A half Carol (probable) prime base 319 (the smallest half Carol (probable) prime base 319 is ((319^5513-1)^2-2)/2)

Carol prime (with even base (b)): (b^n-1)^2-2
Half Carol prime (with odd base (b)): ((b^n-1)^2-2)/2
Kynea prime (with even base (b)): (b^n+1)^2-2
Half Kynea prime (with odd base (b)): ((b^n+1)^2-2)/2

To be completed...