[Math Game] The Four Fours

Z[P(φ(P(Γ(4)))) * P(P(Γ(4)))] = Z[37 * 41] = Z[1517]
which covers 1317-1717

see definition of Z[expression] on page 5 of this thread

Z[P(4!!)*P(4!)] = Z[19*89] = Z[1691]

Covers everything from 1491 to 1891.

Z[P(4!!)*P(4)!!] = Z[19*105] = Z[1995]

Covers everything from 1795 to 2195.

I'm recalling a topic I seem to remember from an Abstract Algebra class long ago and mapping transforms for integers. In general the length of a cycle of a mapping transform for integers is that integer--somehow I seem to be connecting this in my foggy memory to Fermat's Little Theorem, and about how these mapping transforms and their inverses being one to one.

So using these transforms you can use a one to one mapping from any number to any other as long as the integral cycle isn't completed.

example of a 13 digits cycle
1>3>5>7>9>11>0>2>4>6>8>10>12

So if I have a 4 and I need a 11 I apply this mapping 10 times (or the inverse 3 times).

Morover; by definition these mappings transforms and their inverses, are one to one and are functions.(?)

Rap

Are you looking to add these cycles as functions to use? How would you express them? Seems to me you'd need 4 integers:
- length of cycle (13 in your example)
- delta between values (2 in your example)
- starting value (4 in your example)
- number of steps to take (10 in your example)

This would seem to be an arbitrary function - as opposed to the, perhaps obscure, but at least documented in multiple places, functions we've been using.

My 2 cents.

Z[P(P(P(4))) * φ(P(P(Γ(4))))] = Z[59 * 40] = Z[2360]
which covers 2160-2560

see definition of Z[expression] on page 5 of this thread

Z[P(P(4!!)) * P(P(Γ(4)))] = Z[67 * 41] = Z[2747]
which covers 2547-2947

see definition of Z[expression] on page 5 of this thread

Z[φφPφφφφPφφPTPΓ(4) * P(P(P(P(√4))))] = Z[100*31] = Z[3100]


where T(n) is the nth "triangular number" from the sequence 1,3,6,10,...

This takes care of numbers 2900 through 3300.


====

Note:

φφPφφφφPφφPTPΓ(4) = φφPφφφφPφφPTP(6) = φφPφφφφPφφPT(13) = φφPφφφφPφφP(91) = φφPφφφφPφφ(467) = φφPφφφφPφ(466)
= φφPφφφφP(232) = φφPφφφφ(1459) = φφPφφφ(1458) = φφPφφ(486) = φφPφ(162) = φφP(54) = φφ(251) = φ(250) = 100

P(P(P(P(√4)))) = P(P(P(P(2)))) = P(P(P(3))) = P(P(5)) = P(11) = 31

Z[φ(4$) * T(4!!)] = Z[96*36] = Z[3456]

which covers 3256 through 3656.

...where T(n) is the nth triangular number; and "n$" is superfactorial.

Z[φ(P(P(P(√4)))) * (4!!)!!] = Z[10 * 384] = Z[3840]
which covers 3640-4040

see definition of Z[expression] on page 5 of this thread

Z[P(4)!! * φ(T(T(4)))] = Z[7!! * φ(55)] = Z[105 * 40] = Z[4200]
which covers 4000-4400

T(n) is the nth triangular number;
see definition of Z[expression] on page 5 of this thread

Here are the numbers 0-400 expressed with two fours. Henceforth, the function Z[expression containing two fours that equals N] represents N-400 through N+400.

