A non-uniform distribution property of most orbits, in case the 3x+1\n conjecture is true

2015/12/17 by Alain Thomas, Thomas, Alain · 1 citation
Computer Science · Mathematics · #11A99 #11B37 #11B83 #Benford’s Law and Fraud Detection #Computability, Logic, AI Algorithms #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1512.05852

openalex publication_date 2015/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let T(n)= \\3n+1 · amp;(n
hbox odd)
frac n2 · amp;(n
hbox\neven) . (n\∈ mathbb Z). We call "the orbit of the integer\nn", the set
mathcal O_n:=
m
in
mathbb Z
;:
;
exists k
ge0,
m=T^k(n)
\n and we put c_i(n):= # m\∈ mathcal O_n ;: ;m\≡ i hbox mod.18 . Let\nW be the set of the integers whose orbit contains 1 and is, in the\nfollowing sense, about well distributed modulo 18 between the six elements of\nthe set I:= 1,5,7,11,13,17 (the elements of 1,\…,18 that are odd\nand not divisible by 3). More precisely: W:=
Big
n
in
mathbb\nN
;:
;
exists k
ge0,
T^k(n)=1
hbox and
forall i
in I,
\n
fracc_i(n)
sum_{i
in I}c_i(n)
le
frac16+0.0215
Big
. We prove that W\nhas density 0 in mathbb N. Consequently, if the 3x+1 conjecture is true,\nmost of the positive integers n satisfy
frac
max_i
in\nIc_i(n)
sum_{i
in I}c_i(n)gt;
frac16+0.0215.\n

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