A conjecture on class numbers

Titus Piezas III · science forum Guru Wannabe Joined: 10 Mar 2005 Posts: 102

Hello all,

Just two curious conjectures here:

Let m,n be any element of {0,1,2,....}

1. Let F(n) be a square-free number of form 8n+3. If F(n) is prime,
then -8F(n) has class number h(d) of form 4m+2. If composite, then of
form 4m.

Ex. F(n) = 11 with h(-8*11) = 2;
F(n) = 35 with h(-8*35) = 4.

2. Let F(n) be a square-free number of form 8n+5. If F(n) is prime,
then both -4F(n) and -8F(n) have class numbers of form 4m+2. If
composite, then of form 4m.

Ex. F(n) = 13 with h(-4*13) = 2, h(-8*13) = 6;
F(n) = 85 with h(-4*85) = 4, h(-8*85) = 12.

True or not?

P.S. Numbers 8n+1 and 8n+7 are not that well-behaved.

--Titus

victor_meldrew_666@yahoo. · science forum beginner Joined: 19 May 2006 Posts: 17

titus_piezas@yahoo.com wrote:

Titus Piezas III · science forum Guru Wannabe Joined: 10 Mar 2005 Posts: 102

Wonderful! So glad to know. Here's two more, on the j-function (Note:
Discriminant d is understood to be d < 0):

1. Given a fundamental d (not a multiple of 3) with class number h(d),
then j-function j(tau) = x^3, where x is an algebraic integer of degree
h(d). In other words, it is a perfect "cube".

2. If d = 3m with m not a square and h(d) a power of 2, then j(tau) =
v^2x^3 where v is a fundamental unit of m and x is an algebraic integer
of degree h(d).

Note: We will define the fundamental unit v here as p+q*Sqrt[m] such
that p^2-mq^2 = +/-1, hence uses either the + or - case of Sqrt[m].

Example with tau = (1+Sqrt[d])/2, d = -3*17, h(d) = 2:

J(tau) = -(4+Sqrt[17])^2(48^3)(5+Sqrt[17])^3

Are both true?

-Titus

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