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)))%))')