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Old 2021-06-28, 19:20   #12
mart_r
 
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Default

Quote:
Originally Posted by henryzz View Post
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.
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 43rd prime

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.
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Old 2021-08-05, 17:50   #13
mart_r
 
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Default That playground is gigantonormous!

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...


Quote:
Originally Posted by mart_r View Post
I would expect one more prime to be cancelled out at level 797, leaving 225 "stable" primes at this point.
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;
read(Str("p#Y Level "p".txt"));
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");
print("ready.")
}
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)
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];
I'm having way too much fun with this crazy sequence...
(Then again, some say I'm also having way too much fun watching cartoon shows...)
Attached Thumbnails
Click image for larger version

Name:	trajectories 601#, 607#.jpg
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Attached Files
File Type: pdf Project p#Y.pdf (1.02 MB, 118 views)
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Old 2021-08-06, 08:18   #14
sweety439
 
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(P^81993)SZ base 36

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Can floor (y*n!) always be prime?
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Old 2021-08-06, 14:07   #15
mart_r
 
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you know...around...

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Default

Quote:
Originally Posted by sweety439 View Post
Can floor (y*n!) always be prime?
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.
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Old 2021-12-31, 10:45   #16
mart_r
 
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Default tl;dr: some more results, potentially interesting further computations & a question

First, the question:
Calling y*p# the "slowest exponentially growing prime number sequence", would that be formally correct?


Further results:
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-
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.


Up for grabs:

- 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
)
}
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.
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*1012, 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)))
)
}
- 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.
Attached Thumbnails
Click image for larger version

Name:	sequential.jpg
Views:	63
Size:	99.0 KB
ID:	26323   Click image for larger version

Name:	p#Y bifurcations @ 547# split#.jpg
Views:	67
Size:	361.1 KB
ID:	26324  
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