[ 4095 = {\sqrt{\sqrt{4}}}^{4!}-\frac{4}{4} \]
\[ 4096 = \left(4\times{4}^{4}\right)\times4 \]
\[ 4097 = \frac{4}{4}+{\sqrt{\sqrt{4}}}^{4!} \]
\[ 4098 = {\left(4+4\right)}^{4}+\sqrt{4} \]
\[ 4099 = {\sqrt{\sqrt{4}}}^{4!}+\sqrt{\frac{4}{.\dot{4}}} \]
\[ 4100 = 4+{\left(4+4\right)}^{4} \]
\[ 4101 = \frac{\sqrt{4}}{.4}+{\sqrt{\sqrt{4}}}^{4!} \]
\[ 4102 = \left(\sqrt{4}+{\sqrt{\sqrt{4}}}^{4!}\right)+4 \]
\[ 4104 = \left(4+{\sqrt{\sqrt{4}}}^{4!}\right)+4 \]
\[ 4105 = {\sqrt{\sqrt{4}}}^{4!}+\frac{4}{.\dot{4}} \]
\[ 4106 = \frac{4}{.4}+{\sqrt{\sqrt{4}}}^{4!} \]
\[ 4108 = {\sqrt{\sqrt{4}}}^{4!}+\frac{4!}{\sqrt{4}} \]
\[ 4112 = {\sqrt{\sqrt{4}}}^{4!}+\left(4\times4\right) \]
\[ 4116 = {\sqrt{\sqrt{4}}}^{4!}+\left(4!-4\right) \]
\[ 4118 = {\sqrt{\sqrt{4}}}^{4!}-\left(\sqrt{4}-4!\right) \]
\[ 4120 = 4!+{\left(4+4\right)}^{4} \]
\[ 4122 = {\sqrt{\sqrt{4}}}^{4!}+\left(\sqrt{4}+4!\right) \]
\[ 4124 = 4+\left(4!+{\sqrt{\sqrt{4}}}^{4!}\right) \]
\[ 4132 = \frac{4!}{\sqrt{.\dot{4}}}+{\sqrt{\sqrt{4}}}^{4!} \]
\[ 4140 = {\sqrt{\sqrt{4}}}^{4!}+44 \]
\[ 4144 = \left(4!+{\sqrt{\sqrt{4}}}^{4!}\right)+4! \]
\[ 4150 = {\sqrt{\sqrt{4}}}^{4!}+\frac{4!}{.\dot{4}} \]
\[ 4156 = {\sqrt{\sqrt{4}}}^{4!}+\frac{4!}{.4} \]
\[ 4160 = {\sqrt{\sqrt{4}}}^{4!}+\sqrt{{\sqrt{\sqrt{4}}}^{4!}} \]
\[ 4192 = {\sqrt{\sqrt{4}}}^{4!}+\left(4!\times4\right) \]
\[ 4224 = \left(4!\times44\right)\times4 \]
\[ 4225 = {\sqrt{\frac{\left(\sqrt{4}+4!\right)}{.4}}}^{4} \]
\[ 4320 = \frac{{\sqrt{\sqrt{\sqrt{\frac{4!}{\sqrt{4}}}}}}^{4!}}{.4} \]
\[ 4352 = {\sqrt{\sqrt{4}}}^{4!}+{4}^{4} \]
\[ 4356 = \frac{{\sqrt{44}}^{4}}{.\dot{4}} \]
\[ 4374 = \sqrt{.\dot{4}}\times{\frac{4}{.\dot{4}}}^{4} \]
\[ 4394 = \frac{{\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+4!\right)}}}}^{4!}}{4} \]
\[ 4444 = 4444 \]
\[ 4608 = \left(4!\times4!\right)\times\left(4+4\right) \]
\[ 4624 = {\left(4!+44\right)}^{\sqrt{4}} \]
\[ 4672 = {\sqrt{\sqrt{4}}}^{4!}+\left(4!\times4!\right) \]
\[ 4840 = \frac{{\sqrt{44}}^{4}}{.4} \]
\[ 4900 = {\frac{\left(4+4!\right)}{.4}}^{\sqrt{4}} \]
\[ 5000 = \frac{{\frac{4}{.4}}^{4}}{\sqrt{4}} \]
\[ 5120 = \left(4!-4\right)\times{4}^{4} \]
\[ 5184 = {\left(4+\sqrt{4}\right)}^{4}\times4 \]
\[ 5324 = \frac{{\sqrt{\sqrt{\sqrt{\left(4!-\sqrt{4}\right)}}}}^{4!}}{\sqrt{4}} \]
\[ 5400 = \frac{{\sqrt{\frac{4!}{.4}}}^{4}}{\sqrt{.\dot{4}}} \]
\[ 5488 = \frac{{\sqrt{\sqrt{\sqrt{\left(4+4!\right)}}}}^{4!}}{4} \]
\[ 5568 = 4!\times\left({4}^{4}-4!\right) \]
\[ 5625 = \frac{{\frac{\sqrt{4!}}{.4}}^{4}}{4} \]
\[ 5632 = {4}^{4}\times\left(4!-\sqrt{4}\right) \]
\[ 5760 = \frac{{4}^{4}}{\left(.\dot{4}-.4\right)} \]
\[ 5832 = \sqrt{4}\times{\frac{4!}{.\dot{4}}}^{\sqrt{4}} \]
\[ 6048 = 4!\times\left({4}^{4}-4\right) \]
\[ 6084 = {\left(\frac{4!}{.\dot{4}}+4!\right)}^{\sqrt{4}} \]
\[ 6096 = 4!\times\left({4}^{4}-\sqrt{4}\right) \]
\[ 6108 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-4!\right)}{\sqrt{.\dot{4}}} \]
\[ 6120 = \left({4}^{4}\times4!\right)-4! \]
\[ 6128 = 4!\times\left({4}^{4}-\sqrt{.\dot{4}}\right) \]
\[ 6138 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-4\right)}{\sqrt{.\dot{4}}} \]
