MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS

Luis H. Gallardo

doi:10.24330/ieja.768086

Research Article

Year 2020, Volume: 28 Issue: 28, 1 - 8, 14.07.2020

Luis H. Gallardo

https://doi.org/10.24330/ieja.768086

Abstract

References

J. T. B. Beard, Jr., J. R. O'Connell, Jr. and K. I. West, Perfect polynomials over GF(q), Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 62(8) (1977), 283-291.
E. F. Canaday, The sum of the divisors of a polynomial, Duke Math. J., 8 (1941), 721-737.
U. C. Cengiz, P. Pollack and E. Trevino, Counting perfect polynomials, Finite Fields Appl., 47 (2017), 242-255.
L. H. Gallardo, Question: Even perfect numbers n with n + 1 prime, https://mathoverow.net/questions/62797/even-perfect-numbers-n-with-n1-prime.
L. H. Gallardo, Sequence A189373 in The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2017.
L. H. Gallardo and O. Rahavandrainy, Odd perfect polynomials over F2, J. Theor. Nombres Bordeaux, 19 (2007), 165-174.
L. H. Gallardo and O. Rahavandrainy, Even perfect polynomials over F2 with four prime factors, Int. J. Pure Appl. Math., 52(2) (2009), 301-314.
L. H. Gallardo and O. Rahavandrainy, There is no odd perfect polynomial over F2 with four prime factors, Port. Math., 66(2) (2009), 131-145.
L. H. Gallardo and O. Rahavandrainy, Characterization of sporadic perfect polynomials over F2, Funct. Approx. Comment. Math., 55(1) (2016), 7-21.
L. H. Gallardo, P. Pollack and O. Rahavandrainy, On a conjecture of Beard, O'Connell and West concerning perfect polynomials, Finite Fields Appl., 14(1) (2008), 242-249.
S. Lang, Algebra, 2nd ed., Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984.

MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS

Year 2020, Volume: 28 Issue: 28, 1 - 8, 14.07.2020

Luis H. Gallardo

https://doi.org/10.24330/ieja.768086

Abstract

We prove that there is no perfect binary polynomial $R$ that is the sum of two appropriate powers, besides, possibly $R=P+1$ with $P$ irreducible. The proofs follow from analogue results involving the ABC-theorem for polynomials and a classical identity. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

Keywords

Polynomial ABC-theorem, binary polynomial, perfect polynomial

References

J. T. B. Beard, Jr., J. R. O'Connell, Jr. and K. I. West, Perfect polynomials over GF(q), Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 62(8) (1977), 283-291.
E. F. Canaday, The sum of the divisors of a polynomial, Duke Math. J., 8 (1941), 721-737.
U. C. Cengiz, P. Pollack and E. Trevino, Counting perfect polynomials, Finite Fields Appl., 47 (2017), 242-255.
L. H. Gallardo, Question: Even perfect numbers n with n + 1 prime, https://mathoverow.net/questions/62797/even-perfect-numbers-n-with-n1-prime.
L. H. Gallardo, Sequence A189373 in The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2017.
L. H. Gallardo and O. Rahavandrainy, Odd perfect polynomials over F2, J. Theor. Nombres Bordeaux, 19 (2007), 165-174.
L. H. Gallardo and O. Rahavandrainy, Even perfect polynomials over F2 with four prime factors, Int. J. Pure Appl. Math., 52(2) (2009), 301-314.
L. H. Gallardo and O. Rahavandrainy, There is no odd perfect polynomial over F2 with four prime factors, Port. Math., 66(2) (2009), 131-145.
L. H. Gallardo and O. Rahavandrainy, Characterization of sporadic perfect polynomials over F2, Funct. Approx. Comment. Math., 55(1) (2016), 7-21.
L. H. Gallardo, P. Pollack and O. Rahavandrainy, On a conjecture of Beard, O'Connell and West concerning perfect polynomials, Finite Fields Appl., 14(1) (2008), 242-249.
S. Lang, Algebra, 2nd ed., Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984.

There are 11 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Articles
Authors	Luis H. Gallardo This is me
Publication Date	July 14, 2020
Published in Issue	Year 2020 Volume: 28 Issue: 28

Cite

APA	Gallardo, L. H. (2020). MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS. International Electronic Journal of Algebra, 28(28), 1-8. https://doi.org/10.24330/ieja.768086
AMA	Gallardo LH. MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS. IEJA. July 2020;28(28):1-8. doi:10.24330/ieja.768086
Chicago	Gallardo, Luis H. “MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS”. International Electronic Journal of Algebra 28, no. 28 (July 2020): 1-8. https://doi.org/10.24330/ieja.768086.
EndNote	Gallardo LH (July 1, 2020) MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS. International Electronic Journal of Algebra 28 28 1–8.
IEEE	L. H. Gallardo, “MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS”, IEJA, vol. 28, no. 28, pp. 1–8, 2020, doi: 10.24330/ieja.768086.
ISNAD	Gallardo, Luis H. “MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS”. International Electronic Journal of Algebra 28/28 (July 2020), 1-8. https://doi.org/10.24330/ieja.768086.
JAMA	Gallardo LH. MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS. IEJA. 2020;28:1–8.
MLA	Gallardo, Luis H. “MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS”. International Electronic Journal of Algebra, vol. 28, no. 28, 2020, pp. 1-8, doi:10.24330/ieja.768086.
Vancouver	Gallardo LH. MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS. IEJA. 2020;28(28):1-8.