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-   -   A prime number "game of life": can floor(y*p#) always be prime? (https://www.mersenneforum.org/showthread.php?t=26729)

 mart_r 2021-04-23 21:56

A prime number "game of life": can floor(y*p#) always be prime?

1 Attachment(s)
An old pet project of mine, revamped.

If it's in any way possible I would like to turn this into a fully-fledged arXiv paper, so any suggestion on how to proceed with the work in the attachment is highly appreciated.

[I] "In this paper, the author discusses the existence of a real number y such that q = [\$]\lfloor[/\$]p#*y[\$]\rfloor[/\$] is a prime number for every p [\$]\geq[/\$] 2."[/I]

You may berate me for any technical or formal errors or glitches, obsolete or false statements, inappropriate verbiage or lack thereof, or for opening a new thread [SIZE=1](in my defense: this time with a fitting title, and gravedigging is considered rude just as well)[/SIZE] - but remember that [I]beraten[/I] in German means to discuss or give advice.:wink:

 Batalov 2021-04-24 20:40

[QUOTE=mart_r;576673]...but remember that [I]beraten[/I] in German means to discuss or give advice.:wink:[/QUOTE]
But the ensuing discussion can also be a [I]gift[/I]. :rolleyes:

 henryzz 2021-04-24 21:59

It would be nice to see a version of table 7 with primes removed if their branch has terminated in a future iteration(numbers should be monotone increasing in this case).

It is a shame that the branching rate seems to increase as the numbers get larger.

 mart_r 2021-04-25 09:42

[QUOTE=Batalov;576767]But the ensuing discussion can also be a [I]gift[/I]. :rolleyes:[/QUOTE]

:lol:

[QUOTE=henryzz;576776]It would be nice to see a version of table 7 with primes removed if their branch has terminated in a future iteration(numbers should be monotone increasing in this case).

It is a shame that the branching rate seems to increase as the numbers get larger.[/QUOTE]

I've already had an idea that leads into this direction. A version of the table including the monotonely increasing number of surviving primes shouldn't pose much of a problem to me.

It would definitely be more of a discovery if there was a unique solution for a surviving branch.

 xilman 2021-04-25 17:14

[QUOTE=Batalov;576767]But the ensuing discussion can also be a [I]gift[/I]. :rolleyes:[/QUOTE]The humour is going from bath to sausage.

 mart_r 2021-05-14 16:30

Wuthering heights

1 Attachment(s)
[QUOTE=henryzz;576776]It would be nice to see a version of table 7 with primes removed if their branch has terminated in a future iteration(numbers should be monotone increasing in this case).
[/QUOTE]
Here's an updated version of Table 7, currently at level 1723. (Preliminary, as you can tell n* is backtracked from the currently calculated stage, not stage 294 as mentioned in the PDF.)
I let one laptop crunch the numbers en passant, and plan to reach level 1931 by mid June. I wouldn't go that far if it wasn't just to fill the rightmost columns.
Plus I'd also like to find a prime with ten descendants (cf. page 18 of PDF in OP).

[QUOTE=xilman;576852]The humour is going from bath to sausage.[/QUOTE]
I believe I spider...

 mart_r 2021-06-23 16:05

1 Attachment(s)
Here's the updated table 7 for you, including the Pari program where I previously forgot to adjust the variable names to the text.

And here's a decuplet too: (193,057[SUP]th[/SUP] prime of s = 289) * 1,889 + {220, 238, 378, 624, 934, 1048, 1414, 1612, 1678, 1750}
where the 193,057[SUP]th[/SUP] prime of s = 289 is
[CODE]31622342567119297681076089891955816714567540280951897365387154057445488363818198016103621097928354615483039036120985961432365652251514164595044179903724253210032238290899811114046504503777054229122725692603678959245884842428958171279396512793345739562301201289205906157466250260424464347285786720763386343358970437066090084387214846787188810038323800022249321968498343734757162251150197635271251591141913929920578564323411668207833015231803312413278100502072679946962206773893320130362397698036981816468531752935920178488432680654626988222331983116303339502122373584059843024086090134096811878534903483185031614489646009414820924548284793618591561552678558457275689527829154639089241831525443509583812990724843963675707936160940952072970697432023165485601359791369860896277765474897467473349879301[/CODE]Yay! *Throws up a single confetti*[COLOR=LemonChiffon] Or is that called a confetto?[/COLOR]
Oh! Look at how the number starts off... wait... contfrac((log(a)/log(10)) = 796; 2, 41939, 2, 101, 1, 7, ... - nice!

Now at p=1931... but, you know, the year 1931 was not an especially nice one in terms of history. I'll continue to crunch the numbers down. I am unstoppable!!