\[ 6140 = \left({4}^{4}\times4!\right)-4 \]
\[ 6141 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-\sqrt{4}\right)}{\sqrt{.\dot{4}}} \]
\[ 6142 = \left({4}^{4}\times4!\right)-\sqrt{4} \]
\[ 6143 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-\sqrt{.\dot{4}}\right)}{\sqrt{.\dot{4}}} \]
\[ 6144 = \frac{{\left(4+4\right)}^{4}}{\sqrt{.\dot{4}}} \]
\[ 6145 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}+\sqrt{.\dot{4}}\right)}{\sqrt{.\dot{4}}} \]
\[ 6146 = \sqrt{4}+\left({4}^{4}\times4!\right) \]
\[ 6147 = \frac{\left(\sqrt{4}+{\sqrt{\sqrt{4}}}^{4!}\right)}{\sqrt{.\dot{4}}} \]
\[ 6148 = 4+\left({4}^{4}\times4!\right) \]
\[ 6150 = \frac{\left(4+{\sqrt{\sqrt{4}}}^{4!}\right)}{\sqrt{.\dot{4}}} \]
\[ 6160 = \left({4}^{4}+\sqrt{.\dot{4}}\right)\times4! \]
\[ 6168 = \left({4}^{4}\times4!\right)+4! \]
\[ 6180 = \frac{\left(4!+{\sqrt{\sqrt{4}}}^{4!}\right)}{\sqrt{.\dot{4}}} \]
\[ 6192 = 4!\times\left(\sqrt{4}+{4}^{4}\right) \]
\[ 6240 = 4!\times\left(4+{4}^{4}\right) \]
\[ 6250 = .4\times{\sqrt{\sqrt{\frac{\sqrt{4}}{.4}}}}^{4!} \]
\[ 6400 = 4\times{\frac{4}{\sqrt{.4}}}^{4} \]
\[ 6537 = {\frac{4}{.\dot{4}}}^{4}-4! \]
\[ 6557 = {\frac{4}{.\dot{4}}}^{4}-4 \]
\[ 6559 = {\frac{4}{.\dot{4}}}^{4}-\sqrt{4} \]
\[ 6561 = {\frac{\left(4-.4\right)}{.4}}^{4} \]
\[ 6563 = {\frac{4}{.\dot{4}}}^{4}+\sqrt{4} \]
\[ 6565 = {\frac{4}{.\dot{4}}}^{4}+4 \]
\[ 6585 = 4!+{\frac{4}{.\dot{4}}}^{4} \]
\[ 6656 = \left(\sqrt{4}+4!\right)\times{4}^{4} \]
\[ 6720 = 4!\times\left({4}^{4}+4!\right) \]
\[ 6859 = {\sqrt{\sqrt{\sqrt{\left(4!-\frac{\sqrt{4}}{.4}\right)}}}}^{4!} \]
\[ 6912 = 4!\times\sqrt{\frac{{4!}^{4}}{4}} \]
\[ 7056 = {\left(4!+\frac{4!}{.4}\right)}^{\sqrt{4}} \]
\[ 7168 = \left(4+4!\right)\times{4}^{4} \]
\[ 7200 = \sqrt{4}\times{\sqrt{\frac{4!}{.4}}}^{4} \]
\[ 7290 = \frac{{\frac{4!}{.\dot{4}}}^{\sqrt{4}}}{.4} \]
\[ 7744 = 4\times{\sqrt{44}}^{4} \]
\[ 7776 = {\sqrt{\left(4+\sqrt{4}\right)}}^{\frac{4}{.4}} \]
\[ 7976 = {\sqrt{\sqrt{\sqrt{\left(4!-4\right)}}}}^{4!}-4! \]
\[ 7996 = {\sqrt{\sqrt{\sqrt{\left(4!-4\right)}}}}^{4!}-4 \]
\[ 7998 = {\sqrt{\sqrt{\sqrt{\left(4!-4\right)}}}}^{4!}-\sqrt{4} \]
\[ 8000 = {\left(4!-4\right)}^{\sqrt{\frac{4}{.\dot{4}}}} \]
\[ 8002 = \sqrt{4}+{\sqrt{\sqrt{\sqrt{\left(4!-4\right)}}}}^{4!} \]
\[ 8004 = 4+{\sqrt{\sqrt{\sqrt{\left(4!-4\right)}}}}^{4!} \]
\[ 8024 = {\sqrt{\sqrt{\sqrt{\left(4!-4\right)}}}}^{4!}+4! \]
\[ 8100 = {\frac{4}{\left(.\dot{4}-.4\right)}}^{\sqrt{4}} \]
\[ 8144 = \sqrt{4}\times\left({\sqrt{\sqrt{4}}}^{4!}-4!\right) \]
\[ 8168 = \left({\sqrt{\sqrt{4}}}^{4!}\times\sqrt{4}\right)-4! \]
\[ 8184 = \sqrt{4}\times\left({\sqrt{\sqrt{4}}}^{4!}-4\right) \]
\[ 8188 = \left({\sqrt{\sqrt{4}}}^{4!}\times\sqrt{4}\right)-4 \]
\[ 8190 = \left({\sqrt{\sqrt{4}}}^{4!}\times\sqrt{4}\right)-\sqrt{4} \]
\[ 8192 = {\left(4+4\right)}^{4}\times\sqrt{4} \]
\[ 8194 = \sqrt{4}+\left({\sqrt{\sqrt{4}}}^{4!}\times\sqrt{4}\right) \]
\[ 8196 = 4+\left({\sqrt{\sqrt{4}}}^{4!}\times\sqrt{4}\right) \]
\[ 8200 = \sqrt{4}\times\left(4+{\sqrt{\sqrt{4}}}^{4!}\right) \]
