Addendum and corrigenda to the paper “Infinitary superperfect numbers”

Yamada, Tomohiro

doi:10.33039/ami.2018.06.001

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2023

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Cited by 2 publications

(2 citation statements)

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“…The arithmetic function σ * * (n) denotes the sum of positive bi-unitary divisors of the integer n. Wall [3] proved that there are only three bi-unitary perfect numbers (σ * * (n) = 2n), namely, 6, 60, and 90. Yamada [4] proved that 2 and 9 are the only bi-unitary superperfect numbers, that is, σ * * 2 (n) = 2n if and only if n ∈ {2, 9}.…”

Section: Introductionmentioning

confidence: 99%

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Bi-Unitary Superperfect Polynomials over 𝔽2 with at Most Two Irreducible Factors

Chehade,

Miari,

Alkhezi

2023

Symmetry

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A divisor B of a nonzero polynomial A, defined over the prime field of two elements, is unitary (resp. bi-unitary) if gcd(B,A/B)=1 (resp. gcdu(B,A/B)=1), where gcdu(B,A/B) denotes the greatest common unitary divisor of B and A/B. We denote by σ**(A) the sum of all bi-unitary monic divisors of A. A polynomial A is called a bi-unitary superperfect polynomial over F2 if the sum of all bi-unitary monic divisors of σ**(A) equals A. In this paper, we give all bi-unitary superperfect polynomials divisible by one or two irreducible polynomials over F2. We prove the nonexistence of odd bi-unitary superperfect polynomials over F2.

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Section: Introductionmentioning

confidence: 99%

“…A polynomial M of the form 1 + x a (x + 1) b is called Mersenne. The first five Mersenne polynomials over F 2 are 4 , and M 5 = 1 + x 3 + x 4 . Note that all these polynomials are irreducible, so we call them Mersenne primes.…”

Section: Introductionmentioning

confidence: 99%