Fixed points of the sum of divisors function on $F_2[x]$

Abstract

We work an analogue of a classical arithmetic problem over polynomials. More precisely, we study the fixed points F of the sum of divisors function σ : F 2 [x] → F 2 [x] (defined mutatis mutandi like the usual sum of divisors over the integers) of the form F := A 2 • S, S square-free, with ω(S) ≤ 3, coprime with A, for A even, of whatever degree, under some conditions. This gives a characterization of 5 of the 11 known fixed points of σ in F 2 [x]. so that σ(120) = 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120 = 360. Already here we see that we can compute σ(120) more efficiently as follows: Since 120 = 2 3 • 3 • 5 and σ(x • y) = σ(x) • σ(y) provided that x, y has no common factors, we can compute:In a nutshell, in the present paper we study some arithmetic properties of an analogue to the function n → σ(n), in which we replace n by a polynomial A(x) with coefficients 0 and 1 only, and compute with 0, 1 as usual, besides the rule 1 + 1 = 0 that replaces the usual rule 1 + 1 = 2. The field F 2 = {0, 1} in which we compute the coefficients of A(x) is the simplest of all finite fields.For readers less familiar with finite fields, we recommend to look first at section 2 for a simple computation with binary polynomials. Then, to look at subsections 1.1, and 1.2 below. And, finally, come back to look at the rest of this Introduction.For all readers, we added some information about our choice of the finite field F 2 for the coefficients of our polynomials (see subsections 1.1, and 1.2) at the end of this Introduction. We also added a few comments about the role played by some small degree irreducible binary polynomials as prime factors of our perfect polynomials. This comes from an observation of one of the referees.The paper being a little technical, we hope the following considerations will be helpful for the reader.We now introduce some definitions and notation to explain the original arithmetic problem over the integers that motivated the study of our variant over the binary polynomials in F 2 [x], and the link between them as well.Let A ∈ F 2 [x] be an irreducible polynomial, then we say that A is prime. A polynomial M ∈ F 2 [x] is Mersenne (an analogue of a Mersenne number: 2 n − 1) if M + 1 is a product of powers of x and powers of x + 1. We say that M + 1 splits. When a Mersenne polynomial M is irreducible, we say that M is a Mersenne prime. Given a binary polynomial B, a binary polynomial A in the sub-ring, if all coefficients of A are equal to 1; when B = x, we say simply that A is complete. A binary polynomial B is odd if B(0) = B(1) = 1, otherwise B is even. More standard notation follows. We let ω(P ) denote the number of pairwise distinct prime factors of P ∈ F q [x]. Likewise, we let v P (A) denote the valuation of the prime P in the binary polynomial A, i.e., the least positive integer m, such that P m | A but P m+1 ∤ A, we also write this as P m ||A. Finally, we let F 2 denote a fixed algebraic closure of F 2 .We recall that a binary perfect polynomial A (see [11,14,16,19,26,29,31,32,33]) is defined by the ...