Symbol used

a+b: Addition (e.g. 5+3 = 8)

a-b: Subtraction (e.g. 5-3 = 2)

a*b: Multiplication (e.g. 3*5 = 15)

a/b: Division (e.g. 3/5 = 0.2)

a^b: Exponentiation (e.g. 5^3 = 125)

-a: Additive inverse (e.g. -3)

a': Multiplicative inverse (e.g. 5' = 0.2)

.: Decimal separator (e.g. .3 = 3/10)

R: Repeating decimal (e.g. .5R = 5/9)

(): parentheses (e.g. (3+5)*7 = 56)

%: Percentage (e.g. 3% = 3/100)

sq(a): Square function (e.g. sq(5) = 25)

sqrt(a): Square root function (e.g. sqrt(9) = 3)

cb(a): Cube function (e.g. cb(5) = 125)

cbrt(a): Cube root function (e.g. cbrt(27) = 3)

a!: Factorial (e.g. 5! = 120)

gamma(a): Gamma function (e.g. gamma(5) = 24)

We use the least number of 4's to write the nonnegative integers.

These symbols cannot be used:

Logarithm function (e.g. log(3,9) = 2), since if we allow logarithm, then we can use three 4's to write every nonnegative integer: n = -log(sqrt(4),log(4,sqrt(sqrt(sqrt(...sqrt(sqrt(sqrt(4)))...)))))

Trigonometric functions (the unit is turn, e.g. sin(1/6) = sqrt(3)/2)

a#: Primorial (e.g. 7# = 210)

a!!: Double factorial (e.g. 6!! = 48)

a!!!, a!!!!, a!!!!!, etc. Multifactorial (e.g. 8!!! = 80)

!a: Subfactorial (e.g. !6 = 265)

H(n): Hyperfactorial

sf(n): Superfactorial

Euler's totient function (e.g. eulerphi(60) = 16)

Divisor function (e.g. sigma(1,60) = 168)

Floor function (e.g. floor(3.7) = 3)

Ceiling function (e.g. ceiling(3.7) = 4)

Absolute value function (e.g. |-5| = 5)

Sign function (e.g. sgn(3) = 1)

Summation (e.g. sigma(n=1)^5 = 15)

Tetration, Pentation, Hexation, etc. (e.g. 2^^4 = 65536)

Hyperoperation (e.g. 2[4]4 = 65536)

Fourth power function, fifth power function, sixth power function, etc.

Fourth root function, fifth root function, sixth root function, etc.

Other numbers (including 0 and 1, and math constants such as pi and e)

Except 4, 44, 444, 4444, other numbers cannot be used with decimal numbers, for example, ".(sqrt(4))" = 1/5, ".(sqrt(4))R" = 2/9, ".(sq(4))" = 4/25, ".(sq(4))R" = 16/99, ".(4!)" = 6/25, ".(4!)R" = 8/33, etc. are not allowed, like that we do not use "4(sqrt(4))" = 42 and "4(sq(4))" = 416 and "4(4!)" = 424

Data

0 = 4-4

1 = gamma(sqrt(4))

2 = sqrt(4)

3 = sqrt(4/.4R)

4 = 4

5 = (sqrt(4%))'

6 = gamma(4)

7 = gamma(4)+gamma(sqrt(4))

8 = cb(sqrt(4))

9 = 4/.4R

10 = (sqrt((gamma(sqrt(4)))%))'

11 = gamma(4)+(sqrt(4%))'

12 = 4!/sqrt(4)

13 = cb(sqrt(4))+(sqrt(4%))'

14 = sq(4)-sqrt(4)

15 = sq(4)-gamma(sqrt(4))

16 = sq(4)

17 = sq(4)+gamma(sqrt(4))

18 = sq(4)+sqrt(4)

19 = (4%)'-gamma(4)

20 = sqrt(((4')%)')

21 = (4%)'-4

22 = 4!-sqrt(4)

23 = (4%)'-sqrt(4)

24 = 4!

25 = (4%)'

26 = 4!+sqrt(4)

27 = (4%)'+sqrt(4)

28 = 4!+4

29 = (4%)'+4

30 = 4!+gamma(4)

31 = (4%)'+gamma(4)

32 = sq(4)*sqrt(4)

33 = (4%)'+cb(sqrt(4))

34 = sq(gamma(4))-sqrt(4)

35 = sq(gamma(4))-gamma(sqrt(4))

36 = sq(gamma(4))

37 = sq(gamma(4))+gamma(sqrt(4))

38 = sq(gamma(4))+sqrt(4)

39 = (((.4)')%)'-gamma(sqrt(4))

40 = (((.4)')%)'

41 = (4%)'+sq(4)

42 = ((sqrt(4))%)'-cb(sqrt(4))

