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hexagon:2611.00031v1

math.NT — Number Theory

The binary Collatz dynamics of the trinomials xn+x+1x^n+x+1: proof of two conjectures of Gallardo and Rahavandrainy

Contributed by David Niedbala Giraudin

Gallardo and Rahavandrainy studied an analogue of the Collatz map on F2[x]\mathbb{F}_2[x], in which x(x+1)x(x+1) plays the role of 22 and x2+x+1x^2+x+1 that of 33: an odd polynomial AA is sent to the odd part of 1+(x2+x+1)A1+(x^2+x+1)A. They proved that every orbit reaches 11, and conjectured that the orbit of the trinomial xn+x+1x^n+x+1 has length 2s+12^s+1, where ss is the largest integer with 2s+1<n2^{s+1}<n, and that for 1≤t≤s−11\le t\le s-1 it contains 2t2^t terms of the same degree. We prove both statements for every n≥3n\ge3. The proof rests on the identity (x2+x+1)+x(x+1)=1(x^2+x+1)+x(x+1)=1: it reduces the dynamics to a map from one plateau to the next, characterises the last plateau of any orbit, and shows that along the orbit of xn+x+1x^n+x+1 the plateaus start at the 2j2^j-th powers of the odd parts of the polynomials xm(x2+x+1)+1x^m(x^2+x+1)+1, the index mm following an explicit binary recursion. We also note that a later restatement of the conjecture, with a non-strict inequality, fails exactly when nn is a power of 22. A script checks all the statements numerically.

Primarily AI-generated textHuman understanding: no parts
Nov 03, 2026PDF ↗View work →
hexagon:2611.00030v1

math.NT — Number Theory

On the largest irreducible factor of an odd perfect polynomial over F2\mathbb{F}_2

Contributed by David Niedbala Giraudin

A polynomial AA over F2\mathbb{F}_2 is perfect if it equals the sum σ(A)\sigma(A) of all its divisors, and odd if it is coprime to x2+xx^2+x. Canaday conjectured in 1941 that 1 is the only odd perfect polynomial. Known necessary conditions on a counterexample bound its number of distinct irreducible factors (ω≥5\omega\ge 5, Gallardo and Rahavandrainy), its number of irreducible factors counted with multiplicity and its degree (Ω≥12\Omega\ge 12 and deg⁡A>200\deg A>200, Cengiz, Pollack and Trevino), and, when all exponents equal 2, the degree of its smallest irreducible factor (at least 40, Gallardo). We bound a different invariant: every odd perfect polynomial A≠1A\ne 1 has an irreducible factor of degree at least 20. The proof reduces the question, for each DD, to a finite system: if all irreducible factors of AA have degree at most DD, the exponents of AA belong to an explicit finite set, through the cyclotomic factorisation of σ(Ph)\sigma(P^h) and the multiplicative order of PP modulo the factors of σ(Ph)\sigma(P^h). An iterated sieve on the resulting production relations empties for every D≤19D\le 19. The computation was carried out twice, by two independent programs, with identical results; the second program is attached as an ancillary file. As a cross-check of the method, the same programs reproduce, by the orbit argument of Gallardo and Rahavandrainy, the exclusion of every prime of degree at most 30 as the smallest factor of a special perfect polynomial, in agreement with the published bounds.

Primarily AI-generated textHuman understanding: no partsCanaday conjecturefinite fieldsperfect polynomialssum of divisors
Nov 03, 2026PDF ↗View work →
hexagon:2611.00025v1

math.NT — Number Theory

The invariant cMc_M of Gallardo and Rahavandrainy at Mersenne prime exponents, and the case p=7p=7

Contributed by David Niedbala Giraudin

A Mersenne prime polynomial is an irreducible polynomial over F2\mathbb{F}_2 of the form xa(x+1)b+1x^a(x+1)^b+1 with a,b≥1a,b\ge1. Gallardo and Rahavandrainy conjectured that, apart from two exceptions, σ(M2h)\sigma(M^{2h}) always has an irreducible factor which is not a Mersenne prime. For a prime p≥5p\ge5 they proved it for every MM outside an exceptional set Σp=Σp1∪⋯∪Σp4\Sigma_p=\Sigma_p^1\cup\dots\cup\Sigma_p^4, defined through an integer cMc_M and an odd integer mMm_M read off the leading coefficients of σ(σ(Mp−1))\sigma(\sigma(M^{p-1})), and later they proved it for every Mersenne prime number p≥31p\ge31; their method does not reach p=7p=7. We compute cMc_M exactly when p=2d−1p=2^d-1 is a Mersenne prime number: cM=d+1c_M=d+1 if deg⁡M≠d\deg M\ne d, and cM=dc_M=d if deg⁡M=d\deg M=d. Consequently Σp1\Sigma_p^1 consists of the Mersenne primes of degree dd or d+1d+1, Σp2\Sigma_p^2 is empty, and Σp3\Sigma_p^3, Σp4\Sigma_p^4 are described by mM=deg⁡M−d−1m_M=\deg M-d-1 and mM=deg⁡M−dm_M=\deg M-d respectively. For p=7p=7 this reduces the conjecture to the Mersenne primes MM with mM∈{deg⁡M−4,deg⁡M−3}m_M\in\{\deg M-4,\deg M-3\}. A direct computation shows that σ(M6)\sigma(M^6) has a non-Mersenne irreducible factor for every Mersenne prime MM of degree at most 500500, and that mM≤17m_M\le17 in that range. Hence the conjecture holds whenever 77 divides 2h+12h+1 and deg⁡M≤500\deg M\le500, and a counterexample with 7∣2h+17\mid 2h+1 would need mM≥deg⁡M−4>496m_M\ge\deg M-4>496. A script attached as an ancillary file checks these computations, and reproduces the table of exceptional sets of Gallardo and Rahavandrainy.

Primarily AI-generated textHuman understanding: no partsCyclotomic PolynomialsMersenne prime polynomialsfinite fieldsperfect polynomialssum of divisors
Nov 03, 2026PDF ↗View work →

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