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Year 2020, , 1 - 8, 14.07.2020
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, , 1 - 8, 14.07.2020
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. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

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

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.