\[ 8216 = 4!+\left({\sqrt{\sqrt{4}}}^{4!}\times\sqrt{4}\right) \]
\[ 8240 = \sqrt{4}\times\left(4!+{\sqrt{\sqrt{4}}}^{4!}\right) \]
\[ 8464 = {\sqrt{\left(\left(4!\times4\right)-4\right)}}^{4} \]
\[ 8788 = \frac{{\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+4!\right)}}}}^{4!}}{\sqrt{4}} \]
\[ 8836 = {\left(\left(4!\times4\right)-\sqrt{4}\right)}^{\sqrt{4}} \]
\[ 9000 = {\frac{\sqrt{4!}}{.4}}^{4}\times.4 \]
\[ 9162 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-4!\right)}{.\dot{4}} \]
\[ 9192 = {\sqrt{\left(4!\times4\right)}}^{4}-4! \]
\[ 9207 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-4\right)}{.\dot{4}} \]
\[ 9212 = {\sqrt{\left(4!\times4\right)}}^{4}-4 \]
\[ 9214 = {\sqrt{\left(4!\times4\right)}}^{4}-\sqrt{4} \]
\[ 9215 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-.\dot{4}\right)}{.\dot{4}} \]
\[ 9216 = \frac{{\left(4+4\right)}^{4}}{.\dot{4}} \]
\[ 9217 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}+.\dot{4}\right)}{.\dot{4}} \]
\[ 9218 = \sqrt{4}+{\sqrt{\left(4!\times4\right)}}^{4} \]
\[ 9220 = 4+{\sqrt{\left(4!\times4\right)}}^{4} \]
\[ 9225 = \frac{\left(4+{\sqrt{\sqrt{4}}}^{4!}\right)}{.\dot{4}} \]
\[ 9240 = {\sqrt{\left(4!\times4\right)}}^{4}+4! \]
\[ 9261 = {\sqrt{\sqrt{\sqrt{\left(4!-\sqrt{\frac{4}{.\dot{4}}}\right)}}}}^{4!} \]
\[ 9270 = \frac{\left(4!+{\sqrt{\sqrt{4}}}^{4!}\right)}{.\dot{4}} \]
\[ 9600 = 4!\times\sqrt{{\left(4!-4\right)}^{4}} \]
\[ 9604 = {\sqrt{\left(\sqrt{4}+\left(4!\times4\right)\right)}}^{4} \]
\[ 9801 = {\sqrt{\frac{44}{.\dot{4}}}}^{4} \]
\[ 9976 = {\frac{4}{.4}}^{4}-4! \]
\[ 9996 = {\frac{4}{.4}}^{4}-4 \]
\[ 9998 = {\frac{4}{.4}}^{4}-\sqrt{4} \]
\[ 10000 = {\frac{\left(4+.\dot{4}\right)}{.\dot{4}}}^{4} \]
\[ 10002 = {\frac{4}{.4}}^{4}+\sqrt{4} \]
\[ 10004 = 4+{\frac{4}{.4}}^{4} \]
\[ 10024 = {\frac{4}{.4}}^{4}+4! \]
\[ 10180 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-4!\right)}{.4} \]
\[ 10216 = \frac{{\sqrt{\sqrt{4}}}^{4!}}{.4}-4! \]
\[ 10230 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-4\right)}{.4} \]
\[ 10235 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-\sqrt{4}\right)}{.4} \]
\[ 10236 = \frac{{\sqrt{\sqrt{4}}}^{4!}}{.4}-4 \]
\[ 10238 = \frac{{\sqrt{\sqrt{4}}}^{4!}}{.4}-\sqrt{4} \]
\[ 10239 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}-.4\right)}{.4} \]
\[ 10240 = \frac{{\left(4+4\right)}^{4}}{.4} \]
\[ 10241 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}+.4\right)}{.4} \]
\[ 10242 = \sqrt{4}+\frac{{\sqrt{\sqrt{4}}}^{4!}}{.4} \]
\[ 10244 = 4+\frac{{\sqrt{\sqrt{4}}}^{4!}}{.4} \]
\[ 10245 = \frac{\left(\sqrt{4}+{\sqrt{\sqrt{4}}}^{4!}\right)}{.4} \]
\[ 10250 = \frac{\left(4+{\sqrt{\sqrt{4}}}^{4!}\right)}{.4} \]
\[ 10264 = 4!+\frac{{\sqrt{\sqrt{4}}}^{4!}}{.4} \]
\[ 10300 = \frac{\left(4!+{\sqrt{\sqrt{4}}}^{4!}\right)}{.4} \]
\[ 10368 = \frac{{\frac{4!}{\sqrt{4}}}^{4}}{\sqrt{4}} \]
\[ 10624 = {\sqrt{\sqrt{\sqrt{\left(4!-\sqrt{4}\right)}}}}^{4!}-4! \]
\[ 10644 = {\sqrt{\sqrt{\sqrt{\left(4!-\sqrt{4}\right)}}}}^{4!}-4 \]
\[ 10646 = {\sqrt{\sqrt{\sqrt{\left(4!-\sqrt{4}\right)}}}}^{4!}-\sqrt{4} \]
