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If the condition holds, prime(n-1) and prime(n) are twin primes. These
are of the form 10k+9 and 10k+1. Ie., the last digits are 9 and 1. This
is true because a square number must end in 0,1,4,5,6,9. So prime(n-1)+7
is square => it must end in one of these numbers. So to find the ending
of prime(n-1), we subtract 7 form 0,1,4,5,6,9 to get last digit 3,4,7,8,9,2.
Since prime(n-1) is prime endings 2,4,5,8 are not allowed. This leaves
us with 3,7,9 as possible endings of prime(n-1). Now to get prime(n)
for which the condition states is 2 greater than prime(n-1), we add
2 and 3+2=5 => prime(n) not prime, impossible. So the possible
endings of prime(n-1) are dwindled to 7 or 9. Now the condition
prime(n-1)+7 = prime(n+1)-1 => prime(n-1)+8 = prime(n+1). Then adding 7
=> prime(n+1) ends in 5, impossible. So prime(n-1) must end in 9 and
adding 2, makes prime(n) end in 1. This sequence is a calculation of the
conjecture provided in the link. The Pari script provides for the generai
investigation of numbers of the form prime(n-1)+a and prime(n+1)-b. The
values a=5,7; b=1 consistently yield twin primes when the condition holds.
Notice we test for square of the first prime(n-1) retrevial before calling
the second prime(n+1). This cuts the search time in half. A far superior
search routine is the gcc program found in the link which reads a huge 300
gig file of the primes < 1 trillion+1billion.
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