Primality test for Twin Prime numbers.

[Davide Rotondo]

Dear sequence fan, yesterday I was looking to test for twin primes on the web and I noticed https://vixra.org/pdf/2108.0109v3.pdf by professor Gabriel Zeolla. The first formula is equal to in https://oeis.org/A192297 by Vladimir Shevelev.I think the most important part of this paper is where he talk about tests with other bases.

I tried with base 3 reinterpreting his formula: (((3^(n+2))-12*n-27)/(n*(n+2))

The values for which (((3^(n+2))-12*n-27) is divisible by (n*(n+2)) are:

[3, 5, 9, 11, 17, 27, 29, 41, 45, 59, 71, 89, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 561, 569, 599, 617, 641, 659, 671, 701, 809, 821, 827, 857, 881, 947, 1019, 1031, 1049, 1061, 1091, 1103, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1541, 1607, 1619, 1667, 1697, 1721, 1787, 1871, 1877, 1889, 1931, 1949, 1997, 2027, 2081, 2087, 2111, 2129, 2141, 2237, 2267, 2309, 2339, 2381, 2465, 2549, 2591, 2657, 2663, 2687, 2699, 2711, 2729, 2789, 2801, 2819, 2969, 2999, 3119, 3167, 3251, 3257, 3299, 3329, 3359, 3371, 3389, 3461, 3467, 3527, 3539, 3557, 3581, 3671, 3767, 3821, 3851, 3917, 3929, 4001, 4019, 4049, 4091, 4127, 4157, 4217, 4229, 4241, 4259, 4271, 4337, 4421, 4481, 4517, 4547, 4637, 4649, 4721, 4787, 4799, 4931, 4967, 5009, 5021, 5099, 5231, 5279, 5417, 5441, 5477, 5501, 5519, 5639, 5651, 5657, 5741, 5849, 5867, 5879, 6089, 6131, 6197, 6269, 6299, 6359, 6449, 6551, 6569, 6599, 6659, 6689, 6701, 6761, 6779, 6791, 6827, 6869, 6947, 6959, 7107, 7127, 7211, 7307, 7331, 7349, 7457, 7487, 7547, 7559, 7589, 7757, 7877, 7949, 8009, 8087, 8219, 8231, 8291, 8387, 8429, 8537, 8597, 8627, 8819, 8837, 8861, 8969, 8999, 9011, 9041, 9239, 9281, 9341, 9419, 9431, 9437, 9461, 9629, 9677, 9719, 9767, 9857, 9929, 10007, 10037, 10067, 10091, 10139, 10271, 10301, 10331, 10427, 10457, 10499, 10529, 10709, 10859, 10889, 10937, 11057, 11069, 11117, 11159, 11171, 11351, 11489, 11549, 11699, 11717, 11777, 11831, 11939, 11969, 12041, 12071, 12107, 12161, 12239, 12251, 12377, 12401, 12539, 12611, 12821, 12917, 13001, 13007, 13217, 13337, 13397, 13679, 13691, 13709, 13721, 13757, 13829, 13877, 13901, 13931, 13997, 14009, 14081, 14249, 14321, 14387, 14447, 14549, 14561, 14591, 14627, 14867, 15137, 15269, 15287, 15329, 15359, 15581, 15641, 15647, 15731, 15737, 15887, 15971, 16061, 16067, 16139, 16187, 16229, 16361, 16451, 16529, 16631, 16649, 16691, 16829, 16901, 16979, 17027, 17189, 17207, 17291, 17387, 17417, 17489, 17579, 17597, 17657, 17681, 17747, 17789, 17837, 17909, 17921, 17957, 17987, 18041, 18047, 18059, 18119, 18131, 18251, 18287, 18311, 18521, 18539, 18719, 18911, 18917, 19079, 19139, 19181, 19211, 19379, 19421, 19427, 19469, 19541, 19697, 19751, 19841, 19889, 19961, 19991, 20021, 20147, 20231, 20357, 20441, 20477, 20507, 20549, 20639, 20717, 20747, 20771, 20807, 20897, 20981, 21011, 21017, 21059, 21191, 21317, 21377, 21491, 21521, 21557, 21587, 21599, 21611, 21647, 21737, 21839, 22037, 22091, 22109, 22157, 22271, 22277, 22367, 22481, 22541, 22571, 22619, 22637, 22697, 22739, 22859, 22961, 23027, 23039, 23057, 23201, 23291, 23369, 23537, 23561, 23627, 23669, 23687, 23741, 23831, 23909, 24107, 24179, 24371, 24419, 24659, 24917, 24977, 25031, 25169, 25301, 25307, 25409, 25469, 25577, 25601, 25799, 25847, 25931, 25997, 26111, 26249, 26261, 26681, 26699, 26711, 26729, 26861, 26879, 26891, 26951, 27059, 27107, 27239, 27281, 27407, 27479, 27527, 27539, 27581, 27689, 27737, 27749, 27791, 27917, 27941, 28097, 28109, 28181, 28277, 28307, 28349, 28409, 28547, 28571, 28619, 28661, 28751, 29021, 29129, 29207, 29339, 29387, 29399, 29567, 29669, 29759, 29879]

I noticed that these numbers are lesser of twin primes pair or lesser of a pair (n,n+2) such that n is prime while n+2 is a Feramt pseudoprime to base 27, or lesser of a pair (n,n+2) such that n is a Fermat pseudoprime to base 27 while n+2 is prime.

It's important to take into account that this is not a test for twin primes union pseudo-twin primes to base 27, because not all pseudoprimes to base 27 are part of the list.

See you soon

Davide

P.S. Python

def is_integer_expression(n):
    numerator = 3**(n + 2) - 12 * n - 27
    denominator = n * (n + 2)
    return numerator % denominator == 0

# Lista dei valori validi di n
valid_n_values = []

for n in range(3, 30001):
    if is_integer_expression(n):
        valid_n_values.append(n)

print(valid_n_values) 

[Davide Rotondo]

P.s. i forgot these are weak pseudoprimes to base 27.

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