In relation to Hall's conjecture, a new algorithm is presented to search for small nonzero í µí± = ∣í µí±¥ 3 −í µí±¦ 2 ∣ values. Seventeen new values of í µí± < í µí±¥ 1/2 are reported. Dealing with natural numbers, the difference (1.1) í µí± = í µí±¥ 3 − í µí±¦ 2 is zero when í µí±¥ = í µí±¡ 2 and í µí±¦ = í µí±¡ 3 but, in other cases, it seems… (More)
This paper develops an algorithm to search for approximations to a real ξ which finds rationals of the form a/b 2 , such that ˛ ˛ ˛ξ − a b 2 ˛ ˛ ˛ < c(ξ) b 3 , in B 1/2 polynomial time operations, for b ≤ B. An heuristic approach and computational data give support to the conjecture of O. Karpenkov stating that there are infinitely many solutions in… (More)
In relation to Hall's conjecture , a new algorithm is presented to search for small nonzero k = |x 3 − y 2 | values. Thirteen new values of k < x 1/2 are reported. Dealing with natural numbers, the difference (1.1) k = x 3 − y 2 is zero when x = t 2 and y = t 3 but, in other cases, it seems difficult to achieve small absolute values. For a given k = 0,… (More)
Plots of quadratic residues display some visual features that are analyzed mathematically in this paper. The graphical patterns of quadratic residues modulo m depend only on the residues of m modulo the lowest positive integers.