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In this paper the following result is obtained. THEOREM. Let r be any positive integer; in all but finitely many finite fields k, of odd characteristic, for every polynomial f(x) e k[x] of degree r that is not of the form a(Q(x)) 2 or ax(9(x)) 2 , there exists a primitive root β £ k such that f(β) is a square in k. As a result of this and some computation we shall see that for every finite field k of characteristic Φ 2 or 3, there exists a primitive root aek such that-(a 2 + a + 1) = β 2 for some βek; also every linear polynomial with nonzero constant term in the finite field k of odd characteristic represents both nonzero squares and nonsquares at primitive roots of k unless k = GF(S), GF(S) or GF(Ί).

A structure-activity relationship study for a series of vitamin D3-based (VD3) analogues that incorporate aromatic A-ring mimics with varying functionality has provided key insight into scaffold features that result in potent, selective Hedgehog (Hh) pathway inhibition. Three analogue subclasses containing (1) a single substitution at the ortho or para position of the aromatic A-ring, (2) a heteroaryl or biaryl moiety, or (3) multiple substituents on the aromatic A-ring were prepared and evaluated. Aromatic A-ring mimics incorporating either single or multiple hydrophilic moieties on a six-membered ring inhibited the Hh pathway in both Hh-dependent mouse embryonic fibroblasts and cultured cancer cells (IC50 values 0.74-10 μM). Preliminary studies were conducted to probe the cellular mechanisms through which VD3 and 5, the most active analogue, inhibit Hh signaling. These studies suggested that the anti-Hh activity of VD3 is primarily attributed to the vitamin D receptor, whereas 5 affects Hh inhibition through a separate mechanism.

This paper describes a method of constructing an unlimited number of infinite families of continued fraction expansions of the square root of D, an integer. The periods of these continued fractions all have identifiable sub patterns repeated a number of times according to certain parameters. For example, it is possible to construct an explicit family for the square root of D(k, l) where the period of the continued fraction has length 2kl − 2. The method is recursive and additional parameters controlling the length can be added.