[SIZE=1][COLOR=LemonChiffon]You know, it only recently occured to me that those numbers have a subtle crude sense of humour, looking at the point where the first sextuplet appears.[/COLOR][/SIZE]

 mart_r 2021-06-23 20:10

[QUOTE=mart_r;581628]*Throws up a single confetti*[COLOR=LemonChiffon] Or is that called a confetto?[/COLOR]
[/QUOTE]

And that was me relying too much on German grammar, which makes it funny in a non-intentional way - and because I'm not able to edit, I just make a new post.
It should read *Throws a single confetti in the air*
See, now it's not funny anymore:davieddy:

 Nick 2021-06-23 21:13

[QUOTE=mart_r;581648]
It should read *Throws a single confetti in the air*
See, now it's not funny anymore:davieddy:[/QUOTE]
That's called a confetto!

 mart_r 2021-06-23 21:22

[QUOTE=Nick;581658]That's called a confetto![/QUOTE]

:davar55: I knew it! Highlight my post # 7 to check...

 henryzz 2021-06-28 10:34

[QUOTE=mart_r;581628]Here's the updated table 7 for you, including the Pari program where I previously forgot to adjust the variable names to the text.

And here's a decuplet too: (193,057[SUP]th[/SUP] prime of s = 289) * 1,889 + {220, 238, 378, 624, 934, 1048, 1414, 1612, 1678, 1750}
where the 193,057[SUP]th[/SUP] prime of s = 289 is
[CODE]31622342567119297681076089891955816714567540280951897365387154057445488363818198016103621097928354615483039036120985961432365652251514164595044179903724253210032238290899811114046504503777054229122725692603678959245884842428958171279396512793345739562301201289205906157466250260424464347285786720763386343358970437066090084387214846787188810038323800022249321968498343734757162251150197635271251591141913929920578564323411668207833015231803312413278100502072679946962206773893320130362397698036981816468531752935920178488432680654626988222331983116303339502122373584059843024086090134096811878534903483185031614489646009414820924548284793618591561552678558457275689527829154639089241831525443509583812990724843963675707936160940952072970697432023165485601359791369860896277765474897467473349879301[/CODE]Yay! *Throws up a single confetti*[COLOR=LemonChiffon] Or is that called a confetto?[/COLOR]
Oh! Look at how the number starts off... wait... contfrac((log(a)/log(10)) = 796; 2, 41939, 2, 101, 1, 7, ... - nice!

Now at p=1931... but, you know, the year 1931 was not an especially nice one in terms of history. I'll continue to crunch the numbers down. I am unstoppable!!

[SIZE=1][COLOR=LemonChiffon]You know, it only recently occured to me that those numbers have a subtle crude sense of humour, looking at the point where the first sextuplet appears.[/COLOR][/SIZE][/QUOTE]

It's a shame that extending the search made so little difference lower down. From 599 to 797 the difference is only 1. I was hoping that more could be eliminated. Maybe some more could be if the easiest targets are attacked rather than everything.

 mart_r 2021-06-28 19:20

[QUOTE=henryzz;582136]It's a shame that extending the search made so little difference lower down. From 599 to 797 the difference is only 1. I was hoping that more could be eliminated. Maybe some more could be if the easiest targets are attacked rather than everything.[/QUOTE]

Yeah, but I don't hold a grudge against those primes, it's still interesting enough that there's a unique solution up to level 191. I may compare this to the luck that Euler's number 41 has (i.e. the polynomial 41+n(n+1) that produces primes until 41²), and claim that my sequence is even a bit luckier because 191 is the [I]43[SUP]rd[/SUP][/I] prime:smile:

I know for a fact that, at level 797, two more primes bite the dust after 2089#, so at this point only 226 will be left. I'd have to check how far-reaching exactly the consequences are, but at level 541 the 47 surviving primes remain unchanged. I've checked the weakest branches of level 797 already in 2015, and the weakest of those that I haven't checked further has 42 primes at level 1597. Puzzling together the probabilities, I would expect one more prime to be cancelled out at level 797, leaving 225 "stable" primes at this point.

 mart_r 2021-08-05 17:50

That playground is gigantonormous!

2 Attachment(s)
Updated and polished, just a little. Added "higher-order descendants" (page 19).
This is a "MS print to PDF" version, while the one in the OP was a "FreePDF print" version; each version comes with its own minor glitches. If I'm ever able to convert it to proper LaTeX, maybe next year/decade, this issue will be taken care of.

After 1987#, the number of surviving primes n* in table 7 appears to be fixed for p<=601. The following level p=607 contains a split with a branch still present at level 1987, but dissipating at level 2089, so there are only 71 surviving primes at stage 111. You can see the "trajectories" for these weakest branches in the attachment. That split branch at level 607 started out like a flash in the pan! It's like a prime number stock market out there... :smile:

[QUOTE=mart_r;582164]I would expect one more prime to be cancelled out at level 797, leaving 225 "stable" primes at this point.[/QUOTE]
Down to 223 now. There's still a >1% chance it reaches 222, but <1% that the number drops even below that.

Supplemental tool for the backtracking process (only semi-general-purpose):
[CODE]print("backtrack(b[,c]): track level b back to level c [or every level with < 512 surviving primes]");
backtrack(b,c)=
{
p=b;
t=p;
o=vector(primepi(p)-40);
n=#d;
o[1]=n;
k=1;
while(n>1&&p>c,
z=vector(512);
y=1;x=1;
for(i=2,#d,
if(d[i]<t,
if(d[i]>t/p,n--);y++;if(i==#d,z[x]=y;x++),
z[x]=y;y=1;if(x<512,x++)
)
);
k++;
o[k]=n;
p=precprime(p-2);
t*=p;
print("backtrack... p="p);
if(n<512&&(p==c||c==0),
write("backtrack_"b"-"p".txt",vecextract(z,Str("1.."x-1)))
)
);
o=vecextract(o,Str("1.."k));
forstep(i=#o,1,-1,
write("backtrack_"b".txt",o[i])
);
print("data stored in backtrack_{p-c}.txt");
}
[/CODE]Example: backtrack(1987,197) = "[96637, 113792, 156942]" - meaning: 3 surviving branches at level 197 with respectively 96637, 113792, 156942 primes at level 1987. (I'll upload the level 1987 data depending on the resonance, since it has 3.74 MB zipped.)

