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Abstract. The Biquadratic Reciprocity Law is used to produce a determin-

No abstract

Let qðtÞ ¼ c n1 ðtÞ· · ·c nk ðtÞ, where c ni ðtÞ is the n i -th cyclotomic polynomial. Let z q ðtÞ ¼ qðtÞð1 2 tÞ 22 or z q ðtÞ ¼ qðtÞð1 2 t 2 Þ 21 , depending if the leading coefficient of the polynomial qðtÞ is ' þ 1' or ' 2 1', respectively. The rational function z q ðtÞ can be written as, the r i 's are positive integers, m i 's are integers and N z is a positive integer depending on z q . In the present paper, we study the set L z :¼ >{r 1 ; . . . ; r Nz } where the intersection is considered over all the possible decompositions of z q ðtÞ of the type mentioned above. Here, we describe the set L z in terms of the arithmetic properties of the integers n 1 ; . . . ; n k . We also study the question: given S a finite subset of the natural numbers, does exists a z q ðtÞ, such that L z ¼ S? The set L z is called the minimal set of Lefschetz periods associated with q(t). The motivation of these problems comes from differentiable dynamics, when we are interested in describing the minimal set of periods for a class of differentiable maps on orientable surfaces. In this class of maps, the Morse -Smale diffeomorphisms are included (cf.

Abstract. We use a result of E. Lehmer in cubic residuacity to find an algorithm to determine primality of numbers of the form A 2 3 n + 1, n odd, A 2 < 4(3 n + 1). The algorithm represents an improvement over the more general algorithm that determines primality of numbers of the form A.3 n ± 1, A/2 < 4.3 n − 1, presented by Berrizbeitia and Berry (1999).