43 = 44-gamma(sqrt(4))

44 = 44

45 = (((.4)')%)'+(sqrt(4%))'

46 = (((.4)')%)'+gamma(4)

47 = 4!*sqrt(4)-gamma(sqrt(4))

48 = 4!*sqrt(4)

49 = sq(gamma(4)+gamma(sqrt(4)))

50 = ((sqrt(4))%)'

51 = ((sqrt(4))%)'+gamma(sqrt(4))

52 = ((sqrt(4))%)'+sqrt(4)

53 = sq(gamma(4)+gamma(sqrt(4)))+4

54 = ((sqrt(4))%)'+4

55 = ((sqrt(4))%)'+(sqrt(4%))'

56 = cb(4)-cb(sqrt(4))

57 = sq(4)*sqrt(4)+(4%)'

58 = cb(4)-gamma(4)

59 = cb(4)-(sqrt(4%))'

60 = (sqrt(((sq(gamma(4)))')%))'

61 = (cb(sqrt(4%)))'-cb(4)

62 = cb(4)-sqrt(4)

63 = cb(4)-gamma(sqrt(4))

64 = cb(4)

65 = cb(4)+gamma(sqrt(4))

66 = cb(4)+sqrt(4)

67 = cb(4)+sqrt(4/.4R)

68 = cb(4)+4

69 = cb(4)+(sqrt(4%))'

70 = cb(4)+gamma(4)

71 = cb(4)+gamma(4)+gamma(sqrt(4))

72 = cb(4)+cb(sqrt(4))

73 = cb(4)+4/.4R

74 = (cb(sqrt(4%)))'+4!

75 = (((gamma(sqrt(4))-4')')%)'

76 = ((gamma(sqrt(4)))%)'-4!

77 = sq(4/.4R)-4

78 = cb(sqrt(((4')%)'))%-sqrt(4)

79 = cb(sqrt(((4')%)'))%-gamma(sqrt(4))

80 = cb(sqrt(((4')%)'))%

81 = sq(4/.4R)

82 = cb(sqrt(((4')%)'))%+sqrt(4)

83 = sq(4/.4R)+sqrt(4)

84 = cb(sqrt(((4')%)'))%+4

85 = cb(sqrt(((4')%)'))%+(sqrt(4%))'

86 = cb(sqrt(((4')%)'))%+gamma(4)

87 = cb(4)+4!-gamma(sqrt(4))

88 = cb(4)+4!

89 = cb(4)+(4%)'

90 = cb(sqrt(((4')%)'))%+(sqrt((gamma(sqrt(4)))%))'

91 = cb(4)+(4%)'+sqrt(4)

92 = ((gamma(sqrt(4)))%)'-cb(sqrt(4))

93 = cb(4)+(4%)'+4

94 = ((gamma(sqrt(4)))%)'-gamma(4)

95 = ((gamma(sqrt(4)))%)'-(sqrt(4%))'

96 = 4!*4

97 = (4!+4')*4

98 = ((gamma(sqrt(4)))%)'-sqrt(4)

99 = ((gamma(sqrt(4)))%)'-gamma(sqrt(4))

100 = ((gamma(sqrt(4)))%)'

101 = (cb(sqrt(4%)))'-4!

102 = ((gamma(sqrt(4)))%)'+sqrt(4)

103 = ((4%)'+sqrt((.4R)'))*4

104 = gamma(gamma(4))-sq(4)

105 = cb(sqrt(((4')%)'))%+(sqrt(4%))'

106 = ((gamma(sqrt(4)))%)'+gamma(4)

107 = gamma(gamma(4))-cb(sqrt(4))-(sqrt(4%))'

108 = ((gamma(sqrt(4)))%)'+cb(sqrt(4))

109 = ((4%)'+(.4R)')*4

110 = gamma(gamma(4))-(sqrt((gamma(sqrt(4)))%))'

111 = gamma(gamma(4))-4/.4R

112 = (4!+4)*4

113 = (cb(sqrt(4%)))'-4!/sqrt(4)

114 = gamma(gamma(4))-gamma(4)

115 = gamma(gamma(4))-(sqrt(4%))'

116 = gamma(gamma(4))-4

117 = (cb(sqrt(4%)))'-cb(sqrt(4))

118 = gamma(gamma(4))-sqrt(4)

119 = gamma(gamma(4))-gamma(sqrt(4))

120 = gamma(gamma(4))

121 = gamma(gamma(4))+gamma(sqrt(4))

122 = gamma(gamma(4))+sqrt(4)

123 = (cb(sqrt(4%)))'-sqrt(4)

124 = gamma(gamma(4))+4

125 = (cb(sqrt(4%)))'

126 = gamma(gamma(4))+gamma(4)

127 = (cb(sqrt(4%)))'+sqrt(4)

