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Old 2019-04-17, 10:04   #276
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.........
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Old 2019-04-17, 10:05   #277
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searched up to 20 (decimal 24) digits
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Old 2019-04-22, 07:52   #278
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............
There are 59 (69 in decimal) bases n<=1000 (1728 in decimal) without primes of the form 2*n^k-1 with k<=1000 (1728 in decimal):

{8E, 122, 175, 192, 1E0, 1E2, 20X, 213, 278, 27X, 27E, 36E, 376, 39E, 3E8, 402, 405, 412, 436, 443, 45E, 472, 479, 4X2, 4X8, 4E8, 555, 5X5, 608, 662, 688, 689, 6X5, 6X8, 70E, 738, 74E, 752, 75E, 765, 800, 835, 83E, 852, 855, 856, 864, 8XE, 918, 91X, 92E, 942, 972, 982, 9E5, X3E, X72, X95, XXE, E09, E0E, E24, E28, E52, E5E, E72, E78, EE8, EEE}

Last fiddled with by sweety439 on 2019-04-22 at 08:05
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Old 2019-04-22, 07:58   #279
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Quote:
Originally Posted by sweety439 View Post
There are 59 (69 in decimal) bases n without primes of the form 2*n^k-1 with k<=1000 (1728 in decimal):

{8E, 122, 175, 192, 1E0, 1E2, 20X, 213, 278, 27X, 27E, 36E, 376, 39E, 3E8, 402, 405, 412, 436, 443, 45E, 472, 479, 4X2, 4X8, 4E8, 555, 5X5, 608, 662, 688, 689, 6X5, 6X8, 70E, 738, 74E, 752, 75E, 765, 800, 835, 83E, 852, 855, 856, 864, 8XE, 918, 91X, 92E, 942, 972, 982, 9E5, X3E, X72, X95, XXE, E09, E0E, E24, E28, E52, E5E, E72, E78, EE8, EEE}
Most of the bases end with 2, 5, 8 or E (i.e. = 2 mod 3)

There are..

2 bases end with 0: 1E0, 800
no bases end with 1
14 bases end with 2: 122, 192, 1E2, 402, 412, 472, 4X2, 662, 752, 852, 942, 972, 982, X72, E52, E72
2 bases end with 3: 213, 443
2 bases end with 4: 864, E24
X bases end with 5: 175, 405, 555, 5X5, 6X5, 765, 835, 855, 9E5, X95
3 bases end with 6: 376, 436, 856
no bases end with 7
10 bases end with 8: 278, 3E8, 4X8, 4E8, 608, 688, 6X8, 738, 918, E28, E78, EE8
3 bases end with 9: 479, 689, E09
3 bases end with X: 20X, 27X, 91X
14 bases end with E: 8E, 27E, 36E, 39E, 45E, 70E, 74E, 75E, 83E, 8XE, 92E, X3E, XXE, E0E, E5E, EEE

Last fiddled with by sweety439 on 2019-04-22 at 08:13
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Old 2019-04-22, 08:05   #280
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............
There are 71 (85 in decimal) bases n<=1000 (1728 in decimal) without primes of the form 2*n^k+1 with k<=1000 (1728 in decimal):

{32, 85, 11E, 152, 162, 178, 195, 1EE, 215, 255, 265, 268, 27E, 28E, 2X8, 325, 32E, 368, 372, 392, 402, 442, 44E, 45E, 49E, 4X2, 528, 532, 545, 558, 569, 598, 5X2, 605, 612, 61E, 62E, 638, 642, 645, 658, 66E, 676, 6E8, 738, 742, 755, 75E, 762, 775, 7XE, 7E2, 82E, 835, 868, 892, 8X5, 8E2, 90E, 915, 932, 935, 938, 96E, 972, 98E, 998, 9E5, X02, X05, X15, X2E, X3E, X42, X58, X82, X95, XXE, XE8, E08, E3E, E42, E95, EE2, EEE}

(not consider the bases end with 1, 4, 7 or X (i.e. = 1 mod 3), since for these bases n, all numbers of the form 2*n^k+1 are divisible by 3)