Additional Table 7 info not really fit for the paper:
The number of surviving primes n* appears to have settled for...
[CODE]s = 30 to 36 after s = 48
s = 37 to 65 after s = 95
s = 66 to 70 after s = 162
s = 71 to 92 after s = 189
s = 93 after s = 191
s = 94 to 108 after s = 215
s =109 to 110 after s = 277
s =111 to ??? after s = 316
(s=139 after s>= 365)
[/CODE]So, when I give you these 65 primes of level 599 (s=109), you'll (hopefully) end up with all primes of the sequence after level 1787 (s=277):
[CODE]a=39624069013936965087096009433866594830980870998888700289477858045283065746616452803046877051058323608232360514007056743887671907849763652319988387476334527612284794568776090626036384425467238802263863077912348447009215427300887220486657625693147;
d=[0, 466455760294422823953804039675640047201077790734154654780965233768051077060385506551407568084738361988997027851963959610818647415718916847152, 44895563831368704616159353930540997828272453622716098834658210530168205692523959883757599112304579276962544182213994859302153464025610, 26706100635873975723434507798935828965926546552594370, 5349424105738534, 901898922830014190586001959240919331648326160928257099022394892456073146632010, 8995009362764739858959068329200056018893114699905074080917303621788351313861729098338404578, 335544104419153579655358597659804721070678961457899223253722433765243412023981115104923784391687635420326417875238, 4523471714186028497063758655500264647233710442107467015046892971714017309602121083966315477490178979402201052, 13558128246316966557562438271899907310750726822, 4934323886885543883827676375700166702424555062100227255395720919371207047219908212443254804028675171530193258025833342400752532251499752430581830, 388743572956706723444269225383396603443044134640778117599062700920205080348, 7131662087452, 33194405695790798430238872743598395294, 269730, 135320865050066733911728570, 11221001162, 6090567717995320705159336935158027964523675153732520176913426535613320656808731739785868047152936659580764487743815834176652322675806820457799664407012544927967378642, 966178811730010342231796148, 950218907960867236355409812416967381787168349447008, 186, 120950, 73131703534393817971440867435305555941325599875579554494296006281538000190675946442663191144668123588, 9500734976302631650958, 239805326438620473888522605578, 121757244991625966205374175349841785534615242815048191144254, 66687483923550353133932113715210392764921421988484986752527792068, 55963473253722215455307053540553065129204623773008139755208417335670283622191194014140, 374220210121664223149161671575248005142427307251864790690209608349390524597485609165745496013651149975393673161307792844136375404294244730453984728505635072279055652, 1209221770576241494563186529310, 28686868, 513195473075718020212004909660243183839001052209643336479222370958923901013785264992668591798528873698138934, 28832971082846920, 48533460370249984502739595104518, 41532609059904971662727814316530245330921320936385322890063242134040854418808228566236415441515393074907863930376015390, 493668696637110236477409740705170, 1808806165895014602770, 2261804119760347690745179746, 9311195034, 99434466581329648062553784651274547675969070085859370489887908418, 11225742884389569244962733297207525945870303941562175609336782988922, 112530771834007266493561947078591371253972340193864418633152934638854286685371266094730252277266033265826678554646900216277609060540241061184441929856119915131915176, 103350, 44740191674963654791540507699261670144839100857418098222434989798279884602883087809574557987137117510760, 9254712663124863672244531853950613218514667190892855782913340344365740049591921814953513360801840244416196898101508293163612904791809371838669710474565685029165706842604, 10033871640846413419993982882804977302146190667092951612473356704835668, 773645088515448882592365176248227130967319190028631391234006175499055409359482, 6309191045489857456582839123180771159412906264449006658980416499498, 805748915562249520203749088487780366455469723998114574728718044699745578442712351844994056568611864541729304569354505304688218, 1871911391108669620366209771942252280517428081404, 281083455286058273331808398475392, 4187509955264496840236, 10812133799323198, 9428157516571371362264424890339613308558429717606124684073379249324365172732443776189794620110086458638386, 203548, 40180857345840, 2060098878522283566900456084770695449091282256414174589864543021569432, 670291401847140231046228944641480671586531001527350972352877129757263171304541394348129386594411256, 2452831165494447922493076252, 71305177701894194705286338516277103466839112851099161326083402585320861927804314198838482177449447105067911900878227962, 158419166957241279541816714465231159780430892544161477518866140, 1284912443358386721676929857796019465275702641192465355368497364626176221916, 17223504, 318, 154391504732234881359906941526553166298281939591864102142977844101440097198716489216556904069919213710];
[/CODE]I'm having way too much fun with this crazy sequence...
[COLOR=LemonChiffon](Then again, some say I'm also having way too much fun watching cartoon shows...)[/COLOR]

 sweety439 2021-08-06 08:18

Can floor (y*n!) always be prime?

 mart_r 2021-08-06 14:07

[QUOTE=sweety439;584949]Can floor (y*n!) always be prime?[/QUOTE]

I'm afraid not. n! grows slower per step, leaving not enough room for primes of size y*n!.
With this regard, p# has just the "right" growth rate. Any slower rate would not result in an infinite sequence of this kind.

 mart_r 2021-12-31 10:45

tl;dr: some more results, potentially interesting further computations & a question

2 Attachment(s)
[B]First, the question:[/B]
Calling y*p# the "slowest exponentially growing prime number sequence", would that be formally correct?

