Four 4's

Remark

There is a convention I applied to identities below, especially for the numbers beyond 100. As the table grew, some awkward formulas became more common. In order to simplify the entries and shorten the file that contains this page, I introduced several shorthands as listed below. These are used to express numbers with a single 4:

[√√4] = 1
[√√√√4!!] = [24!^1/16] = 30
[√30] = 5
5! = 120
[√120] = 10
[√10] = 3
3! = 6
[√(6!)] = 26

where brackets denote the whole part function. ([x] is the largest integer not exceeding x.)

Now, let's agree to use {1},{30},{5},{120},{3} as substitutes for the corresponding formulas. Thus for example, {3} means 3 expressed as [√[√[√√√√√4!!]!]]. Similarly, {5} and {120} stand for 5 and 120, respectively, in terms of the just described expressions.

1	4·4/4/4 = (4 + 4)/(4 + 4) = 44/44^(¹)
2	4/4 + 4/4 = 4/√4 + 4 - 4
3	(4 + 4 + 4)/4 = 4!!/4 + 4/4 = 4!/4!!·4/4
4	4 + 4·(4 - 4) = 4!!/(4 + 4)·4
5	(4 + 4·4)/4 = 4 + (√4 + √4)/4
6	4 + (4 + 4)/4 = 4!/4 + 4 - 4
7	4 + 4 - 4/4 = 4!/4 + 4/4
8	4 + 4 - 4 + 4 = 4!! + (4 - 4)/4 = (4 + 4)·4/4
9	4 + 4 + 4/4 = 4/.4 - 4/4 = 4/.(4) + 4 - 4 = (4!/4/√4)^√4^(¹)
10	(44 - 4)/4 = 4/.4 - 4 + 4 = 44/4.4^(¹)
11	44/(√4 + √4) = 4/.4 + 4/4
12	(44 + 4)/4 = 4/.4 + 4/√4
13	44/4 + √4 = 4!! + 4 + 4/4
14	4!/4 + 4 + 4 = 4!! + 4 + 4/√4
15	4·4 - 4/4 = 44/4 + 4^(¹) = ((√√√4)^4! - 4)/4^(²)
16	4 + 4 + 4 + 4 = 4·4·4/4 = 4⁴/4/4 = 4! - 4 - 4
17	4·4 + 4/4 = ((√√√4)^4! + 4)/4^(²)
18	4·4 + 4 - √4
19	4! - 4 - 4/4 = 4!!/.4 - 4/4
20	(4! - 4)·4/4 = (4! - 4) + 4 - 4
21	4! - 4 + 4/4 = 4!!/.4 + 4/4
22	4·4 + 4 + √4 = 4/.(4)·√4 + 4
23	4! - 4 + 4/4 = 4!!/.4 + 4/4
24	4·4 + 4 + 4 = 4/.4·√4 + 4
25	4! + (√4 + √4)/4
26	4! + (4 + 4)/4
27	4! + 4 - 4/4 = (4·4 - 4)/.4^(¹)
28	4! + 4 + 4 - 4 = (4! + 4)·4/4
29	4! + 4 + 4/4
30	4·4·√4 - √4 = ((√√√4)^4! - 4)/√4^(²)
31	4!!·4 - 4/4 = ((√√√4)^4! - √4)/√4^(²)
32	4!!·4 + 4 - 4
33	4!!·4 + 4/4 = ((√√√4)^4! + √4)/√4^(²)
34	4/.(4)·4 - √4 = 4·4·√4 + √4 = ((√√√4)^4! + 4)/√4^(²)
35	4!!/.(√4) - 4/4 = 4! + 44/4
36	4·(4 + 4) + 4 = 4·4·√4 + 4
37	4!!/.(√4) + 4/4
38	4!!/.(√4) + 4/√4
39	4!!/.√4 - 4/4
40	4·4/.(4) + 4 = 4!!·(4 + 4/4) = 4!!/.√4 + 4 - 4
41	4!!/.√4 + 4/4
42	44 - 4/√4
43	44 - 4/4
44	44 + 4 - 4
45	44 + 4/4
46	44 + 4/√4
47	4!·√4 - 4/4
48	44 + √4 + √4
49	4!·√4 + 4/4
50	4!·√4 + 4/√4
51	4!·√4 + 4 - {1} = (4! - 4 + .4)/.4^(¹)
