P^2+Σ2P

H is the upper half-plane which contains the set of complex numbers with positive imaginary part. This is modular group . ad − bc = 1 This includes point at infinity .

a,b,c,d are integers. ad-bc=1 This is modular group which is also isomorphic of PSL(2, Z). a/b, a/c, c/d and b/d are irreducible . (a,b,c,d)=(7,5,4,3),(4,11,17,47),(4,7,13,23),(2,5,17,43),(6,19,23,73),(8,13,19,31),(48,13,11,3) You see 3 prime numbers and 1 even number in it. Τ=3P+N (P:prime number, N:even number) 2 is prime number, but it is also even number. This has special meaning in my blog . This is also irreducible . You see (7,5,4,3). 7p+5q 4p+3q (p=2, q=3) 14+15=29 8+9=17 29/17=1.705882353 PSL(V)=SL(V)/SZ(V) PSL is projective linear group, and V is vector space. Moreover, SZ(V) is the center, and SL(V) is factor group. This is order. d is the center.