0 = 4-4
1 = 4/4
2 = 4/√4
<SNIP: for 3 to 197, see THIS post.>
198 = !4*φ(P(!4))
199 = P(Γ(4)!!)-4!
200 = P(Γ(4)!!)-P(!4)
(the expressions in red below are calculated in detail in the footnotes)
201 = P(P([P(P(√4))]!!)) - T(4) ("T(n)" is nth triangular number)
202 = P(P([P(P(√4))]!!)) - !4
203 = P(P([P(P(√4))]!!)) - 4!!
204 = P(P([P(P(√4))]!!)) - P(4)
205 = P(P([P(P(√4))]!!)) - Γ(4)
206 = P(P([P(P(√4))]!!)) - P(P(√4))
207 = P(P([P(P(√4))]!!)) - 4
208 = P(P([P(P(√4))]!!)) - P(√4)
209 = P(P([P(P(√4))]!!)) - √4
210 = P(P([P(P(√4))]!!)) - !√4
211 = P(P([P(P(√4))]!!)) * !√4
212 = P(P([P(P(√4))]!!)) + !√4
213 = P(P([P(P(√4))]!!)) + √4
214 = P(P([P(P(√4))]!!)) + P(√4)
215 = P(P([P(P(√4))]!!)) + 4
216 = P(P([P(P(√4))]!!)) + P(P(√4))
217 = P(P([P(P(√4))]!!)) + Γ(4)
218 = P(P([P(P(√4))]!!)) + P(4)
219 = P(P([P(P(√4))]!!)) + 4!!
220 = P(P([P(P(√4))]!!)) + !4
221 = P(P([P(P(√4))]!!)) + T(4)
222 = T(T(Γ(4))) - !4
223 = T(T(Γ(4))) - 4!!
224 = T(T(Γ(4))) - P(4)
225 = T(T(Γ(4))) - Γ(4)
226 = T(T(Γ(4))) - P(P(√4))
227 = T(T(Γ(4))) - 4
228 = T(T(Γ(4))) - P(√4)
229 = T(T(Γ(4))) - √4
230 = T(T(Γ(4))) - !√4
231 = T(T(Γ(4))) * !√4
232 = T(T(Γ(4))) + !√4
233 = T(T(Γ(4))) + √4
234 = T(T(Γ(4))) + P(√4)
235 = T(T(Γ(4))) + 4
236 = T(T(Γ(4))) + P(P(√4))
237 = T(T(Γ(4))) + Γ(4)
238 = T(T(Γ(4))) + P(4)
239 = T(T(Γ(4))) + 4!!
240 = T(T(Γ(4))) + !4
241 = T(P(P(4))) - φ(P(Γ(4)))
242 = T(P(P(4))) - P(P(P(√4)))
243 = T(P(P(4))) - T(4)
244 = T(P(P(4))) - !4
245 = T(P(P(4))) - 4!!
246 = T(P(P(4))) - P(4)
247 = T(P(P(4))) - Γ(4)
248 = T(P(P(4))) - P(P(√4))
249 = T(P(P(4))) - 4
250 = T(P(P(4))) - P(√4)
251 = T(P(P(4))) - √4
252 = T(P(P(4))) - !√4
253 = T(P(P(4))) * !√4
254 = T(P(P(4))) + !√4
255 = T(P(P(4))) + √4
256 = T(P(P(4))) + P(√4)
257 = T(P(P(4))) + 4
258 = T(P(P(4))) + P(P(√4))
259 = T(P(P(4))) + Γ(4)
260 = T(P(P(4))) + P(4)
261 = T(P(P(4))) + 4!!
262 = T(P(P(4))) + !4
263 = T(P(P(4))) + T(4)
264 = T(P(P(4))) + P(P(P(√4)))
265 = T(P(P(4))) + φ(P(Γ(4)))
266 = P(P(P(P(4)))) - P(P(P(√4)))
267 = P(P(P(P(4)))) - T(4)
268 = P(P(P(P(4)))) - !4
269 = P(P(P(P(4)))) - 4!!
270 = P(P(P(P(4)))) - P(4)
271 = P(P(P(P(4)))) - Γ(4)
272 = P(P(P(P(4)))) - P(P(√4))
273 = P(P(P(P(4)))) - 4
274 = P(P(P(P(4)))) - P(√4)
275 = P(P(P(P(4)))) - √4
276 = P(P(P(P(4)))) - !√4
277 = P(P(P(P(4)))) * !√4
278 = P(P(P(P(4)))) + !√4
279 = P(P(P(P(4)))) + √4
280 = P(P(P(P(4)))) + P(√4)
281 = P(P(P(P(4)))) + 4
282 = P(P(P(P(4)))) + P(P(√4))
283 = P(P(P(P(4)))) + Γ(4)
284 = P(P(P(P(4)))) + P(4)
285 = P(P(P(P(4)))) + 4!!
286 = P(P(P(P(4)))) + !4
287 = P(P(P(P(4)))) + T(4)
288 = P(P(P(P(4)))) + P(P(P(√4)))
289 = T(4!) - P(P(P(√4)))
290 = T(4!) - T(4)
291 = T(4!) - !4
292 = T(4!) - 4!!
293 = T(4!) - P(4)
294 = T(4!) - Γ(4)
295 = T(4!) - P(P(√4))
296 = T(4!) - 4
297 = T(4!) - P(√4)
298 = T(4!) - √4
299 = T(4!) - !√4
300 = T(4!) * !√4
301 = T(4!) + !√4
302 = T(4!) + √4
303 = T(4!) + P(√4)
304 = T(4!) + 4