52	(4! + √4)·4/√4
53	4!·√4 + 4 + {1}
54	4!·√4 + 4 + √4
55	44/√4/.4^(¹)
56	4!·√4 + 4 + 4 = 4! + 4! + 4 + 4^(¹)
57	44 + [√(4!!·4!)]
58	4!/.4 - 4 + √4^(¹)
59	4!/.4 - 4/4^(¹)
60	4·4·4 - 4^(¹)
61	4!/.4 + 4/4^(¹)
62	4·4·4 - √4^(¹) = 4!!^√4 - 4/√4
63	4!!^√4 - 4/4
64	4! + 44 - 4^(¹)
65	4!!^√4 + 4/4
66	4!/.4 + 4!/4^(¹) = 4·4·4 + √4^(¹)
67	4!!^√4 + 4 - {1}
68	4·4·4 + 4^(¹)
69	4!!^√4 + 4 + {1}
70	44 + 4! + √4^(¹)
71	(4! + 4 + .4 )/.4^(¹)
72	(4!·4!)/(4 + 4)^(¹)
73	4!!/.4 + [√(4!!·4!)]
74	4!·4 - 4! - √4^(¹)
75	(4! + 4 + √4)/.4^(¹)
76	4!·4 - 4! + 4^(¹) = (4! - 4)·4 - 4 = 4!!/.4·4 - 4
77	(4!!)^√4 + [√(4!!·4!)]
78	(4! - 4)·4 - √4^(¹)
79	(4! - 4)·4 - {1}
80	(44 - 4!)·4^(¹) = 4!!/.4·4·{1}
81	(4 - 4/4)⁴^(¹)
82	(4! - 4)·4 + √4^(¹)
83	4!·4 - [√(4!!·4!)]
84	44·√4 - 4^(¹) = (4! - √4)·4 -4^(¹) = (4! - 4)·4 + 4 = 4!!/.4·4 + 4
85	(4! + 4/.4)/.4^(¹)
86	(4! - √4)·4 - √4^(¹)
87	(4! - √4)·4 - {1}
88	44·(4 - √4)^(¹) = 4·4·4 + 4!^(¹)
89	(4! - √4)·4 + {1}
90	(4! - √4)·4 + √4^(¹)
91	4!·4 - √4/.4
92	(4! - 4/4)·4^(¹)
93	4!·4 - [4/√√√4]
94	4!·4 - 4 + √4^(¹)
95	4!·4 - 4/4^(¹)
96	(4 + 4/4)! - 4!^(¹)
97	4!·4 + 4/4^(¹)
98	4!·4 + 4 - √4^(¹)
99	4!·4 + 4 - {1} = 44/.(44)
100	(4! + 4/4)·4^(¹) = 44/.44^(¹)
101	4!·4 + √4/.4
102	4!·4 + 4 + √4
103	4!·4 + 4!! - {1}
104	4!·4 + 4 + 4
105	4!·4 + 4!! + {1}
106	4!·4 + 4!! + √4 = 44/.4 - 4
107	44/.(4) + 4!!
108	4!·4 + 4!! + 4 = 44/.4 - √4
109	44/.4 - {1}
110	44/.4·{1}
111	44/.4 + {1}
112	44/.4 + √4
113	44/.4 + {3}
114	44/.4 + 4
115	44/.4 + {5}
116	4!/.4·√4 - 4
117	4!/.4·√4 - {3}
118	4!/.4·√4 - √4
119	4!/.4·√4 - {1}
120	4!/.4·√4·{1}
121	4!/.4·√4 + {1}
122	4!/.4·√4 + √4
123	4!/.4·√4 + {3}
124	4!/.4·√4 + 4 = (4!!^√4 - √4)·√4 = 4!!^√4·√4 - 4
125	4!!^√4·√4 - {3}
126	4!!^√4·√4 - √4
127	4!!^√4·√4 - {1}
128	4!!^√4·√4·{1}
129	4!!^√4·√4 + {1}
130	4!!^√4·√4 + √4
131	4!!^√4·√4 + {3}
132	4!!^√4·√4 + 4
133	4!!^√4·√4 + {5}
134	{120} + 4/.4 + 4 = {120} + 4·4 - √4
135	{120} + 4·4 - [√√4]
136	{120} + (√4 + √4)·4 = 4!!^√4·√4 + 4!!
137	{120} + 4/.(4) + 4!!
138	{120} + 4·4 + √4
139	{120} + [4·4!/√4!]
140	{120} + 4!! + 4!! + 4 = {120} + 4·4 + 4
141	{120} + 4! - 4 + [√√4]
142	{120} + 4! - 4/√4
143	{120} + 4! - 4/4
144	{120} + 4! + 4 - 4 = {120} + 4!! + 4!! + 4!!
145	{120} + 4! + 4/4
146	{120} + 4! + 4/√4
147	{120} + 4! + 4 - [√√4]
148	{120} + 4! + √(4·4)
149	{120} + 4! + 4 + [√√4]
