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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)
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.
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)