Abstract.A Newman polynomial is a sum of powers of z, with constant term 1. The Newman polynomial of four terms whose minimum modulus on the unit circle is as large as possible is found by examining the expression /(4) = sup inf and determining an extremal system (xx, ... , x4) using a technique that reduces the problem to a finite search.

The principal thrust of this investigation is to provide families of quadratic polynomials fD k ðX Þ ¼ f 2 k X 2 þ 2e k X þ Cg kAN ; where e 2 k À f 2 k C ¼ n (for any given nonzero integer n) satisfying the property that for any X AN; the period length c k ¼ cð ffiffiffiffiffiffiffiffiffiffiffiffiffiffi D k ðX Þ p Þ of the simple continued fraction expansion of ffiffiffiffiffiffiffiffiffiffiffiffiffiffi D k ðX Þ p is constant for fixed k and lim k-N c k ¼ N: This generalizes, and completes, numerous results in the literature, where the primary focus was upon jnj ¼ 1; including the work of this author, and coauthors, in Mollin (Far East

No abstract

ABSTRACT. The primary purpose of this paper is to provide necessary and sufficient conditions for certain quadratic polynomials of negative discriminant (which we call Euler-Rabinowitsch type), to produce consecutive prime values for an initial range of input values less than a Minkowski bound. This not only generalizes the classical work of Frobenius, the later developments by Hendy, and the generalizations by others, but also concludes the line of reasoning by providing a complete list of all such primeproducing polynomials, under the assumption of the generalized Riemann hypothesis (GRH). We demonstrate how this prime-production phenomenon is related to the exponent of the class group of the underlying complex quadratic field. Numerous examples, and a remaining conjecture, are also given.