Let E be a real quadratic field with discriminant d ≡ 0 (mod p) where p is an odd prime. In terms of a Lucas quotient, the fundamental unit and the class number of E, we determine 0<c<d, (d c)=ρ p−1 ⌊pc/d⌋ modulo p 2 where ρ = ±1.

Let A„{a, b) = {ban+(a-l)/b)2+4an with n > 1 and ¿>|a-l . If W is a finite set of primes such that for each n > 1 there exists some q £W for which the Legendre symbol {A„{a, b)/q) ^ -1 , we call <£ a quadratic residue cover (QRC) for the quadratic fields K„{a, b) = Q{^jA„{a, b)). It is shown how the existence of a QRC for any a, b can be used to determine lower bounds on the class number of K„{...

It is well-known since Gauss that infinitely many quadratic fields have even class number. In fact, if K is a quadratic field of discriminant D, having r prime divisors, then the class number hK is divisible by 2 if D < 0 and by 2 if D > 0. See [4, Theorem 3.8.8] for a more precise statement. In 1922 Nagell [17, Satz VI] obtained the following remarkable result: given a positive integer l, ther...

We introduce a ray class invariant X(C) for a totally real field, following Shintani’s work in the real quadratic case. We prove a factorization formula X(C) = X1(C) · · ·Xn(C) where each Xi(C) corresponds to a real place (Theorem 3.5). Although this factorization depends a priori on some choices (especially on a cone decomposition), we can show that it is actually independent of these choices ...

We study the evolution of energy distribution and equation state Universe from end inflation until onset either radiation domination (RD) or a transient period matter (MD). use both analytical techniques lattice simulations. consider two-field models where inflaton $\Phi$ has monomial potential after $V(\Phi) \propto |\Phi - v|^p$ ($p\geq2$), is coupled to daughter field $X$ through quadratic-q...

We prove that almost any pair of real numbers α, β, satisfies the following inhomogeneous uniform version of Littlewood’s conjecture: For any real γ, δ, lim inf |n|→∞ n〈nα− γ〉〈nβ − δ〉 = 0, (0.1) where 〈·〉 denotes the distance from the nearest integer. The existence of even a single pair that satisfies (0.1), solves an open problem of Cassels [Ca] from the 50’s. We then prove that if 1, α, β spa...