[B]Further results:[/B]
Computation complete to 2111# on Sep 27, 2021. There was a delay of three days due to insufficient allocatemem, so net calculation time was 157 days (one Intel i7-10510U @ 2.3 GHz).
The most difficult part after that was figuring out the potential crossover n* > p.
[CODE] s p n n* s*
1 2 1 1 1
2 3 1 1 2
3 5 1 1 3
4 7 1 1 4
5 11 2 1 8
6 13 2 1 8
7 17 3 1 8
8 19 3 1 11
9 23 4 1 11
10 29 6 1 21
11 31 4 1 21
12 37 5 1 21
13 41 5 1 21
14 43 9 1 23
15 47 11 1 23
16 53 10 1 23
17 59 12 1 23
18 61 8 1 23
19 67 6 1 23
20 71 11 1 25
21 73 5 1 25
22 79 4 1 25
23 83 6 1 25
24 89 3 1 25
25 97 2 1 26
26 101 1 1 26
27 103 3 1 28
28 107 1 1 28
29 109 1 1 29
30 113 3 1 48
31 127 2 1 48
32 131 5 1 48
33 137 6 1 48
34 139 12 1 48
35 149 21 1 48
36 151 19 1 48
37 157 15 1 95
38 163 16 1 95
39 167 24 1 95
40 173 18 1 95
41 179 18 1 95
42 181 17 1 95
43 191 14 1 95
44 193 24 2 95
45 197 24 3 95
46 199 28 5 95
47 211 30 5 95
48 223 36 5 95
49 227 49 5 95
50 229 44 5 95
51 233 52 5 95
52 239 53 5 95
53 241 55 5 95
54 251 67 6 95
55 257 69 6 95
56 263 72 7 95
57 269 81 7 95
58 271 79 7 95
59 277 85 8 95
60 281 83 8 95
61 283 93 8 95
62 293 81 9 95
63 307 81 9 95
64 311 67 9 95
65 313 66 11 95
66 317 74 11 162
67 331 91 12 162
68 337 88 12 162
69 347 90 14 162
70 349 95 15 162
71 353 102 16 189
72 359 126 18 189
73 367 152 19 189
74 373 154 19 189
75 379 166 19 189
76 383 187 20 189
77 389 214 20 189
78 397 206 21 189
79 401 201 21 189
80 409 220 21 189
81 419 241 23 189
82 421 249 25 189
83 431 269 25 189
84 433 320 27 189
85 439 354 29 189
86 443 354 31 189
87 449 365 32 189
88 457 369 33 189
89 461 358 33 189
90 463 387 34 189
91 467 413 35 189
92 479 426 36 189
93 487 446 37 191
94 491 454 37 215
95 499 491 37 215
96 503 456 38 215
97 509 490 38 215
98 521 559 41 215
99 523 573 43 215
100 541 594 47 215
101 547 652 47 215
102 557 718 50 215
103 563 757 50 215
104 569 786 53 215
105 571 835 55 215
106 577 839 57 215
107 587 854 59 215
108 593 906 63 215
109 599 916 65 277
110 601 998 69 277
111 607 988 71 316
112 613 1016 74 316
113 617 1078 78 316
114 619 1165 81 316
115 631 1237 84 316
116 641 1295 86 316
117 643 1371 90 316
118 647 1399 93 316
119 653 1523 97 316
120 659 1618 101 316
121 661 1701 105 316
122 673 1773 108 316
123 677 1801 116 316
124 683 1889 121 316
125 691 1885 127 316 <-- n* > s
126 701 1944 132 316
127 709 2008 138 316
128 719 2141 144 316
129 727 2136 150 316
130 733 2241 158 316
131 739 2354 165- 365
132 743 2442 177-
133 751 2548 179-
134 757 2649 184-
135 761 2791 193-
136 769 2998 200-
137 773 3017 207-
138 787 3148 214-
139 797 3132 224-
140 809 3385 231-
141 811 3537 245-
142 821 3698 250-
143 823 3914 261-
144 827 4175 266-
145 829 4345 273-
146 839 4360 284-
147 853 4537 292-
148 857 4722 304-
149 859 4862 308-
150 863 4951 320-
151 877 5099 332-
152 881 5228 345-
153 883 5334 357-
154 887 5460 364-
155 907 5715 379-
156 911 5971 389-
157 919 6366 404-
158 929 6582 414-
159 937 6799 433-
160 941 7006 455-
161 947 7151 463-
162 953 7599 474-
163 967 7920 494-
164 971 8276 511-
165 977 8708 524-
166 983 9086 535-
167 991 9577 553-
168 997 9736 570-
169 1009 10329 586-
170 1013 10722 594-
171 1019 11048 611-
172 1021 11387 639-
173 1031 11769 669-
174 1033 12251 685-
175 1039 12490 711-
176 1049 12905 733-
177 1051 13209 750-
178 1061 13556 775-
179 1063 13918 797-
180 1069 14393 822-
181 1087 15006 852-
182 1091 15472 874-
183 1093 15695 908-
184 1097 16075 937-
185 1103 16548 971-
186 1109 16852 1008-
187 1117 17548 1033-
188 1123 17926 1064-
189 1129 18415 1090-
