[3] **viXra:1508.0201 [pdf]**
*replaced on 2015-08-30 18:35:17*

**Authors:** Marco Ripà

**Comments:** 9 Pages.

We provide an optimal strategy to solve the n X n X n points problem inside the box, considering only 90° turns, and at the same time a pattern able to drastically lower down the known upper bound. We use a very simple spiral frame, especially if compared to the previous plane by plane approach, that significantly reduces the number of straight lines connected at their end-points necessary to join all the n3 dots. In the end, we combine the square spiral frame with the rectangular spiral pattern in the most profitable way, in order to minimize the difference hu(n) − hl(n) between the upper and the lower bound, proving that it is ≤ 0.5 ∙ n ∙ (n + 3), for any n > 1.

**Category:** Combinatorics and Graph Theory

[2] **viXra:1508.0085 [pdf]**
*replaced on 2015-08-27 14:39:38*

**Authors:** Philip Gibbs

**Comments:** 16 Pages.

The Ulam numbers form an increasing sequence beginning 1,2 such that each subsequent number can be uniquely represented as the sum of two smaller Ulam numbers. An algorithm is described and implemented in Java to compute the first billion Ulam numbers.

**Category:** Combinatorics and Graph Theory

[1] **viXra:1508.0045 [pdf]**
*replaced on 2015-08-08 10:45:52*

**Authors:** Philip Gibbs

**Comments:** 4 Pages.

A conjecture on the quasi-periodic behaviour of Ulam sequences

**Category:** Combinatorics and Graph Theory