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 2020-06-01, 22:37 #12 Bobby Jacobs     May 2018 18610 Posts Yes. 282332382833 is a cool number. It reminds me of the maximal prime gap between 352521223451364323 and 352521223451365651, and the maximal prime gap after 6787988999657777797.
2020-06-03, 16:38   #13
sweety439

Nov 2016

5·7·59 Posts

Quote:
 Originally Posted by Bobby Jacobs Yes. 282332382833 is a cool number. It reminds me of the maximal prime gap between 352521223451364323 and 352521223451365651, and the maximal prime gap after 6787988999657777797.
How about searching gaps between primes having primitive root 2? I think 2 is more natural, it is no reason to search gaps between primes having primitive root 10.

For primitive root 2, a gap of 8k+4 does not exist, since all primes having primitive root 2 are == 3, 5 (mod 8)

Code:
gap  prime
2  3
6  5
8  29
10  19
14  149
16  37
18  83
22  421
24  107
26  587
30  317
32  2099
34  619
38  2621
40  1693
42  227
46  1381
48  709
50  3203
54  2477
56  4547
58  12979
62  4157
64  4723
66  1307
70  4021
72  947
74  1787
78  5573
80  12659
82  23251
86  20357
88  9949
90  13523
94  18493
96  15971
98  14243
102  33637
104  3083
106  63667
110  20789
112  24547
114  9059
118  88093
120  11317
122  109619
126  70717
128  46349
130  49891
134  244109
136  70237
138  105691
142  132709
144  18269
146  425387
150  221261
152  266117
154  62323
158  235541
160  31699
162  139907
166  102877
168  65371
170  142211
174  199037
176  265163
178  296299
182  223829
184  411013
186  137723
190  699757
192  191837
194  658643
198  103093
200  339827
202  302227
206  2989757
208  806581
210  425603
214  155797
216  598427
218  184259
222  736469
224  514949
226  2373307
230  1530293
232  1078411
234  512819
238  1427509
240  370133
242  1353371
246  2084653
248  476603
250  2551099
254  4013573
256  2617651

2020-06-03, 18:00   #14
mart_r

Dec 2008
you know...around...

22516 Posts

Quote:
 Originally Posted by sweety439 How about searching gaps between primes having primitive root 2? I think 2 is more natural, it is no reason to search gaps between primes having primitive root 10.
You're on!

You could calculate the merits via Artin*gap/log(p) and look for bounds in the CSG analogue Artin*gap/log²(p). It'd be interesting to see if there are some extraordinarily large gaps. If not in base 2, you could check other bases as well.

2020-06-30, 12:42   #15
sweety439

Nov 2016

206510 Posts

Quote:
 Originally Posted by sweety439 Also see https://de.wikipedia.org/w/index.php...ldid=195978132 for 2<=m<=128, 1<=n<=128 (0 if not exist). (I searched up to p=2^24, and I assume that there are no p>2^24 which is the smallest prime such that znorder(Mod(m,p)) = (p-1)/n for some m,n <= 128, in the list, the largest such prime is 2334251 (for m=126, n=125), and 2334251 is just between 2^21 and 2^22)
They are in https://en.wikipedia.org/w/index.php...ldid=964864733, for 2<=m<=128, 1<=n<=128 (0 if not exist)

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