p*2^{p}+1 cannot be prime if p>3 is a prime twin.
First suppose p=1 (mod 3) and let p=2q+1 (because p is odd). Operating modulo 3 we find:
p*2^{p}+1 = 1*2^{2q+1}+1 = 2^{2q}*2+1 = (2^{2})^{q}*2+1 = 1^{q}*2+1 = 2+1 = 0 (mod 3)
The other possibility is p=2 (mod 3) and p+2 also a prime. Operating modulo p+2 we find:
p*2^{p}+1 = (2)*2^{p}+1 = 2^{p+1}+1 = 1+1 = 0 (mod p+2)
So in both cases p*2^{p}+1 is composite.