190 1151 19145 1124-
191 1153 19690 1147-
192 1163 20364 1179-
193 1171 21154 1216- <-- n* > p ??
194 1181 21954 1262-
195 1187 22688 1287-
196 1193 23528 1333-
197 1201 24295 1363-
198 1213 25085 1401-
199 1217 25717 1442-
200 1223 26680 1492-
201 1229 27569 1527-
202 1231 28242 1572-
203 1237 28917 1614-
204 1249 29871 1659-
205 1259 30707 1708-
206 1277 31555 1754-
207 1279 32633 1808-
208 1283 33494 1871-
209 1289 34556 1921-
210 1291 35744 1981-
211 1297 36546 2034-
212 1301 37363 2087-
213 1303 38018 2150-
214 1307 38936 2218-
215 1319 39633 2303-
216 1321 40152 2375-
217 1327 40822 2456-
218 1361 42524 2541-
219 1367 44138 2626-
220 1373 45674 2704-
221 1381 47308 2791-
222 1399 49203 2876-
223 1409 50999 2966-
224 1423 53578 3047-
225 1427 56009 3139-
226 1429 58245 3213-
227 1433 60614 3340-
228 1439 63012 3442-
229 1447 65622 3520-
230 1451 67609 3633-
231 1453 69411 3747-
232 1459 71677 3844-
233 1471 73978 3960-
234 1481 76816 4078-
235 1483 78753 4173-
236 1487 81051 4292-
237 1489 82647 4411-
238 1493 84106 4551-
239 1499 85756 4684-
240 1511 87469 4811-
241 1523 90333 4958-
242 1531 93284 5084-
243 1543 96279 5233-
244 1549 99150 5376-
245 1553 101723 5539-
246 1559 104560 5675-
247 1567 107041 5832-
248 1571 109795 5999-
249 1579 111912 6155-
250 1583 114508 6341-
251 1597 117842 6528-
252 1601 121087 6716-
253 1607 124407 6901-
254 1609 126382 7113-
255 1613 128949 7305-
256 1619 130732 7493-
257 1621 132088 7720-
258 1627 133678 7965-
259 1637 135308 8188-
260 1657 137970 8439-
261 1663 141747 8687-
262 1667 144129 8937-
263 1669 146278 9203-
264 1693 150011 9484-
265 1697 153719 9799-
266 1699 157438 10097-
267 1709 160935 10460-
268 1721 164713 10819-
269 1723 168209 11187-
270 1733 171914 11552-
271 1741 175208 11966-
272 1747 178924 12392-
273 1753 182442 12861-
274 1759 186286 13363-
275 1777 190466 13889-
276 1783 196294 14432-
277 1787 200525 14980-
278 1789 204944 15548-
279 1801 209420 16163-
280 1811 214672 16789-
281 1823 218990 17483-
282 1831 224272 18194-
283 1847 230486 18982-
284 1861 238339 19837-
285 1867 246127 20704-
286 1871 253236 21653-
287 1873 259946 22631-
288 1877 266707 23698-
289 1879 272114 24815-
290 1889 278512 26035-
291 1901 285821 27356-
292 1907 292121 28714-
293 1913 298404 30241-
294 1931 305779 31927-
295 1933 314060 33679-
296 1949 323493 35595-
297 1951 332369 37696-
298 1973 344077 40005-
299 1979 355840 42551-
300 1987 367371 45250-
301 1993 379303 48293-
302 1997 391133 51726-
303 1999 402306 55407-
304 2003 413306 59585-
305 2011 424348 64450-
306 2017 435084 69845-
307 2027 446383 76167-
308 2029 457067 83297-
309 2039 468311 91883-
310 2053 482068 102217-
311 2063 494789 114639-
312 2069 509095 130467-
313 2081 524247 150449-
314 2083 538695 177304-
315 2087 551951 214976-
316 2089 564045 271897-
317 2099 577252 370885-
318 2111 592642 592642-
[/CODE]n* = surviving primes backtracked from stage 318; "-" indicates that this number will decrease further in the process (e.g. n* = 164 for p = 739)
s* = stage after which n* appears to have settled
The point where n* > p is in a state of "touch and go": checking the weakest branches for s > 318, at p = 1171 n* will settle between 1168 and 1173 with probability > 0.99; I'm still running calculations to determine whether or not three more branches with around 50 or more primes are becoming extinct. Some manual work is involved, and I have other projects, so in the worst case, it may take a couple more weeks to determine a solution to this question. In any case, n* will be > 1181 for p = 1181.