\[ 10648 = {\left(4!-\sqrt{4}\right)}^{\sqrt{\frac{4}{.\dot{4}}}} \]
\[ 10650 = {\sqrt{\sqrt{\sqrt{\left(4!-\sqrt{4}\right)}}}}^{4!}+\sqrt{4} \]
\[ 10652 = 4+{\sqrt{\sqrt{\sqrt{\left(4!-\sqrt{4}\right)}}}}^{4!} \]
\[ 10656 = 444\times4! \]
\[ 10672 = {\sqrt{\sqrt{\sqrt{\left(4!-\sqrt{4}\right)}}}}^{4!}+4! \]
\[ 10816 = {\sqrt{\left(4\times\left(\sqrt{4}+4!\right)\right)}}^{4} \]
\[ 10976 = \frac{{\sqrt{\sqrt{\sqrt{\left(4+4!\right)}}}}^{4!}}{\sqrt{4}} \]
\[ 11250 = \frac{{\frac{\sqrt{4!}}{.4}}^{4}}{\sqrt{4}} \]
\[ 11264 = {4}^{4}\times44 \]
\[ 11520 = 4!\times\left(4!\times\left(4!-4\right)\right) \]
\[ 11616 = 4!\times{\left(4!-\sqrt{4}\right)}^{\sqrt{4}} \]
\[ 11664 = 4\times{\frac{4!}{.\dot{4}}}^{\sqrt{4}} \]
\[ 12000 = \frac{{\sqrt{\sqrt{\sqrt{\left(4!-4\right)}}}}^{4!}}{\sqrt{.\dot{4}}} \]
\[ 12100 = {\frac{44}{.4}}^{\sqrt{4}} \]
\[ 12150 = \frac{4!}{{\left(.\dot{4}-.4\right)}^{\sqrt{4}}} \]
\[ 12167 = {\sqrt{\sqrt{\sqrt{\left(4!-\frac{4}{4}\right)}}}}^{4!} \]
\[ 12288 = \left(\sqrt{4}\times{4}^{4}\right)\times4! \]
\[ 12544 = {\left(4\times\left(4+4!\right)\right)}^{\sqrt{4}} \]
\[ 12672 = \left(\left(4!-\sqrt{4}\right)\times4!\right)\times4! \]
\[ 12696 = {\sqrt{\left(\left(\sqrt{4!}\times4!\right)-\sqrt{4!}\right)}}^{4} \]
\[ 12960 = 4!\times\frac{4!}{\left(.\dot{4}-.4\right)} \]
\[ 13122 = \sqrt{4}\times{\frac{4}{.\dot{4}}}^{4} \]
\[ 13248 = 4!\times\left(\left(4!\times4!\right)-4!\right) \]
\[ 13440 = \left(\left(4!-\sqrt{.\dot{4}}\right)\times4!\right)\times4! \]
\[ 13568 = 4!\times\left(\left(4!-.\dot{4}\right)\times4!\right) \]
\[ 13728 = \left(\left(4!\times4!\right)-4\right)\times4! \]
\[ 13776 = \left(\left(4!\times4!\right)-\sqrt{4}\right)\times4! \]
\[ 13800 = \frac{{4!}^{4}}{4!}-4! \]
\[ 13808 = \left(\left(4!\times4!\right)-\sqrt{.\dot{4}}\right)\times4! \]
\[ 13820 = \frac{{4!}^{4}}{4!}-4 \]
\[ 13822 = \frac{{4!}^{4}}{4!}-\sqrt{4} \]
\[ 13823 = \frac{\left({4!}^{4}-4!\right)}{4!} \]
\[ 13824 = {4!}^{\left(4-\frac{4}{4}\right)} \]
\[ 13825 = \frac{\left(4!+{4!}^{4}\right)}{4!} \]
\[ 13826 = \sqrt{4}+\frac{{4!}^{4}}{4!} \]
\[ 13828 = 4+\frac{{4!}^{4}}{4!} \]
\[ 13840 = 4!\times\left(\left(4!\times4!\right)+\sqrt{.\dot{4}}\right) \]
\[ 13848 = 4!+\frac{{4!}^{4}}{4!} \]
\[ 13872 = 4!\times\left(\sqrt{4}+\left(4!\times4!\right)\right) \]
\[ 13920 = 4!\times\left(\left(4!\times4!\right)+4\right) \]
\[ 14080 = \left(4!\times4!\right)\times\left(.\dot{4}+4!\right) \]
\[ 14208 = \left(4!\times\left(\sqrt{.\dot{4}}+4!\right)\right)\times4! \]
\[ 14400 = {\sqrt{\frac{4!}{.4}}}^{4}\times4 \]
\[ 14641 = {\frac{44}{4}}^{4} \]
\[ 14976 = \left(\left(\sqrt{4}+4!\right)\times4!\right)\times4! \]
\[ 15000 = \frac{{\frac{4}{.4}}^{4}}{\sqrt{.\dot{4}}} \]
\[ 15360 = 4!\times\frac{{4}^{4}}{.4} \]
\[ 15601 = {\sqrt{\sqrt{\frac{\sqrt{4}}{.4}}}}^{4!}-4! \]
\[ 15621 = {\sqrt{\sqrt{\frac{\sqrt{4}}{.4}}}}^{4!}-4 \]
\[ 15623 = {\sqrt{\sqrt{\frac{\sqrt{4}}{.4}}}}^{4!}-\sqrt{4} \]
\[ 15625 = {\frac{\sqrt{4}}{.4}}^{\left(4+\sqrt{4}\right)} \]
\[ 15627 = {\sqrt{\sqrt{\frac{\sqrt{4}}{.4}}}}^{4!}+\sqrt{4} \]