128 = cb(4)*sqrt(4)

129 = (cb(sqrt(4%)))'+4

130 = gamma(gamma(4))+(sqrt((gamma(sqrt(4)))%))'

131 = (cb(sqrt(4%)))'+gamma(4)

132 = cb(4)*sqrt(4)+4

133 = (cb(sqrt(4%)))'+cb(sqrt(4))

134 = cb(4)*sqrt(4)+gamma(4)

135 = (cb(sqrt(4%)))'+(sqrt((gamma(sqrt(4)))%))'

136 = gamma(gamma(4))+sq(4)

137 = gamma(gamma(4))+sq(4)+gamma(sqrt(4))

138 = gamma(gamma(4))+sq(4)+sqrt(4)

139 = gamma(gamma(4))+(4%)'-gamma(4)

140 = gamma(gamma(4))+sqrt(((4')%)')

141 = (cb(sqrt(4%)))'+sq(4)

142 = gamma(gamma(4))+4!-sqrt(4)

143 = gamma(gamma(4))+4!-gamma(sqrt(4))

144 = gamma(gamma(4))+4!

145 = gamma(gamma(4))+(4%)'

146 = gamma(gamma(4))+4!+sqrt(4)

147 = gamma(gamma(4))+(4%)'+sqrt(4)

148 = gamma(gamma(4))+4!+4

149 = (cb(sqrt(4%)))'+4!

150 = 4!/sq(.4)

151 = gamma(gamma(4))+(4%)'+gamma(4)

152 = cb(4)*sqrt(4)+4!

153 = gamma(gamma(4))+(4%)'+cb(sqrt(4))

154 = 4!/sq(.4)+4

155 = (cb(4)-sqrt(4))/.4

156 = gamma(gamma(4))+sq(gamma(4))

157 = (cb(sqrt(4%)))'+(sq(4)*sqrt(4))

158 = 4!/sq(.4)+cb(sqrt(4))

159 = cb(4)/.4-gamma(sqrt(4))

160 = (((sqrt((sq(sq(4)))%))')%)'

161 = (cb(sqrt(4%)))'+sq(gamma(4))

162 = (((sqrt((sq(sq(4)))%))')%)'+sqrt(4)

163 = sq(cb(sqrt(4))+(sqrt(4%))')-gamma(4)

164 = gamma(gamma(4))+44

165 = (cb(sqrt(4%)))'+(((.4)')%)'

166 = (((sqrt((sq(sq(4)))%))')%)'+gamma(4)

167 = (cb(sqrt(4%)))'+((sqrt(4))%)'-cb(sqrt(4))

168 = (((sqrt((sq(sq(4)))%))')%)'+cb(sqrt(4))

169 = sq(cb(sqrt(4))+(sqrt(4%))')

170 = gamma(gamma(4))+((sqrt(4))%)'

171 = sq(cb(sqrt(4))+(sqrt(4%))')+sqrt(4)

172 = gamma(gamma(4))+((sqrt(4))%)'+sqrt(4)

173 = sq(cb(sqrt(4))+(sqrt(4%))')+4

174 = gamma(gamma(4))+((sqrt(4))%)'+4

175 = sq(cb(sqrt(4))+(sqrt(4%))')+gamma(4)

176 = (((sqrt((sq(sq(4)))%))')%)'+sq(4)

177 = sq(cb(sqrt(4))+(sqrt(4%))')+cb(sqrt(4))

178 = gamma(gamma(4))+cb(4)-gamma(4)

179 = gamma(gamma(4))+cb(4)-(sqrt(4%))'

180 = (((sqrt((sq(sq(4)))%))')%)'+sqrt(((4')%)')

181 = cb(4)+(cb(sqrt(4%)))'-cb(sqrt(4))

182 = gamma(gamma(4))+cb(4)-sqrt(4)

183 = cb(4)+(cb(sqrt(4%)))'-gamma(4)

184 = gamma(gamma(4))+cb(4)

185 = cb(4)+(cb(sqrt(4%)))'-4

186 = (((sqrt((sq(sq(4)))%))')%)'+sqrt(((4')%)')+gamma(4)

187 = cb(4)+(cb(sqrt(4%)))'-sqrt(4)

188 = cb(4)+(cb(sqrt(4%)))'-gamma(sqrt(4))

189 = cb(4)+(cb(sqrt(4%)))'

190 = (((sqrt(4))')%)'-(sqrt((gamma(sqrt(4)))%))'

191 = cb(gamma(4))-4!-gamma(sqrt(4))

192 = cb(gamma(4))-4!

