This is my answer, but someone already posted the right answer to the original question.
A and B: length=X and width=X+1.
C: 2X+1 is prime.
D: perimeter=4X+2=p1-3 and p2+1 where p1 and p2 are prime.
E and F: X(X+1)=X^2+X cannot be prime and 64<X(X+1)<100.
X^2+X-64>0; X^2+X-100<0; X^2+X+1/4>64.25; (X+1/2)^2>64.25, so X>7.51; (X+1/2)^2<100.25, so X<9.51. If X is a whole number, it must be 8 or 9. Both satisfy C. 4X+5 is 37 or 41. 4X+1 is 33 or 37. Since 37 is prime, X=9. The dimensions of the rectangle are 9 and 10 with perimeter 38.
Check list:
A and B: length=9 width=10.
C: 19.
D: 38 is 3 less than 41 and 1 more than 37.
E: 90 is not a perfect square.
F: Area=90 is between 64 and 100.