\[ 15629 = {\sqrt{\sqrt{\frac{\sqrt{4}}{.4}}}}^{4!}+4 \]
\[ 15649 = 4!+{\sqrt{\sqrt{\frac{\sqrt{4}}{.4}}}}^{4!} \]
\[ 15972 = \frac{{\sqrt{\sqrt{\sqrt{\left(4!-\sqrt{4}\right)}}}}^{4!}}{\sqrt{.\dot{4}}} \]
\[ 16000 = \frac{{\sqrt{\sqrt{\frac{4}{\sqrt{.4}}}}}^{4!}}{4} \]
\[ 16128 = \left(\left(4+4!\right)\times4!\right)\times4! \]
\[ 16224 = 4!\times\sqrt{{\left(\sqrt{4}+4!\right)}^{4}} \]
\[ 16288 = 4\times\left({\sqrt{\sqrt{4}}}^{4!}-4!\right) \]
\[ 16360 = \left(4\times{\sqrt{\sqrt{4}}}^{4!}\right)-4! \]
\[ 16368 = \left({\sqrt{\sqrt{4}}}^{4!}-4\right)\times4 \]
\[ 16376 = 4\times\left({\sqrt{\sqrt{4}}}^{4!}-\sqrt{4}\right) \]
\[ 16380 = \left(4\times{\sqrt{\sqrt{4}}}^{4!}\right)-4 \]
\[ 16382 = \left(4\times{\sqrt{\sqrt{4}}}^{4!}\right)-\sqrt{4} \]
\[ 16384 = {\left(4+4\right)}^{4}\times4 \]
\[ 16386 = \left(4\times{\sqrt{\sqrt{4}}}^{4!}\right)+\sqrt{4} \]
\[ 16388 = 4+\left(4\times{\sqrt{\sqrt{4}}}^{4!}\right) \]
\[ 16392 = 4\times\left(\sqrt{4}+{\sqrt{\sqrt{4}}}^{4!}\right) \]
\[ 16400 = 4\times\left(4+{\sqrt{\sqrt{4}}}^{4!}\right) \]
\[ 16408 = 4!+\left(4\times{\sqrt{\sqrt{4}}}^{4!}\right) \]
\[ 16480 = \left(4!+{\sqrt{\sqrt{4}}}^{4!}\right)\times4 \]
\[ 17496 = 4!\times{\sqrt{\sqrt{\sqrt{\frac{4}{.\dot{4}}}}}}^{4!} \]
\[ 17552 = {\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+4!\right)}}}}^{4!}-4! \]
\[ 17572 = {\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+4!\right)}}}}^{4!}-4 \]
\[ 17574 = {\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+4!\right)}}}}^{4!}-\sqrt{4} \]
\[ 17576 = {\left(\sqrt{4}+4!\right)}^{\sqrt{\frac{4}{.\dot{4}}}} \]
\[ 17578 = \sqrt{4}+{\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+4!\right)}}}}^{4!} \]
\[ 17580 = 4+{\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+4!\right)}}}}^{4!} \]
\[ 17600 = {\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+4!\right)}}}}^{4!}+4! \]
\[ 18000 = \frac{{\sqrt{\sqrt{\sqrt{\left(4!-4\right)}}}}^{4!}}{.\dot{4}} \]
\[ 18225 = {\sqrt{\frac{4!}{\left(.4\times.\dot{4}\right)}}}^{4} \]
\[ 18432 = {\sqrt{\left(4!\times4\right)}}^{4}\times\sqrt{4} \]
\[ 18816 = 4!\times\sqrt{{\left(4+4!\right)}^{4}} \]
\[ 19683 = {\frac{4}{.\dot{4}}}^{\frac{\sqrt{4}}{.\dot{4}}} \]
\[ 20000 = \sqrt{4}\times{\frac{4}{.4}}^{4} \]
\[ 20480 = \frac{\left({\sqrt{\sqrt{4}}}^{4!}\times\sqrt{4}\right)}{.4} \]
\[ 20712 = {\frac{4!}{\sqrt{4}}}^{4}-4! \]
\[ 20732 = {\frac{4!}{\sqrt{4}}}^{4}-4 \]
\[ 20734 = {\frac{4!}{\sqrt{4}}}^{4}-\sqrt{4} \]
\[ 20736 = {\left(4+\left(4+4\right)\right)}^{4} \]
\[ 20738 = {\frac{4!}{\sqrt{4}}}^{4}+\sqrt{4} \]
\[ 20740 = {\frac{4!}{\sqrt{4}}}^{4}+4 \]
\[ 20760 = 4!+{\frac{4!}{\sqrt{4}}}^{4} \]
\[ 21296 = \frac{{\sqrt{\sqrt{\sqrt{44}}}}^{4!}}{4} \]
\[ 21928 = {\sqrt{\sqrt{\sqrt{\left(4+4!\right)}}}}^{4!}-4! \]
\[ 21948 = {\sqrt{\sqrt{\sqrt{\left(4+4!\right)}}}}^{4!}-4 \]
\[ 21950 = {\sqrt{\sqrt{\sqrt{\left(4+4!\right)}}}}^{4!}-\sqrt{4} \]
\[ 21952 = {\left(4+4!\right)}^{\sqrt{\frac{4}{.\dot{4}}}} \]
\[ 21954 = {\sqrt{\sqrt{\sqrt{\left(4+4!\right)}}}}^{4!}+\sqrt{4} \]