Last fiddled with by sweety439 on 2019-04-22 at 10:29
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Old 2019-04-22, 08:07   #281
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Quote:
Originally Posted by sweety439 View Post
There are 71 (85 in decimal) bases n<=1000 (1728 in decimal) without primes of the form 2*n^k+1 with k<=1000 (1728 in decimal):

{32, 85, 11E, 152, 162, 178, 195, 1EE, 215, 255, 265, 268, 27E, 28E, 2X8, 325, 32E, 368, 372, 392, 402, 442, 44E, 45E, 49E, 4X2, 528, 532, 545, 558, 569, 598, 5X2, 605, 612, 61E, 62E, 638, 642, 645, 658, 66E, 676, 6E8, 738, 742, 755, 75E, 762, 775, 7XE, 7E2, 82E, 835, 868, 892, 8X5, 8E2, 90E, 915, 932, 935, 938, 96E, 972, 98E, 998, 9E5, X02, X05, X15, X2E, X3E, X42, X58, X82, X95, XXE, XE8, E08, E3E, E42, E95, EE2, EEE}
Most of the bases end with 2, 5, 8 or E (i.e. = 2 mod 3)

There are..

no bases end with 0
no bases end with 3
1 base ends with 6: 676
1 base ends with 9: 569
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Old 2019-04-22, 08:10   #282
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Quote:
Originally Posted by sweety439 View Post
There are 71 (85 in decimal) bases n<=1000 (1728 in decimal) without primes of the form 2*n^k+1 with k<=1000 (1728 in decimal):

{32, 85, 11E, 152, 162, 178, 195, 1EE, 215, 255, 265, 268, 27E, 28E, 2X8, 325, 32E, 368, 372, 392, 402, 442, 44E, 45E, 49E, 4X2, 528, 532, 545, 558, 569, 598, 5X2, 605, 612, 61E, 62E, 638, 642, 645, 658, 66E, 676, 6E8, 738, 742, 755, 75E, 762, 775, 7XE, 7E2, 82E, 835, 868, 892, 8X5, 8E2, 90E, 915, 932, 935, 938, 96E, 972, 98E, 998, 9E5, X02, X05, X15, X2E, X3E, X42, X58, X82, X95, XXE, XE8, E08, E3E, E42, E95, EE2, EEE}
Those bases end with E are related to Divides Phi(n^k,2) category

There are 1X such bases: 11E, 1EE, 27E, 28E, 32E, 44E, 45E, 49E, 61E, 62E, 66E, 75E, 7XE, 82E, 90E, 96E, 98E, X2E, X3E, XXE, E3E, EEE

Last fiddled with by sweety439 on 2019-04-22 at 08:11
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Old 2019-04-22, 08:25   #283
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Quote:
Originally Posted by sweety439 View Post
Those bases end with E are related to Divides Phi(n^k,2) category

There are 1X such bases: 11E, 1EE, 27E, 28E, 32E, 44E, 45E, 49E, 61E, 62E, 66E, 75E, 7XE, 82E, 90E, 96E, 98E, X2E, X3E, XXE, E3E, EEE
11E is prime
1EE = 7 × 35
27E is prime
28E = 5 × 67
32E is prime
44E = 5 × X7
45E is prime
49E = 5 × E7
61E is prime
62E = 25 × 27
66E is prime
75E = 11 × 6E
7XE = 15 × 57
82E is prime
90E is prime
96E = 7 × 145
98E = 1E × 51
X2E = 52 × 4E
X3E is prime
XXE is prime
E3E = 7 × 175
EEE = E × 111
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Old 2019-04-22, 10:26   #284
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............
There are 11 (decimal 13) bases n<=1000 (decimal 1728) with both sides (2*n^k-1 and 2*n^k+1) remaining:

{27E, 402, 45E, 4X2, 738, 75E, 835, 972, 9E5, X3E, X95, XXE, EEE}

Last fiddled with by sweety439 on 2019-04-22 at 10:27
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Old 2019-06-03, 07:48   #285
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.......
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File Type: txt even pseudoprime.txt (39.1 KB, 11 views)
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Old 2019-06-12, 03:33   #286
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the title should be "least k such that a063994(k)=n"

searched up to k=10^6, for n up to 400
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