Conjecture: If the condition holds, prime(k-1) and prime(k) are twin primes of the form 10m+1 and 10m+3. I.e., the last digits are 1 and 3.
This is so because prime(k-1)+5 square => possible ending digits of 0,1,4,5,6,9. To get ending digits for prime(k-1) we subtract 5 to get ending digits 5,6,9,0,1,4. So 1,9 are the only possible endings since 0,4,5,6 => prime(k-1) not prime, impossible. Now by the condition of equality, prime(k-1)+6 = prime(k+1). So if prime(k-1) ends in 9, prime(k-1)+6 ends in 5 => prime(k+1) not prime, impossible. Therefore prime(k-1) ends in 1 and by the condition of prime(k-1) and prime(k) being twin primes, prime(k) ends in 3. This sequence is a variation of the conjecture provided in the link.
The PARI script allows for general investigation of numbers of the form prime(k-1)+a and prime(k+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(k-1) retrieval before calling the second prime(k+1). This cuts the search time in half. A much faster and more inclusive program is in the Link.