\[ 21956 = {\sqrt{\sqrt{\sqrt{\left(4+4!\right)}}}}^{4!}+4 \]
\[ 21976 = 4!+{\sqrt{\sqrt{\sqrt{\left(4+4!\right)}}}}^{4!} \]
\[ 22476 = {\frac{\sqrt{4!}}{.4}}^{4}-4! \]
\[ 22496 = {\frac{\sqrt{4!}}{.4}}^{4}-4 \]
\[ 22498 = {\frac{\sqrt{4!}}{.4}}^{4}-\sqrt{4} \]
\[ 22500 = \frac{{\frac{4}{.4}}^{4}}{.\dot{4}} \]
\[ 22502 = {\frac{\sqrt{4!}}{.4}}^{4}+\sqrt{4} \]
\[ 22504 = 4+{\frac{\sqrt{4!}}{.4}}^{4} \]
\[ 22524 = {\frac{\sqrt{4!}}{.4}}^{4}+4! \]
\[ 23040 = \frac{{\sqrt{\left(4!\times4\right)}}^{4}}{.4} \]
\[ 23328 = \frac{{\sqrt{\sqrt{\left(4+\sqrt{4}\right)}}}^{4!}}{\sqrt{4}} \]
\[ 23958 = \frac{{\sqrt{\sqrt{\sqrt{\left(4!-\sqrt{4}\right)}}}}^{4!}}{.\dot{4}} \]
\[ 24000 = \sqrt{{\sqrt{\sqrt{\frac{4}{.4}}}}^{4!}}\times4! \]
\[ 24389 = {\sqrt{\sqrt{\sqrt{\left(\frac{\sqrt{4}}{.4}+4!\right)}}}}^{4!} \]
\[ 24576 = \left({4}^{4}\times4!\right)\times4 \]
\[ 25000 = \frac{{\frac{4}{.4}}^{4}}{.4} \]
\[ 25344 = 4!\times\left(4!\times44\right) \]
\[ 25600 = {\frac{\left(4+4\right)}{\sqrt{.4}}}^{4} \]
\[ 26244 = 4\times{\frac{4}{.\dot{4}}}^{4} \]
\[ 26364 = \frac{{\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+4!\right)}}}}^{4!}}{\sqrt{.\dot{4}}} \]
\[ 26620 = \frac{{\sqrt{\sqrt{\sqrt{\left(4!-\sqrt{4}\right)}}}}^{4!}}{.4} \]
\[ 27000 = {\sqrt{\sqrt{\sqrt{\frac{4!}{\left(.4+.4\right)}}}}}^{4!} \]
\[ 27648 = \left(\left(4!\times4!\right)\times\sqrt{4}\right)\times4! \]
\[ 28561 = {\left(4+\frac{4}{.\dot{4}}\right)}^{4} \]
\[ 30625 = {\frac{\sqrt{\left(4+4!\right)}}{.4}}^{4} \]
\[ 30976 = {\left(4\times44\right)}^{\sqrt{4}} \]
\[ 31104 = 4!\times{\left(4+\sqrt{4}\right)}^{4} \]
\[ 31250 = {\sqrt{\sqrt{\frac{\sqrt{4}}{.4}}}}^{4!}\times\sqrt{4} \]
\[ 32000 = {\sqrt{\sqrt{\sqrt{\left(4!-4\right)}}}}^{4!}\times4 \]
\[ 32744 = {\sqrt{\sqrt{\sqrt{4}}}}^{\frac{4!}{.4}}-4! \]
\[ 32764 = {\sqrt{\sqrt{\sqrt{4}}}}^{\frac{4!}{.4}}-4 \]
\[ 32766 = {\sqrt{\sqrt{\sqrt{4}}}}^{\frac{4!}{.4}}-\sqrt{4} \]
\[ 32768 = \frac{{\left(4\times4\right)}^{4}}{\sqrt{4}} \]
\[ 32770 = {\sqrt{\sqrt{\sqrt{4}}}}^{\frac{4!}{.4}}+\sqrt{4} \]
\[ 32772 = {\sqrt{\sqrt{\sqrt{4}}}}^{\frac{4!}{.4}}+4 \]
\[ 32792 = 4!+{\sqrt{\sqrt{\sqrt{4}}}}^{\frac{4!}{.4}} \]
\[ 32928 = \frac{{\sqrt{\sqrt{\sqrt{\left(4+4!\right)}}}}^{4!}}{\sqrt{.\dot{4}}} \]
\[ 33750 = \frac{{\frac{\sqrt{4!}}{.4}}^{4}}{\sqrt{.\dot{4}}} \]
\[ 34560 = \frac{{4!}^{4}}{\left(4!\times.4\right)} \]
\[ 35152 = {\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+4!\right)}}}}^{4!}\times\sqrt{4} \]
\[ 35937 = {\sqrt{\sqrt{\sqrt{\left(\frac{4}{.\dot{4}}+4!\right)}}}}^{4!} \]
\[ 36864 = \frac{{4!}^{4}}{4}\times.\dot{4} \]
\[ 38400 = {\frac{4}{\sqrt{.4}}}^{4}\times4! \]
\[ 38416 = {\left(\frac{4}{.4}+4\right)}^{4} \]
\[ 39304 = {\sqrt{\sqrt{\sqrt{\left(4!+\frac{4}{.4}\right)}}}}^{4!} \]
\[ 39366 = \frac{{\sqrt{\sqrt{\sqrt{\frac{4!}{.\dot{4}}}}}}^{4!}}{4} \]
\[ 39546 = \frac{{\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+4!\right)}}}}^{4!}}{.\dot{4}} \]
\[ 40000 = {\frac{4}{.4}}^{4}\times4 \]
\[ 40960 = \frac{\left(4\times{\sqrt{\sqrt{4}}}^{4!}\right)}{.4} \]
\[ 41472 = \frac{{4!}^{4}}{\left(4+4\right)} \]
