Pattern in a specific range in the prime twin numbers I found this pattern, does it have any mathematical significance or something else?
From twin (5,7) to twin (101,103) there is a common pattern in that area.
(P+1)^2 - 6=A
A/6=E
E: Prime number
the pattern
6^2 - 6=30
30/6=5 prime

12^2 - 6=138
138/6=23 Prime

18^2 -6=318
318/6=53 prime

30^2 - 6=894
894/6=149 Prime

42^2 - 6=1758
1758/6=293 prime

60^2 -6=3594
3594/6=599

72^2 -6=5178
5178/6=863 Prime

102^2 -6=10398
10398/6=1733 Prime
 A: All twin prime pairs except $(4,6)$ have the form $(6k-1, 6k+1)$.  If you pick the lower one ($P=6k-1$), then $(P+1)^2 = (6k)^2$ will obviously be a multiple of 6, and thus so will $A=(P+1)^2-6$.
It seems that you're asking if $E = \frac{A}{6}$ is “usually” prime.  This can be tested with a simple Python program:
import math

def is_prime(n):
    limit = int(math.sqrt(n) + 2)
    for divisor in range(2, limit):
        if n % divisor == 0:
            return False
    return True

def primes(limit):
    '''
    Return a list of all prime numbers 2 <= n < limit.
    '''
    result = []
    potential = range(2, limit)
    while potential:
        next_prime = potential[0]
        result.append(next_prime)
        potential = [n for n in potential if n % next_prime]
    return result

LIMIT = 1000
PRIME_LIST = primes(LIMIT)
TWIN_PRIMES = [
    pair for pair in (
        (PRIME_LIST[i], PRIME_LIST[i+1]) for i in range(0, len(PRIME_LIST)-1)
    ) if pair[1] - pair[0] == 2
]

examples = []
counterexamples = []
for (p, _) in TWIN_PRIMES:
    a = (p + 1) ** 2 - 6
    e = a // 6
    if is_prime(e):
        examples.append(p)
    else:
        counterexamples.append(p)

print('examples = ', examples)
print('counterexamples = ', counterexamples)

I get the output:

*

*examples =  [3, 5, 11, 17, 29, 41, 59, 71, 101, 179, 191, 227, 269, 311, 347, 419, 461, 521, 599, 857]

*counterexamples =  [107, 137, 149, 197, 239, 281, 431, 569, 617, 641, 659, 809, 821, 827, 881]

