A $3$-adic Recurrence for the Fixed Points of the Josephus Function $J_4$

Summary · qwen2.5:32b

The paper establishes a $3$-adic recurrence to explain erratic fluctuations in fixed points of the Josephus function for stepsize four, where the survivor is initially positioned last; this alternation between two types of near-misses (falling short by one or two seats) differentiates it from solved cases with smaller stepsizes. The length of each block of near-misses corresponds to the number of times three divides a quantity derived from preceding circle sizes, enabling an efficient computation of the survivor's position for any circle size proportional to counting near-misses rather than iterating through all smaller circles.

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Discover the hidden pattern in the Josephus problem with a new $3$-adic recurrence for fixed points of the Josephus function J4.

Excerpt

arXiv:2607.01270v2 Announce Type: replace Abstract: In the Josephus problem with stepsize four, the participants in a circle are eliminated one by one, every fourth person leaving, until a single survivor remains. A fixed point occurs when the survivor turns out to be the person who began in the last seat. The circle sizes with this property form the sequence 1; 21; 38; 51; 122; 163; 689; 919; 2,906; and so on, whose gaps fluctuate erratically. This paper explains the fluctuation and turns it into a recurrence. Between consecutive fixed points, the circle sizes at which the survivor falls exactly one or two seats short of the last one, the near-misses, group into alternating blocks of the two kinds, and the length of every block is the number of times three divides a simple quantity built from the circle size that precedes the block. Iterating these divisibility counts carries each fixed point to the next. Stepsize four is the first case in which two kinds of near-miss coexist, and the alternation they force is what separates it from the solved cases of stepsizes two and three. As a byproduct, the survivor's position for an arbitrary circle size can be computed by walking the near-misses of a single interval, in a number of steps proportional to their count, rather than stepping through every smaller circle as the defining recursion does.