\[ 42592 = \frac{{\sqrt{\sqrt{\sqrt{44}}}}^{4!}}{\sqrt{4}} \]
\[ 42875 = {\sqrt{\sqrt{\sqrt{\frac{\left(4!-\sqrt{.\dot{4}}\right)}{\sqrt{.\dot{4}}}}}}}^{4!} \]
\[ 43904 = \sqrt{4}\times{\sqrt{\sqrt{\sqrt{\left(4+4!\right)}}}}^{4!} \]
\[ 43940 = \frac{{\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+4!\right)}}}}^{4!}}{.4} \]
\[ 45000 = {\frac{\sqrt{4!}}{.4}}^{4}\times\sqrt{4} \]
\[ 46464 = 4!\times{\sqrt{44}}^{4} \]
\[ 46632 = {\sqrt{\sqrt{\left(4+\sqrt{4}\right)}}}^{4!}-4! \]
\[ 46652 = {\sqrt{\sqrt{\left(4+\sqrt{4}\right)}}}^{4!}-4 \]
\[ 46654 = {\sqrt{\sqrt{\left(4+\sqrt{4}\right)}}}^{4!}-\sqrt{4} \]
\[ 46656 = {\left(4+\sqrt{4}\right)}^{\left(4+\sqrt{4}\right)} \]
\[ 46658 = {\sqrt{\sqrt{\left(4+\sqrt{4}\right)}}}^{4!}+\sqrt{4} \]
\[ 46660 = {\sqrt{\sqrt{\left(4+\sqrt{4}\right)}}}^{4!}+4 \]
\[ 46680 = 4!+{\sqrt{\sqrt{\left(4+\sqrt{4}\right)}}}^{4!} \]
\[ 46875 = 4!\times{\sqrt{\sqrt{\frac{\sqrt{\sqrt{4}}}{.4}}}}^{4!} \]
\[ 49152 = {\sqrt{\sqrt{\sqrt{4}}}}^{44}\times4! \]
\[ 49392 = \frac{{\sqrt{\sqrt{\sqrt{\left(4+4!\right)}}}}^{4!}}{.\dot{4}} \]
\[ 50625 = {\frac{4!}{\left(4\times.4\right)}}^{4} \]
\[ 50653 = {\sqrt{\sqrt{\frac{\sqrt{\left(\sqrt{.\dot{4}}+4!\right)}}{\sqrt{\sqrt{.\dot{4}}}}}}}^{4!} \]
\[ 51840 = \frac{{\frac{4!}{\sqrt{4}}}^{4}}{.4} \]
\[ 53824 = {\left({4}^{4}-4!\right)}^{\sqrt{4}} \]
\[ 54000 = \frac{{\sqrt{\sqrt{\sqrt{\frac{4!}{.4}}}}}^{4!}}{4} \]
\[ 54872 = {\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+\frac{4!}{\sqrt{.\dot{4}}}\right)}}}}^{4!} \]
\[ 54880 = \frac{{\sqrt{\sqrt{\sqrt{\left(4+4!\right)}}}}^{4!}}{.4} \]
\[ 55296 = 4\times\frac{{4!}^{4}}{4!} \]
\[ 56250 = \frac{{\frac{\sqrt{4!}}{.4}}^{4}}{.4} \]
\[ 57600 = {\sqrt{\frac{\left(4!\times4\right)}{.4}}}^{4} \]
\[ 58564 = \frac{{\left(4!-\sqrt{4}\right)}^{4}}{4} \]
\[ 59049 = {\sqrt{\frac{4}{.\dot{4}}}}^{\frac{4}{.4}} \]
\[ 59319 = {\sqrt{\sqrt{\sqrt{\frac{\left(\sqrt{4}+4!\right)}{\sqrt{.\dot{4}}}}}}}^{4!} \]
\[ 62500 = {\frac{4}{\left(\sqrt{.4}\times.4\right)}}^{4} \]
\[ 63504 = {\left({4}^{4}-4\right)}^{\sqrt{4}} \]
\[ 63976 = {\sqrt{\sqrt{\frac{4}{\sqrt{.4}}}}}^{4!}-4! \]
\[ 63996 = {\sqrt{\sqrt{\frac{4}{\sqrt{.4}}}}}^{4!}-4 \]
\[ 63998 = {\sqrt{\sqrt{\frac{4}{\sqrt{.4}}}}}^{4!}-\sqrt{4} \]
\[ 64000 = .4\times{\left(4!-4\right)}^{4} \]
\[ 64002 = {\sqrt{\sqrt{\frac{4}{\sqrt{.4}}}}}^{4!}+\sqrt{4} \]
\[ 64004 = 4+{\sqrt{\sqrt{\frac{4}{\sqrt{.4}}}}}^{4!} \]
\[ 64024 = 4!+{\sqrt{\sqrt{\frac{4}{\sqrt{.4}}}}}^{4!} \]
\[ 64516 = {\left({4}^{4}-\sqrt{4}\right)}^{\sqrt{4}} \]
\[ 65512 = {\left(4\times4\right)}^{4}-4! \]
\[ 65532 = {\left(4\times4\right)}^{4}-4 \]
\[ 65534 = {\left(4\times4\right)}^{4}-\sqrt{4} \]
\[ 65536 = {4}^{\frac{\left(4-.\dot{4}\right)}{.\dot{4}}} \]
\[ 65538 = {\left(4\times4\right)}^{4}+\sqrt{4} \]
\[ 65540 = 4+{\left(4\times4\right)}^{4} \]
\[ 65560 = {\left(4\times4\right)}^{4}+4! \]
\[ 66564 = {\left(\sqrt{4}+{4}^{4}\right)}^{\sqrt{4}} \]
\[ 67600 = {\left(4+{4}^{4}\right)}^{\sqrt{4}} \]
\[ 69984 = {\frac{4!}{.\dot{4}}}^{\sqrt{4}}\times4! \]
