We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is

In his 1961 paper, Marcel Golay showed how the search for pairs of binary sequences of length n with complementary autocorrelation is at worst a 2 3n 2 âˆ’6 problem. Andres, in his 1977 masterâ€™sâ€¦ (More)

k=l Fermat's little theorem says that if p is a prime, then kP-l 3 lmodp for k = 1,.. .,p-1. Therefore, for each prime p, Sp--lmodp. The question becomes: Does there xist a non-prime n such that Sn 3â€¦ (More)

We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most n with integer coefficients. This is an old and hard problem that cannot beâ€¦ (More)

in certain natural regions in the complex plane where Pc, and q. are polynomials of degree cn and n, respectively. In particular we construct natural maximal regions (as a function of ~ and e) whereâ€¦ (More)

The Liouville function Î»(n) is the completely multiplicative function whose value is âˆ’1 at each prime. We develop some algorithms for computing the sum T (n) = Pn k=1 Î»(k)/k, and use these methods toâ€¦ (More)

Approximation to exp of the form f~m(z) := pro(z) e -2z + qm(7.) e -Z + rm(z) = O(z3C"+t)-l), where p,., qm, and r,, are polynomials of degree at most m and p,, has lead coefficient 1 is considered.â€¦ (More)

We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is a+ =an a 2bn and bâ€¦ (More)

We examine sequences of polynomials with {+1,âˆ’1} coefficients constructed using the iterations p(x) â†’ p(x) Â± xd+1pâˆ—(âˆ’x), where d is the degree of p and pâˆ— is the reciprocal polynomial of p. If p0 = 1â€¦ (More)