Thank you all. I realized the design I am interested in has been studied in a more general setting named intersecting families in extremal set theory.

Thank you so much. With the referene you mentioned, I found a comprehensive treatment on this problem by J. H. E. Cohn ( "The diophantine equation x^2 + C = y^n", Acta Arithmetica,1993 ). More precisely, x^2+1=y^n has no solution for x, y being positive integers and n greater than or equal to 3. Now, this question is closed.