[B]Up for grabs:[/B]

- The currently largest known p for which a prime in the sequence is known is 11131. This Pari program tries to find successively larger primes, thus digging ever deeper into the sequence:
[CODE]{p=88156674359250180889368283104852390376653324509005255568803893544153488606573492283451795473796701738660226070713255672629846721465376085644580955017063493523264965964488014977610709062475593930182010742149002896144249223315628347072100200993715416577222452764712432514595282411581747516210049069325532901205070575067403900529983337708045954385269926574392052015018613449274545772373640756531237519071038132881907743485944743241657725208549917949756917482861433588459752960779949295718050875721885718902616818594321687674612097039579858551689842567715802065873313581552878231763881626853352041337806562464420190245705366722000244890533909798353473854288352625263671767092386446046663746872617431436877522708092198055074286137606507517615889161189440063468872119602987403182756240374662416405110403244986472434522944394685608720161790770988827126081218454910178824492682209522892189964570323329611167040704384725321276555376682215639599309753824323865841657261563227629627183243515850824587771765650886771626150556904792445421944087350105502512405671731005142269409919745644009732650968249065051426844384813966052285981148734593991705752196190750037120917430019904704903068533325935222342209036746862742801205543293249257324598513771419783309582867729070686594637621954375145200496870136503976738117987367365144632703891669130212390189802207075434274532036488795936545848887957512285500184917745544871227574301618621721494759378265116804805925142660895507691726798886262250133924093660130733730790092895751094006025159403131113093550195418779900464105983492057581725185453017959551419365258417627453197381921723482678820157153302027017699690966573492992722855598854683690655126898936817950888579710694637553392196480803839312818022761313116218586542291984860432977931596371516874172200602731438066782588068982257354227703386629520964881711004675627457929209179441673745636527675932771302168760966561399406735795858719639313564866313810561584349363631638227244082235161088948503873770608580651465877549919299592539020668514344660620273432862155169926185304006080465316267243274171379650509749134014148824198072849345215455659789820315583293114802021228744986606302091785576141627727150703368205623789742784782157015911146477085119569196523039065562229035957136848085723755596505360449645219360336645029287105499599877006773316336198567278358658914066953162787975796564082933014149683084165011562143282478622390691931158932805895884408501046067513990937812266627711220062099442955987110695534841247222353797340966716074601898206639629219186567900691845965376960827110305126918026434539255529262244313486929918519147055747664478051557572834585643491143004306603032963704864367969232411748608600777666198987856624522031189894586710130282474379653684524329841532555954762762580699648680329953801846610683746504705437442186689534868592177929971615120163157970360760602017316717907190624734767431023889716351029146698226817170696122638587338808597490470481807268721033951351485569031595645617607497247530807715459414425003694729339274843384105860311780817700968757936324139944546341363552296394135987765438798082527976554378979615285920606275093035156916480434015430349126712389665059042723515407219124322323387623172114343061731356132671850415130064541045233271613222994558622835910268220259022290852189190448619928321960544585628234343829864822464808735371508761885363497014243878794303264198559438085485195201672533446779329435982926779206762647105143931634276341155840551986710904737445140030759666607593059749235936855721075241380423656927597100971952333831852547204816424747336725994817781886565798328109478810872370542031041459052335523128990475739866154407882705083739208687025285204735401805789638884657484240201847273167113359810889569773060652202262983011023341858071427425961994635769485368265365922001497989495413746770108400153509739371174449373177741380692564787832969187795351736253949623505021922763898522505143611098614251743486369638215083492034882040259958615885677059211526063138540792300827182999077134249161541867891872550431587611062831129973035825891243522299599167300228641474978333244404377650896639478214354672751515114968874215304868904821295213561733408105370524339312661044637106067712583432074043059629663037814439276958716678757553450507383549596746386118995655370642316850351813212799205814741488551047505971383838431787888293260611288420461588973354923398976481540364421031632365793752950872737438980993783669271556988277084730386212164650137267702256037524847670889483452532592919509206059507616929837801202235944427718618291899737437711944643762041159935301643976191290010401800904146643419727545279873423651421655832907561776646075898873654015382050771344051581633767382260661881401;
i=0;
o=p;
while(o>1,
i++;
o\=prime(i)
);
q=prime(i);
gettime();
g=1;
h=1;
forprime(i=3,1327,g*=i);
forprime(i=1361,355111,h*=i);
a=vector(4096);
n=p;
o=q;
while(o>2,
a[primepi(o)]=n%o;
n\=o;
o=precprime(o-1)
);
l=primepi(q);
m=1348;
while(l<#a,
l++;
while(a[l]+2<prime(l),
a[l]+=2;
q=p*prime(l)+a[l];
if(gcd(q,g)==1,
if(gcd(q,h)==1,
if(ispseudoprime(q),
write("p#progress.txt",prime(l)" "q);
x=Str(prime(l)" ... +"a[l]" ["floor(gettime()/1000)" s]");
print(x);
write("p#progress_runtime.txt",x);
if(l>m,
m=l;
write("p#first.txt",prime(l)" "q)
);
p=q;
l++
)
)
)
);
p\=prime(l-1);
a[l]=0;
l-=2
)
}[/CODE]The output file "_p#first.txt" includes only the first prime of consecutively larger levels, and "_p#progress.txt" keeps track of all the primes found, each of those pose as an improvement on the value y_min (1.2541961...).
Screen output will look like this (first column = level p / second column = p#Y mod (p-1)#, this may also serve as a cross-check value / third column = time since last prime in seconds):
[CODE]10979 ... +1374 [52 s]
10987 ... +5772 [225 s]
10979 ... +5634 [776 s]
10987 ... +3372 [137 s]
10993 ... +8434 [319 s]
10973 ... +8780 [1073 s]
10979 ... +4998 [171 s]
etc.[/CODE]So within a matter of hours, larger primes may be found. The thrilling element here is the volatility: see the attached graph "sequential". We're practically following one single thread in the calculation. Occasionally, the first prime of a certain level becomes a stable branch - those are the points in the graph that exceed a previously unattained level without ever dropping below it again, with p = {2, 3, 5, 7, 11, 13, 17, 83, 89, 101, 103, 107, 109, 127, 131, 173, 241, 277, 281, 283, 1511, 1523, 1559, 1579, 1583, 1597, 1613, 4327, 4337, 4339, 4349, 4357, 4373, 4421, 4643, 4649, 4651, 4657, 4703, 4721, 6047, 6053, 6067, 6073, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6373, 6389, 6397, 6427, 6449, 6451, 6481, 6491, 6521, 9227, 9239, 9319, 9371, 9397, 9403, 9413, 9419, 9421, 9431, 9433} being the currently known points. Much like e.g. local extrema of Li(x)-pi(x), they tend to come in flocks.

