Class number 2 problem for certain real quadratic fields of Richaud–Degert type

In this paper we will apply Biró's method in [A. Biró, Yokoi's conjecture, Acta Arith. 106 (2003) 85–104; A. Biró, Chowla's conjecture, Acta Arith. 107 (2003) 179–194] to class number 2 problem of real quadratic fields of Richaud–Degert type and will show that there are exactly 4 real quadratic fields of the form with class number 2, where n2+1 is a even square free integer.