150	{120} + 4! + 4 + √4
151	{120} + 4!!·4 - [√√4]
152	{120} + 4! + 4 + 4
153	{120} + 4!!·4 + [√√4] = {120} + 4! + 4/.(4)
154	{120} + 4! + 4/.4
155	{120} + {30} + 4 + {1}
156	{120} + {30} + 4 + √4
157	{120} + {30} + 4 + {3}
158	{120} + {30} + 4 + 4
159	{120} + {30} + {5} + 4
160	{120} + {30} + {10}·{1}
161	{120} + {30} + {10} + {1}
162	{120} + {30} + {10} + √4
163	{120} + {30} + {10} + {3}
164	{120} + {30} + {10} + 4
165	{120} + {30} + {10} + {5}
166	{120} + {30} + 4!! + 4!!
167	{120} + 4! + 4! - {1}
168	{120} + 4! + 4!·{1}
169	{120} + 4! + 4! + {1}
170	{120} + 4! + 4! + √4
171	{120} + 4! + 4! + {3}
172	{120} + 4! + 4! + 4
173	{120} + 4! + 4! + {5}
174	{120} + 4! + 4! + {3}!
175	{120} + {30} + 4! + {1}
176	{120} + 4! + 4! + 4!!
177	{120} + {30} + 4! + {3}
178	{120} + {30} + 4! + 4
179	{120} + {30} + 4! + {5}
180	{120} + {30} + {30}·{1} = {120} + {120} - {30} - {30} = {30}·{10} - {120}·{1}
181	{120} + {30} + {30} + {1}
182	{120} + {30} + {30} + √4
183	{120} + {30} + {30} + {3}
184	{120} + {30} + {30} + 4
185	{120} + {30} + {30} + {5}
186	{120} + {30} + {30} + {3}!
187	{120} + 4! · {3} - {5}
188	{120} + 4! · {3} - 4
189	{120} + 4! · {3} - {3}
190	{120} + 4! · {3} - √4
191	{120} + 4! · {3} - {1}
192	{120} + 4! + 4! + 4!
193	{120} + {26} · {3} - {5}
194	{120} + {26} · {3} - 4
195	{120} + {26} · {3} - {3}
196	{120} + {26} · {3} - √4
197	{120} + {26} · {3} - {1}
198	{120} + {26} · {3} · {1}
199	{120} + {26} · {3} + {1}
200	{120} + {26} · {3} + √4
201	{120} + {26} · {3} + {3}
202	{120} + {26} · {3} + 4
203	{120} + {26} · {3} + {5}
204	{120} + {26} · {3} + {3}!
205	{120} + {30} · {3} - {5}
206	{120} + {30} · {3} - 4
207	{120} + {30} · {3} - {3}
208	{120} + {30} · {3} - √4
209	{120} + {30} · {3} - {1}
210	{120} + {30} · {3} · {1}
211	{120} + {30} · {3} + {1}
212	{120} + {30} · {3} + √4
213	{120} + {30} · {3} + {3}
214	{120} + {30} · {3} + 4
215	{120} + {30} · {3} + {5}
216	{120} + {30} · {3} + {3}!
217	{120} + 4! · 4 + {1}
218	{120} + 4! · 4 + √4
219	{120} + 4! · 4 + {3}
220	{120} + {30} · {3} + {10} = {120} + 4! · 4 + 4
221	{120} + 4! · 4 + {5}
222	{120} + 4! · 4 + {3}!
223	{120} + {26} ·4 - {1}
224	{120} + {120} - 4*4
225	{120} + {120} - {5} · {3}
226	{120} + {120} - 4! + {10}
227	{120} + {26} ·4 + {3}
228	{120} + {26} ·4 + 4
229	{120} + {26} ·4 + {5}
230	{120} + {26} ·4 + {3}!