\[ 70304 = {\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+4!\right)}}}}^{4!}\times4 \]
\[ 73728 = \frac{\left(.\dot{4}\times{4!}^{4}\right)}{\sqrt{4}} \]
\[ 74088 = {\sqrt{\sqrt{\sqrt{\left(44-\sqrt{4}\right)}}}}^{4!} \]
\[ 75625 = {\frac{\sqrt{44}}{.4}}^{4} \]
\[ 78125 = {\sqrt{\sqrt{\frac{\sqrt{4}}{.4}}}}^{\left(4+4!\right)} \]
\[ 78400 = {\sqrt{\left({4}^{4}+4!\right)}}^{4} \]
\[ 78732 = \frac{{\sqrt{\sqrt{\sqrt{\frac{4!}{.\dot{4}}}}}}^{4!}}{\sqrt{4}} \]
\[ 80000 = \frac{{\left(4!-4\right)}^{4}}{\sqrt{4}} \]
\[ 81920 = \left(4!-4\right)\times{\sqrt{\sqrt{4}}}^{4!} \]
\[ 82920 = \frac{{4!}^{4}}{4}-4! \]
\[ 82938 = \frac{\left({4!}^{4}-4!\right)}{4} \]
\[ 82940 = \frac{{4!}^{4}}{4}-4 \]
\[ 82942 = \frac{{4!}^{4}}{4}-\sqrt{4} \]
\[ 82943 = \frac{\left({4!}^{4}-4\right)}{4} \]
\[ 82944 = \frac{{\sqrt{\left(4\times4\right)}!}^{4}}{4} \]
\[ 82945 = \frac{\left(4+{4!}^{4}\right)}{4} \]
\[ 82946 = \sqrt{4}+\frac{{4!}^{4}}{4} \]
\[ 82948 = 4+\frac{{4!}^{4}}{4} \]
\[ 82950 = \frac{\left(4!+{4!}^{4}\right)}{4} \]
\[ 82968 = 4!+\frac{{4!}^{4}}{4} \]
\[ 85160 = {\sqrt{\sqrt{\sqrt{44}}}}^{4!}-4! \]
\[ 85180 = {\sqrt{\sqrt{\sqrt{44}}}}^{4!}-4 \]
\[ 85182 = {\sqrt{\sqrt{\sqrt{44}}}}^{4!}-\sqrt{4} \]
\[ 85184 = {44}^{\sqrt{\frac{4}{.\dot{4}}}} \]
\[ 85186 = {\sqrt{\sqrt{\sqrt{44}}}}^{4!}+\sqrt{4} \]
\[ 85188 = {\sqrt{\sqrt{\sqrt{44}}}}^{4!}+4 \]
\[ 85208 = 4!+{\sqrt{\sqrt{\sqrt{44}}}}^{4!} \]
\[ 86400 = {\sqrt{\frac{4!}{.4}}}^{4}\times4! \]
\[ 86436 = {\left(\frac{\sqrt{4!}}{.4}+\sqrt{4!}\right)}^{4} \]
\[ 87808 = 4\times{\sqrt{\sqrt{\sqrt{\left(4+4!\right)}}}}^{4!} \]
\[ 90000 = 4\times{\frac{\sqrt{4!}}{.4}}^{4} \]
\[ 90112 = \left(4!-\sqrt{4}\right)\times{\sqrt{\sqrt{4}}}^{4!} \]
\[ 91125 = {\sqrt{\sqrt{\sqrt{\frac{\left(4!-4\right)}{.\dot{4}}}}}}^{4!} \]
\[ 92160 = \frac{{4!}^{4}}{\left(4-.4\right)} \]
\[ 93312 = \frac{{4!}^{4}}{\left(4-.\dot{4}\right)} \]
\[ 96000 = {\sqrt{\sqrt{\sqrt{\frac{4!}{.4}}}}}^{4!}\times.\dot{4} \]
\[ 97336 = {\sqrt{\sqrt{\sqrt{\left(\sqrt{4}+44\right)}}}}^{4!} \]
\[ 97728 = \left({\sqrt{\sqrt{4}}}^{4!}-4!\right)\times4! \]
\[ 98208 = \left({\sqrt{\sqrt{4}}}^{4!}-4\right)\times4! \]
\[ 98256 = 4!\times\left({\sqrt{\sqrt{4}}}^{4!}-\sqrt{4}\right) \]
\[ 98280 = \left({\sqrt{\sqrt{4}}}^{4!}\times4!\right)-4! \]
\[ 98288 = 4!\times\left({\sqrt{\sqrt{4}}}^{4!}-\sqrt{.\dot{4}}\right) \]
\[ 98300 = \left({\sqrt{\sqrt{4}}}^{4!}\times4!\right)-4 \]
\[ 98302 = \left({\sqrt{\sqrt{4}}}^{4!}\times4!\right)-\sqrt{4} \]
\[ 98304 = {\left(4+4\right)}^{4}\times4! \]
\[ 98306 = \left({\sqrt{\sqrt{4}}}^{4!}\times4!\right)+\sqrt{4} \]
\[ 98308 = 4+\left({\sqrt{\sqrt{4}}}^{4!}\times4!\right) \]
\[ 98320 = \left({\sqrt{\sqrt{4}}}^{4!}+\sqrt{.\dot{4}}\right)\times4! \]
\[ 98328 = \left({\sqrt{\sqrt{4}}}^{4!}\times4!\right)+4! \]
\[ 98352 = 4!\times\left(\sqrt{4}+{\sqrt{\sqrt{4}}}^{4!}\right) \]
\[ 98400 = 4!\times\left(4+{\sqrt{\sqrt{4}}}^{4!}\right) \]
\[ 98880 = 4!\times\left(4!+{\sqrt{\sqrt{4}}}^{4!}\right) \]
\[ 100000 = {\frac{4}{.4}}^{\frac{\sqrt{4}}{.4}} \]

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