193 = cb(gamma(4))-4!+gamma(sqrt(4))

194 = (((sqrt(4))')%)'-gamma(4)

195 = (((sqrt(4))')%)'-(sqrt(4%))'

196 = sq(sq(4)-sqrt(4))

197 = cb(4)+(cb(sqrt(4%)))'+cb(sqrt(4))

198 = (((sqrt(4))')%)'-sqrt(4)

199 = (((sqrt(4))')%)'-gamma(sqrt(4))

200 = (((sqrt(4))')%)'

201 = (((sqrt(4))')%)'+gamma(sqrt(4))

202 = (((sqrt(4))')%)'+sqrt(4)

203 = (((sqrt(4))')%)'+sqrt(4/.4R)

204 = (((sqrt(4))')%)'+4

205 = (((sqrt(4))')%)'+(sqrt(4%))'

206 = (((sqrt(4))')%)'+gamma(4)

207 = cb(gamma(4))-4/.4R

208 = cb(gamma(4))-cb(sqrt(4))

209 = (((sqrt(4))')%)'+4/.4R

210 = cb(gamma(4))-gamma(4)

211 = cb(gamma(4))-(sqrt(4%))'

212 = cb(gamma(4))-4

213 = cb(gamma(4))-sqrt(4/.4R)

214 = cb(gamma(4))-sqrt(4)

215 = cb(gamma(4))-gamma(sqrt(4))

216 = cb(gamma(4))

217 = cb(gamma(4))+gamma(sqrt(4))

218 = cb(gamma(4))+sqrt(4)

219 = ((.4R)%)'-gamma(4)

220 = cb(gamma(4))+4

221 = ((.4R)%)'-4

222 = cb(gamma(4))+gamma(4)

223 = ((.4R)%)'-sqrt(4)

224 = cb(gamma(4))+cb(sqrt(4))

225 = ((.4R)%)'

226 = ((.4R)%)'+gamma(sqrt(4))

227 = ((.4R)%)'+sqrt(4)

228 = ((.4R)%)'+sqrt(4/.4R)

229 = ((.4R)%)'+4

230 = ((.4R)%)'+(sqrt(4%))'

231 = ((.4R)%)'+gamma(4)

232 = sq(sq(4))-4!

233 = ((.4R)%)'+cb(sqrt(4))

234 = (.4%)'-sq(4)

235 = ((.4R)%)'+(sqrt((gamma(sqrt(4)))%))'

236 = sq(sq(4))-sqrt(((4')%)')

237 = ((.4R)%)'+4!/sqrt(4)

238 = gamma(gamma(4))*sqrt(4)-sqrt(4)

239 = gamma(gamma(4))*sqrt(4)-gamma(sqrt(4))

240 = gamma(gamma(4))*sqrt(4)

241 = gamma(gamma(4))*sqrt(4)+gamma(sqrt(4))

242 = (.4%)'-cb(sqrt(4))

243 = cbrt(sqrt(sqrt(sqrt(sqrt(4/.4R)^gamma(gamma(4))))))

244 = (.4%)'-gamma(4)

245 = (.4%)'-(sqrt(4%))'

246 = (.4%)'-4

247 = sq(sq(4))-4/.4R

248 = sq(sq(4))-cb(sqrt(4))

249 = ((.4R)%)'+4!

250 = (.4%)'

251 = sq(sq(4))-(sqrt(4%))'

252 = sq(sq(4))-4

253 = sq(sq(4))-sqrt(4/.4R)

254 = sq(sq(4))-sqrt(4)

255 = sq(sq(4))-gamma(sqrt(4))

256 = sq(sq(4))

257 = sq(sq(4))+gamma(sqrt(4))

258 = sq(sq(4))+sqrt(4)

259 = sq(sq(4))+sqrt(4/.4R)

260 = sq(sq(4))+4

261 = sq(sq(4))+(sqrt(4%))'

262 = sq(sq(4))+gamma(4)

263 = sq(sq(4))+gamma(4)+gamma(sqrt(4))

264 = sq(sq(4))+cb(sqrt(4))

265 = sq(sq(4))+4/.4R

266 = (.4%)'+sq(4)

267 = (.4%)'+sq(4)+gamma(sqrt(4))

268 = (.4%)'+sq(4)+sqrt(4)

269 = 44+((.4R)%)'

270 = (.4%)'+sqrt(((4')%)')

272 = sq(sq(4))+sq(4)

280 = sq(sq(4))+4!

281 = sq(sq(4))+(4%)'

375 = 4!/cb(.4)

400 = ((4')%)'

481 = (gamma(gamma(4))+4')*4

500 = ((sqrt(4%))%)'

512 = cb(cb(sqrt(4)))

576 = sq(4!)

600 = (((gamma(4))')%)'

625 = sq((4%)')

720 = (gamma(4))!

800 = (((cb(sqrt(4)))')%)'

1000 = cb((sqrt((gamma(sqrt(4)))%))')