231	{120} + {120} - {3} - {3}!
232	{120} + {120} - {5} - {3}
233	{120} + {120} - 4 - {3}
234	{120} + {120} - {3} - {3}
235	{120} + {120} - 4 - {1}
236	{120} + {120} - 4 · {1}
237	{120} + {120} - 4 + {1}
238	{120} + {120} - {1} - {1}
239	{120} + {120} - {1} · {1}
240	{120} · {1} + {120} · {1}
241	{120} + {120} + {1} · {1}
242	{120} + {120} + {1} + {1}
243	{120} + {120} + 4 - {1}
244	{120} + {120} + 4 · {1}
245	{120} + {120} + 4 + {1}
246	{120} + {120} + {3} + {3}
247	{120} + {120} + 4 + {3}
248	{120} + {120} + {5} + {3}
249	{120} + {120} + {3} + {3}!
250	{120} + {120} + {5} + {5}
251	{120} + {120} + {5} + {3}!
252	{120} + {120} + {3} · 4
253	{26} · {5} + {120} + {3}
254	{120} + {120} + 4! - {10}
255	{120} + {120} + {3} · {5}
256	{120} + {120} + 4 · 4
257	{26} · {5} · {2} - {3}
258	{120} + {120} + 4! - {3}!
259	{120} + {120} + 4! - {5}
260	{120} + {120} + {10} + {10}
261	{26} · {10} + {1} · {1}
262	{26} · {10} + {1} + {1}
263	{26} · {10} + {3} · {1}
264	{26} · {10} + 4 · {1}
265	{26} · {10} + {5} · {1}
266	{26} · {10} + {3}! · {1}
267	{26} · {10} + {3} + 4
268	{26} · {10} + 4 + 4
269	{26} · {10} + {5} + 4
270	{26} · {10} + {10} · {1}
271	{26} · {10} + {10} + {1}
272	{26} · {10} + {3}! + {3}!
273	{26} · {10} + {5} + 4!!
274	{26} · {10} + {10} + 4
275	{26} · {10} + {10} + {5}
276	{26} · {10} + {10} + {3}!
278	{26} · {10} + 4! - {3}!
279	{26} · {10} + 4! - {5}
280	{26} · {10} + {10} + {10}
281	{26} · {10} + 4! - {3}
282	{26} · {10} + 4! - √4
283	{26} · {10} + 4! - {1}
284	{26} · {10} + 4! · {1}
285	{26} · {10} + 4! + {1}
286	{26} · {10} + 4! + √4
287	{26} · {10} + 4! + {3}
288	{26} · {10} + 4! + 4
289	{26} · {10} + 4! + {5}
290	{26} · {10} + 4! + {3}!
291	{26} · {10} + {30} + {1}
292	{26} · {10} + 4! + 4!!
293	{26} · {10} + {30} + {3}
294	{26} · {10} + {30} + 4
295	{26} · {10} + {30} + {5}
296	{26} · {10} + {30} + {3}!
297	{30} · {10} - {3} · {1}
298	{26} · {10} + {30} + 4!!
299	{30} · {10} - {1} · {1}
300	{30} · {10} · {1} · {1}
301	{30} · {10} + {1} · {1}
302	{30} · {10} + {1} + {1}
303	{30} · {10} + {3} · {1}
304	{30} · {10} + {3} + {1}
305	{30} · {10} + {5} · {1}
306	{30} · {10} + {5} + {1}
307	{30} · {10} + {10} - {3}
308	{30} · {10} + 4 + 4^(¹)
309	{30} · {10} + {10} - {1}
310	{30} · {10} + {10} · {1}
311	{30} · {10} + {10} + {1}^(¹)
312	{30} · {10} + {3}! + {3}!
