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Old 2020-01-27, 07:31   #1
enzocreti
 
Mar 2018

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Default primes of ther form 19*3*2^k+ or -1

I found only three primes of this form up to k=10.000. (k=2,8,18)


Are they finite?
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Old 2020-01-27, 07:56   #2
CRGreathouse
 
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There should be infinitely many. You can look at the residue classes and do an infinite product to guess how sparse they will be.
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Old 2020-01-27, 13:59   #3
Dr Sardonicus
 
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Looks like the OP missed quite a few. If the intent was to find exponents on both lists, the exponent 10 is missing and the exponent 18 is wrong.

From the List of primes k*2n + 1 for k < 300 we have for k = 19*3 = 57,
Quote:
57 2, 3, 7, 8, 10, 16, 18, 19, 40, 48, 55, 90, 96, 98, 190, 398, 456, 502, 719, 1312, 1399, 1828, 6723, 6816, 10680, 12592, 20742, 25010, 26838, 29623, 45435, 52783, 70950, 89691, 111691, 114400, 136152, 145183, 146223, 177459, 212908, 300910, 342151, 360447, 382156, 411635, 442948, 519862, 519892, 975036, 1158942, 1181438, 1756702, 2033643 [2100000] 2492031 L1230, 2747499 L3514, 2765963 L3262
The exponents in bold give factors of Fermat numbers.

From the List of primes k*2n - 1 for k < 300, again for k = 19*3 = 57,
Quote:
57 1, 2, 4, 5, 8, 10, 20, 22, 25, 26, 32, 44, 62, 77, 158, 317, 500, 713, 1657, 1790, 2761, 2794, 3704, 4174, 6772, 14348, 16132, 16160, 16766, 21097, 29125, 40094, 44824, 49006, 67585, 74650, 94798, 162538, 173585, 239854 [400000] 450430 L171, 565994 L261, 707245 L384, 839446 L80, 1098272 L260, 1110980 L121, 1486214 L1828, 2103370 L2055, 2639528 L2484, 3339932 L3519
(The L's denote "proof codes.")

Last fiddled with by Dr Sardonicus on 2020-01-27 at 14:24 Reason: w, misread multiplier, had to redo.
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