- The first surviving prime of level 9973 is probably 74612...13259 - take the number a in the program below *9973#/11131# to get the full decimal expansion. That's the 35th prime of level 9973 (out of an expected number of about 1.2*10[SUP]12[/SUP], by the way). I thought it would be interesting to see how such a branch deep down in the sequence would evolve, as it is much more volatile there, plus it would also improve the confidence that the eleven numbers between 9227 and 9433 in the list above "first prime of a certain level becomes a stable branch" are valid. So I let my rather slow laptop run for a couple of days on that branch, with the following development:
Level {10007, 10009, 10037 ...} - #primes = {1, 2, 3, 3, 4, 2, 5, 8, 11, 14, 15, 16, 12, 12, 13, 19, 18, 22, 23, 27, 33, 35, 25, 26, 22, 18, 19, 25, 25, 26, 27, 31, 37, 31, 26, 27, 25, 28, 32, 27, 27, 30, 37, 36, 32, 31, 36, 30, 32, 32, 35, 29, 35, 36, 29, 30, 30, 36, 28, 33, 38, 44, 37, 44, 46, 57, 51, 50, 53, 44, 44, 41, 42, 45, 43, 54, 46, 40, 25, 26, 39, 39, 45, 38, 32, 39, 45, 44, 44, 47, 53, 52, 51, 61, 70, 64, 71, 81, 83, 63, 65, 55, 60, 48, 52, 56, 56, 76, 68, 74, 81, 90, 76, 75, 80, 81, 84, 71, 69, 63 ...}
By the heuristics, it's safe enough to assume that the branch survives once the number of primes exceeds 500-ish, so here's the program for continuing here:
[CODE]{a=48934348445650255491429754307422852698745898679725029263835201735053124556394586002511369416690518728507281780065742568204460817345038423170328849559738439493960717469404427064771615915990639032233976443267359117088140191872885044115121710626176895055077688046534070204191253770304243049340907226873109472031918157579377166612356272002144417043050719042993512641484586738871390423045836555550188004964219075554178219293179599768016132800321772798609506232011433916888344936264950621271774072130189563964713381459185002886270893799498648303690792684117836325473478535129204345570709568585491221034795023124287158966674052860611807142520177745149151119207526830512768396393690798704053021609601060968831579801241799897131195040335009278947224983954407773236076435326662072423435430018910059724581773726163035880223110641621447935598114871402639346362457812901080714727120615842143708373609653903998506913059568911208318847217695034809282420336085128117416704689284819361639679766620272334869889744827567079598204629825280093941847494955661065337863036092322963123859704056410742641930976822846255975144630686672209594213184344499762752336104179456128428709569688832807720330443504373831028379990248922168182042102543683856429146289337757774711709074150545793941172575034714755412352925527082918272368725295688636590559250717802409163798057801231750424163416697548439550483184195698606126044724569231968802989725048937538961765283744165620044209282869971306917392520617161628753305274358906058915059179351639858293750773982347517537296662601037669973094931345608966574335940805763688288368105660842081770998120289522527631302574361418974540435215021361143613263866190040287587245087187929978691860028628276514623264117303791256617805037791663211094712313670248958197215505165151091047965272037529959794820863439129095706293464014148275797279208044084218997961518223985239510080695921528118859936870481140095552292445028751037537860698905667439615057735735117339740405973628502445293108543245368679854101530904447269106211265197219128456713970819201513478434561935546499686246048569562469234457099586891315662432912596791475155874502729766771906599755746055000403927428301464690009146939129426868532512407887811907332560562339193665464106250053603686192894741651904298081246577772134797642019215198077489174597002962584512441245063837108889461620804683604950561634552488464203980017884949360903694698230576739156530815475948039934712141155663432660056289365972726592491674471679763872273960596417791850459255854377928058996762107596784040802011770571651147049185414477634335639317649367897323352750949348632894369943442176091052364898275256038695720276011705016242997702845889363879534282982590868911740364206355317805193105636433799121159478424118613204387941952799940303385967498508638955099875686842245379091951909716618485352606854176753122674754862723861456744429321026333075808485472366252831588879397271948834067355181711387533022186248128905715565891227637931836959170913505747662159849736135096232165171114519971112548368534302087785984218507710463763442622480749059745038748739414638955866771861484048208813659019