313	{30} · {10} + {10} + {3}
314	{30} · {10} + {3}! + 4!!
315	{30} · {10} + {3}·{5}
316	{30} · {10} + 4·4
317	4! · {26} / √4 + {5}^(¹)
318	{30} · {10} + {3}·{3}! = ({3}·{6})^√4 - {6}
319	{3}⁴ · 4 - {5}^(¹) = ({3}·{6})^√4 - {5}
320	{30} · {10} + {10} + {10} = ({3}·{6})^√4 - 4
321	{3}⁴ · 4 - {3}^(¹) = ({3}·{6})^√4 - {3}
322	{3}⁴ · 4 - √4^(¹) = ({3}·{6})^√4 - √4
323	{3}⁴ · 4 - {1}^(¹) = ({3}·{6})^√4 - {1}
324	(4 - {1})⁴ · 4^(¹) = {3}·{6}·{3}·{6} = {3} · {120} - {30} - {6}
325	{3}⁴ · 4 + {1}^(¹) = ({3}·{6})^√4 + {1}
326	{3}⁴ · 4 + √4^(¹) = ({3}·{6})^√4 + √4
327	{3}⁴ · 4 + {3}^(¹) = ({3}·{6})^√4 + {3}
328	{3}⁴ · 4 + 4^(¹) = ({3}·{6})^√4 + 4
329	{3}⁴ · 4 + {5}^(¹) = ({3}·{6})^√4 + {5}
330	{3}⁴ · 4 + {6}^(¹) = ({3}·{6})^√4 + {6} = {3}·{120} - {30}·{1}
331	{3}·{120} - {30} + {1}
332	{3}·{120} - {30} + √4
333	{3}·{120} - {30} + {3}
334	{3}·{120} - {30} + 4 = ({3}·{6})^√4 + {10}
335	{3}·{120} - {30} + {5}
336	{3}·{120} - {30} + {6}
337	{3}·{120} - {26} + {3}
338	{3}·{120} - {26} + 4
339	{3}·{120} - {26} + {5}
340	{3}·{120} - {30} + {10} = {3}·{120} - {26} + {6}
341	{3}·{120} - 4! + {5}
342	{3}·{120} - 4! + {6}
343	({6} + {1})^{3}·{1}
344	{3}·{120} - {26} + {10}
345	({6} + {1})^{3} + √4
346	({6} + {1})^{3} + {3}
347	({6} + {1})^{3} + 4
348	({6} + {1})^{3} + {5}
349	({6} + {1})^{3} + {6}
350	{3}·{120} - {10}·{1}
351	{3}·{120} - {10} + {1}
352	{3}·{120} - {10} + √4
353	{3}·{120} - {10} + {3}
354	{3}·{120} - {10} + 4
355	{3}·{120} - {10} + {5}
356	{3}·{120} - {10} + {6}
357	{3}·{120} - {3}·{1}
358	{3}·{120} - {1} - {1}
359	{3}·{120} - {1} · {1}
360	{3}·{120} - {1} + {1}
361	{3}·{120} + {1} · {1}
362	{3}·{120} + {1} + {1}
363	{3}·{120} + {3} · {1}
364	{3}·{120} + 4 · {1}
365	{3}·{120} + {5} · {1}
366	{3}·{120} + {6} · {1}
366	{3}·{120} + {6} · {1}
367	{3}·{120} + {6} + {1} = ({6} + {1})^{3} + 4!
368	{3}·{120} + {6} + √4
369	{3}·{120} + {6} + {3} = ({6} + {1})^{3} + {26}
370	{3}·{120} + {10}·{1}
371	{3}·{120} + {10} + {1}
372	{3}·{120} + {10} + √4
373	{3}·{120} + {10} + {3}
374	{3}·{120} + {10} + 4
375	{3}·{120} + {10} + {5}
376	{3}·{120} + {10} + {6}
377
378	{3}·{120} + 4! - {6}
379	{3}·{120} + 4! - {5}
380	{3}·{120} + {10} + {10}

(¹) By Andre Gustavo dos Santos, Brasil

(²) By Richard Tschumpel, Vienna, Austria

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