Counting perfect polynomials

Cengiz, Ugur Caner; Pollack, Paul; Treviño, Enrique

doi:10.1016/j.ffa.2017.05.006

Cited by 7 publications

(18 citation statements)

References 7 publications

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“…While for any, say T , of the trivial even perfect binary polynomials, with deg(T ) > 2, T + 1 is always reducible (indeed, it has x 2 + x + 1 as a prime factor). Moreover, from computations in [3] it is known that the conjecture holds when deg(R) ≤ 200.…”

Section: Our Conjecture Ismentioning

confidence: 99%

“…2 n −1 where [6,7,8,9,10], Cengiz et al [3]. On the other hand, the polynomial ABC-theorem, i.e., Mason-Stothers theorem (see [11, pp.…”

Section: Canaday Found the Infinite Family Of Even Perfect Polynomialmentioning

confidence: 99%

“…The list of all known sporadic perfect binary polynomials (see [1,2,3]) is:…”

Section: Toolsmentioning

confidence: 99%

“…He says furthermore, that it seems plausible that no odd perfect polynomial can exist but that this is not proved. No new perfect 2 LUIS H. GALLARDO polynomial has been discovered since besides the efforts of Beard et al [1], Gallardo and Rahavandrainy [6,7,8,9,10], Cengiz et al [3]. On the other hand, the polynomial ABC-theorem, i.e., Mason-Stothers theorem (see [11, pp.…”

Section: Introductionmentioning

confidence: 99%

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Mason-Stothers Theorem and Perfect Binary Polynomials

Gallardo¹

2020

International Electronic Journal of Algebra

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Section: Our Conjecture Ismentioning

confidence: 99%

“…2 n −1 where [6,7,8,9,10], Cengiz et al [3]. On the other hand, the polynomial ABC-theorem, i.e., Mason-Stothers theorem (see [11, pp.…”

Section: Canaday Found the Infinite Family Of Even Perfect Polynomialmentioning

confidence: 99%

“…The list of all known sporadic perfect binary polynomials (see [1,2,3]) is:…”

Section: Toolsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Mason-Stothers Theorem and Perfect Binary Polynomials

Gallardo¹

2020

International Electronic Journal of Algebra

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“…There is no other perfect polynomial A with ω(A) < 5 (see [4,5,6]). Recently, Cengiz et al [3] proved by extensive computations that there is no other perfect polynomial A with deg(A) ≤ 200.…”

Section: Introductionmentioning

confidence: 99%

Admissible family for binary perfect polynomials

Gallardo¹,

Rahavandrainy²

2022

Preprint

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On the index of special perfect polynomials

Gallardo

2023

Carpathian Math. Publ.

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We give a lower bound of the degree and the number of distinct prime divisors of the index of special perfect polynomials. More precisely, we prove that $\omega(d) \geq 9$, and $\deg(d) \geq 258$, where $d := \gcd(Q^2,\sigma(Q^2))$ is the index of the special perfect polynomial $A := p_1^2 Q^2$, in which $p_1$ is irreducible and has minimal degree. This means that $ \sigma(A)=A$ in the polynomial ring ${\mathbb{F}}_2[x]$. The function $\sigma$ is a natural analogue of the usual sums of divisors function over the integers. The index considered is an analogue of the index of an odd perfect number, for which a lower bound of $135$ is known. Our work use elementary properties of the polynomials as well as results of the paper [J. Théor. Nombres Bordeaux 2007, 19 (1), 165$-$174].

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Counting perfect polynomials

Cited by 7 publications

References 7 publications

Mason-Stothers Theorem and Perfect Binary Polynomials

Mason-Stothers Theorem and Perfect Binary Polynomials

Admissible family for binary perfect polynomials

On the index of special perfect polynomials

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