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

[enzocreti]

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


Are they finite?

[CRGreathouse]

There should be infinitely many. You can look at the residue classes and do an infinite product to guess how sparse they will be.

[Dr Sardonicus]

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,

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,

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.")