305 = T(4!) + P(P(√4))
306 = T(4!) + Γ(4)
307 = T(4!) + P(4)
308 = T(4!) + 4!!
309 = T(4!) + !4
310 = T(4!) + T(4)
311 = P(T(P(P(P(√4)))) - Γ(4)
312 = P(T(P(P(P(√4)))) - P(P(√4))
313 = P(T(P(P(P(√4)))) - 4
314 = P(T(P(P(P(√4)))) - P(√4)
315 = P(T(P(P(P(√4)))) - √4
316 = P(T(P(P(P(√4)))) - !√4
317 = P(T(P(P(P(√4)))) * !√4
318 = P(T(P(P(P(√4)))) + !√4
319 = P(T(P(P(P(√4)))) + √4
320 = P(T(P(P(P(√4)))) + P(√4)
321 = P(T(P(P(P(√4)))) + 4
322 = P(T(P(P(P(√4)))) + P(P(√4))
323 = P(T(P(P(P(√4)))) + Γ(4)
324 = P(T(P(P(P(√4)))) + P(4)
325 = P(T(P(P(P(√4)))) + 4!!
326 = P(T(P(P(P(√4)))) + !4
327 = P(T(P(P(P(√4)))) + T(4)
328 = P(T(P(P(P(√4)))) + P(P(P(√4)))
329 = P(T(P(P(P(√4)))) + φ(P(Γ(4)))
330 = P(T(P(P(P(√4)))) + P(Γ(4))
331 = P(P(P(4!!))) * !√4
332 = P(P(P(4!!))) + !√4
333 = P(P(P(4!!))) + √4
334 = P(P(P(4!!))) + P(√4)
335 = P(P(P(4!!))) + 4
336 = P(P(P(4!!))) + P(P(√4))
337 = P(P(P(4!!))) + Γ(4)
338 = P(P(P(4!!))) + P(4)
339 = P(P(P(4!!))) + 4!!
340 = P(P(P(4!!))) + !4
341 = P(P(P(4!!))) + T(4)
342 = P(P(P(4!!))) + P(P(P(√4)))
343 = P(P(φ(T(P(P(P(√4)))))) - T(4)
344 = P(P(φ(T(P(P(P(√4)))))) - !4
345 = P(P(φ(T(P(P(P(√4)))))) - 4!!
346 = P(P(φ(T(P(P(P(√4)))))) - P(4)
347 = P(P(φ(T(P(P(P(√4)))))) - Γ(4)
348 = P(P(φ(T(P(P(P(√4)))))) - P(P(√4))
349 = P(P(φ(T(P(P(P(√4)))))) - 4
350 = P(P(φ(T(P(P(P(√4)))))) - P(√4)
351 = P(P(φ(T(P(P(P(√4)))))) - √4
352 = P(P(φ(T(P(P(P(√4)))))) - !√4
353 = P(P(φ(T(P(P(P(√4)))))) * !√4
354 = P(P(φ(T(P(P(P(√4)))))) + !√4
355 = P(P(φ(T(P(P(P(√4)))))) + √4
356 = P(P(φ(T(P(P(P(√4)))))) + P(√4)
357 = P(P(φ(T(P(P(P(√4)))))) + 4
358 = P(P(φ(T(P(P(P(√4)))))) + P(P(√4))
359 = P(P(φ(T(P(P(P(√4)))))) + Γ(4)
360 = P(P(φ(T(P(P(P(√4)))))) + P(4)
361 = P(P(φ(T(P(P(P(√4)))))) + 4!!
362 = P(P(φ(T(P(P(P(√4)))))) + !4
363 = P(P(φ(T(P(P(P(√4)))))) + T(4)
364 = P(P(φ(T(P(P(P(√4)))))) + P(P(P(√4)))
365 = P(P(φ(T(P(P(P(√4)))))) + φ(P(Γ(4)))
366 = P(P(φ(T(P(P(P(√4)))))) + P(Γ(4))
367 = F[P(P(√4))!!] - T(4)
368 = F[P(P(√4))!!] - !4
369 = F[P(P(√4))!!] - 4!!
370 = F[P(P(√4))!!] - P(4)
371 = F[P(P(√4))!!] - Γ(4)
372 = F[P(P(√4))!!] - P(P(√4))
373 = F[P(P(√4))!!] - 4
374 = F[P(P(√4))!!] - P(√4)
375 = F[P(P(√4))!!] - √4
376 = F[P(P(√4))!!] - !√4
377 = F[P(P(√4))!!] * !√4
378 = F[P(P(√4))!!] + !√4
379 = F[P(P(√4))!!] + √4
380 = F[P(P(√4))!!] + P(√4)
381 = F[P(P(√4))!!] + 4
382 = F[P(P(√4))!!] + P(P(√4))
383 = F[P(P(√4))!!] + Γ(4)
384 = F[P(P(√4))!!] + P(4)
385 = F[P(P(√4))!!] + 4!!
386 = F[P(P(√4))!!] + !4
387 = F[P(P(√4))!!] + T(4)
388 = F[P(P(√4))!!] + P(P(P(√4)))
389 = F[P(P(√4))!!] + φ(P(Γ(4)))
390 = F[P(P(√4))!!] + P(Γ(4))
391 = P(P(φ(P(!4)))) - T(4)
392 = P(P(φ(P(!4)))) - !4
393 = P(P(φ(P(!4)))) - 4!!
394 = P(P(φ(P(!4)))) - P(4)
395 = P(P(φ(P(!4)))) - Γ(4)
396 = P(P(φ(P(!4)))) - P(P(√4))
397 = P(P(φ(P(!4)))) - 4
398 = P(P(φ(P(!4)))) - P(√4)
399 = P(P(φ(P(!4)))) - √4
400 = P(P(φ(P(!4)))) - !√4
401 = P(P(φ(P(!4)))) * !√4

