(2+\sqrt{3} )^2

=7+4\sqrt{3}

Let

a =7+4\sqrt{3}

b =7-4\sqrt{3}

Then

a+b = 14

a = 14-b

b =7-4\sqrt{3} =(2-\sqrt{3} )^2<(2-\sqrt{1} )^2 =1

Therefore

the greatest integer less than or equal to a is 13