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2022-08-24, 17:03   #58
mart_r

Dec 2008
you know...around...

22×11×19 Posts

Quote:
 Originally Posted by mart_r As a by-product, a puzzle: Given x, find the next three consecutive primes >= x. Denote the two gaps between them g1 and g2, and let g1 >= g2. Let r = g1/g2. As x becomes larger, the geometric mean rgm of values of r also become larger. Find an asymptotic function f(x) ~ rgm.
$$\lim\limits_{\substack{x\to\infty}} r_{gm} = 4$$, if I may so conjecture (based on a random model similar to CramÃ©r's). Is there a proof available?

I'd like to take the search for T(38,16) in A086153 up to 10^16, which will likely not be enough to find an example, but I still would like to see that case solved. It would take a bit more than a week with my program. I've identified 746 distinct constellations as shown in the attachment. I believe that list to be complete, albeit I'd be more content if that number was divisible by 4, so there's a slight possibility I have overlooked some constellations. If anyone with enough time on their hands feels inclined to do a quick double-check...

For good measure, here's a batch of 79 instances where CSG > 1 for k > 1000:

Code:
p                k     gap    CSG
123146152018999  1152  44280  1.0322718
123146152018933  1154  44346  1.0310658
123146152018999  1127  43378  1.0263032
123146152018999  1126  43342  1.0260889
123146152018933  1129  43444  1.0250767
123146152018933  1156  44394  1.0249425
123146152018933  1128  43408  1.0248616
123146152018921  1155  44358  1.0247205
123146152018999  1157  44428  1.0246186
123146152018999  1133  43582  1.0242866
123146152018993  1153  44286  1.0242773
123146152018999  1138  43758  1.0242719
123146152018933  1159  44494  1.0234272
123146152018933  1135  43648  1.0230694
123146152018933  1140  43824  1.0230603
123146152018823  1158  44456  1.0226588
123146152019071  1151  44208  1.0221944
123146152018999  1125  43288  1.0209057
123146152018999  1139  43780  1.0206462
123146152018933  1141  43846  1.0194401
123146152018933  1160  44514  1.0192932
123146152018933  1130  43458  1.0192328
123146152019521  1113  42856  1.0183353
123146152019521  1112  42820  1.0181224
123146152018853  1132  43524  1.0180185
123146152018853  1131  43488  1.0178006
123146152018999  1094  42184  1.0176670
123146152018999  1091  42078  1.0176016
123146152018921  1136  43660  1.0166975
123146152019521  1143  43906  1.0165963
123146152018853  1162  44574  1.0164815
123146152018933  1096  42250  1.0164132
123146152018933  1093  42144  1.0163444
123146152019521  1119  43060  1.0163041
123146152019521  1124  43236  1.0162807
123146152018993  1134  43588  1.0162592
123146152018999  1095  42210  1.0150911
123146152018823  1163  44604  1.0150789
123146152018823  1144  43934  1.0146314
123146152019071  1137  43686  1.0141775
123146152018933  1097  42276  1.0138418
123146152019419  1142  43860  1.0136414
123146152018823  1161  44528  1.0135410
123146152019521  1111  42766  1.0129308
123146152019507  1114  42870  1.0124697
123146152018801  1164  44626  1.0115109
123146152019521  1116  42936  1.0112458
123146152018801  1145  43956  1.0110334
123146152019507  1120  43074  1.0104594
123146152019521  1080  41662  1.0097438
123146152019521  1077  41556  1.0096842
123146152018993  1092  42084  1.0094570
123146152019483  1115  42894  1.0093765
123146152018853  1099  42330  1.0092682
123146152018999  1088  41938  1.0080624
123146152018823  1100  42360  1.0078183
123146152019419  1117  42958  1.0076073
123146152018801  1146  43978  1.0074438
123146152018801  1165  44646  1.0074105
123146152018921  1098  42288  1.0073935
123146152019483  1121  43098  1.0073785
123146152019071  1090  42006  1.0073695
123146152019521  1081  41688  1.0071605
123146152019521  1079  41616  1.0067457
123146152019419  1147  44008  1.0060364
123146152019167  1150  44112  1.0056258
123146152019419  1123  43162  1.0056203
123146152018823  1101  42386  1.0052645
123146152019207  1149  44072  1.0043139
123146152018999  1089  41958  1.0038262
123146152019461  1122  43120  1.0037558
123146152019507  1078  41570  1.0037494
123146152019521  1118  42976  1.0028794
123146152018583  1167  44696  1.0019334
123146152019419  1148  44028  1.0019166
123146152018801  1102  42408  1.0016116
123146152019507  1082  41702  1.0012434
123146152019521  1074  41416  1.0001307
123146152018793  1166  44654  1.0000862
Attached Files
 17primes76apart.txt (36.1 KB, 27 views)