FOOTNOTES:
------------------------------
The "... * !√4" part in what follows is just is multiplication by 1, so we just want to show the red terms equal the numbers to the left...

211 = P(P([P(P(√4))]!!)) * !√4

P(P([P(P(√4))]!!)) = P(P([P(P(2))]!!)) = P(P(5!!)) = P(P(15)) = P(47) = 211

231 = T(T(Γ(4))) * !√4

T(T(Γ(4))) = T(T(6)) = T(21) = 231

253 = T(P(P(4))) * !√4

T(P(P(4))) = T(P(7)) = T(17) = 253

277 = P(P(P(P(4)))) * !√4

P(P(P(P(4)))) = P(P(P(7))) = P(P(17)) = P(59) = 277

300 = T(4!) * !√4

T(4!) = T(24) = 300

317 = P(T(P(P(P(√4)))) * !√4

P(T(P(P(P(√4))))) = P(T(P(P(P(2))))) = P(T(P(5))) = P(T(11)) = P(66) = 317

331 = P(P(P(4!!))) * !√4

P(P(P(4!!))) = P(P(P(8))) = P(P(19)) = P(67) = 331

353 = P(P(φ(T(P(P(P(√4)))))) * !√4

P(P(φ(T(P(P(P(√4)))))) = P(P(φ(T(P(P(P(2)))))) = P(P(φ(T(P(5)))) = P(P(φ(T(11))) = P(P(φ(66)) = P(P(20)) = P(71) = 353

377 = F[P(P(√4))!!] * !√4

F[P(P(√4))!!] = F[P(P(2))!!] = F[5!!] = F[15] = (15th number in the sequence 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,...) = 377

Some people would consider 377 to be the 14th term in the Fibonacci sequence,
but that's not a problem, because in that case we have:
377 = F[14] = F[√196] = F[√[φ(197)]] = F[√[φ(P(45))]] = F[√[φ(P(T(9)))]] = F[√[φ(P(T(!4)))]]

401 = P(P(φ(P(!4)))) * !√4

P(P(φ(P(!4)))) = P(P(φ(P(9)))) = P(P(φ(23))) = P(P(22)) = P(79) = 401

Z[P(4)!! * T(!4)] = Z[105 * 45] = Z[4725]
which covers 4325 to 5125 (because of the new improvements to Z[...])

<> T(n) is the nth triangular number.
<> See definition of Z[expression] on page 5 of this thread.
<> See the long post on page 6 for how the power of Z[expression] has grown.

Nice work. Now if rap can provide the next 200...

I noticed your note about the Fibonacci function. I tried to avoid it for that very reason.

Z[P(Γ(4)! + φ(P(P(P(√4)))))] = Z[P(720+10)] = Z[P(730)] = Z[5521]
which covers 5121 to 5921 because of the new Z[...] function described on page 6 of this thread

Z[P(P(φ(4$) + T(!4)))] = Z[P(P(96 + 45))] = Z[P(P(141))] = Z[P(811)] = Z[6229]

which covers 5829 to 6629 because of the new Z[...] function described on page 6 of this thread

Z[P((!4)!! - P(P(Γ(4))))] = Z[P(945 - 41)] = Z[P(904)] = Z[7027]
which covers 6627 to 7427 because of the new Z[...] function described on page 6 of this thread

Z[T(Γ(Γ(4)) + 4)] = Z[T(Γ(6) + 4)] = Z[T(120 + 4)] = Z[T(124)] = Z[7750]

Covers 7350 to 8150 because of the new Z[...] function described on p.6

44
4 x 4 ...~

Z[!4 * (!4)!!] = Z[9 * 945] = Z[8505]

Covers 8105 to 8905 because of the new Z[...] function described on p.6