059605877908333366950486646537916224248473332600152131603468321920062220329995902715259801333475774392627767061114519818855837369943661405325604240993560876410444728774457239098727764756373656178806902420531112673420691525465840020822003662595335479525202546064232041234655466767345479040616675040330058007582288400577093614533923189131143239571812373872440875229067446568767620052352927050439243539731936943674580307504269518217221825405464856367248244180321347249299936153790421666507823000577309081110812649575493418078425220272883428702858666274228030092908106198761696063179628805831752804785693661958147773717778157137229304720064975999914414684554280123199200011655199755526851435063376117950434651084005425002995754916649293601271825911591667437447690278632347003982061227756445989218965438431025530317604148263010554685724737872964922480016882509054163746864484619428429984031068159637528549743689129956357689339474190216599804346505336186949991363496307901988759085295866476030508819033094505873087348475915656323986006974564312999034974653484747634668268517853758114069459701941188197523149799754269165914614635580495505997983206355203635284169705202017458318365729422469143547777937447244298454796890168689469557332693148988787176224962148272074428233539613291694637141232256096568994921186045198149039113343541832835221586197274707504762114258278574162420236299306473877483935342976864091293729830387379072807161592582977075717203304609698952079636200640423866969860664079349742943800554421780199529384736587703628546640622849644231267518993251206991384803507212797859856612407248552265310431577845822776993545253608643379224956801611311023737462623; d=[0, 2778003193391157627089323520755728579061443990119777710451723526790349339554005245488, 63887669758118978106, 1017920472699345500061756403080892302858203744771338715984167130830462592030300664597346250683103732164134000509861036123909901908619653265570128688091862, 1219677150505529264465486013704043532243042001014, 398, 2146, 6067874708925443592, 3248, 255956618666133897542791663437458747828377110317078268415383760053402150092863021246787241866047152028317514213131809996755517774500899337900397516453831138087965904414362267358879597985493454919462, 2050, 3834, 126, 46852074603681593539240838283300, 3887738522335914, 92827286, 57951728362, 403690535312, 949503011758563778866834, 1155896951927506114088679552457132600, 568430922959533130642288139672030296572633912649966496789630073728956675221221762814053399072, 1457976159686761813703226647435487891188, 510429129482610506475400, 780, 4704, 52740880470424034664231997774080667213005258305149885360506482547998059832208816288738521417345004655013202371952677270934863215425234750653583649070136549654563582899475887328352205316648196484818927416041469293796172123437744846, 22177604531727053380, 107800812888170728652896590663686, 16604026, 78, 690, 8934, 4135519530, 5920655143599653575357224613979876727030207367795308, 5010711041691654472851287568, 407344034506394, 416636449138, 3939326235211838642, 42969706, 72935321292762714102, 139798175202280184401082980410254, 3778, 84146490, 9348, 7534866756524099872482709477334651583879236253167550, 28688136814322883572997745026930951659205644981389934607623095153589496942822, 3410888, 1410445057272208022540693755074072759714937893432, 92634822, 664, 6564, 139513674552, 204, 74027960, 1444, 7786050, 972304457115227170906085753604152838, 5720, 74362059357914437901903881861640572048440388, 23104584, 4110, 3300, 948];
s=0;
b=a;
while(b>1,s++;b\=prime(s));
i=#d;
gettime();
while(1,
s++;
p=prime(s);
o=a*p;
c=d;
e=floor(256+i*(1+2/sqrt(p)));
d=vector(e);
m=i;
i=0;
for(j=1,m,
o+=c[j]*p;
y=vector(p);
forprime(b=3,p-2,
r=b-lift(Mod(o,b));
forstep(l=r,p,b,y[l]=1)
);
forprime(b=p+2,floor(p^2.1),
r=b-lift(Mod(o,b));
if(r<p,y[r]=1)
);
forstep(k=2,p-1,2,
if(!y[k],
q=o+k;
if(ispseudoprime(q),
i++;
if(i>1,d[i]=q-z,a=q);
z=q
)
)
)
);
g=floor(gettime()/1000);
x="[";
f=floor(g/3600); if(f,x=Str(x,f"h "));
f=floor(g/60); if(f,x=Str(x,f%60"m "));
x=Str(x,g%60"s]");
t=Str("Level "p);
print(t": "i" possibilities "x);
t=Str("p#Y "t".txt");
write(t,"a="a"; d="vecextract(d,Str("1.."i)))
)
}[/CODE]- I can't upload the full level 2111 data (6 MB zipped), but, depending on the resonance, smaller batches can be provided, so in case, just let me know which branch number (1..47) - or split number (1..46) - from the attached bifurcation picture you'd like.

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