Research Article
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Year 2020, Volume 28, Issue 28, 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, Volume 28, Issue 28, 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.

Details

Primary Language English
Subjects Mathematics
Journal Section Articles
Authors

Luis H. GALLARDO This is me (Primary Author)
Univ Brest, UMR CNRS 6205 Laboratoire de Mathematiques de Bretagne Atlantique
France

Publication Date July 14, 2020
Published in Issue Year 2020, Volume 28, Issue 28

Cite

Bibtex @research article { ieja768086, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2020}, volume = {28}, number = {28}, pages = {1 - 8}, doi = {10.24330/ieja.768086}, title = {MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS}, key = {cite}, author = {Gallardo, Luis H.} }
APA Gallardo, L. H. (2020). MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS . International Electronic Journal of Algebra , 28 (28) , 1-8 . DOI: 10.24330/ieja.768086
MLA Gallardo, L. H. "MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS" . International Electronic Journal of Algebra 28 (2020 ): 1-8 <https://dergipark.org.tr/en/pub/ieja/issue/55997/768086>
Chicago Gallardo, L. H. "MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS". International Electronic Journal of Algebra 28 (2020 ): 1-8
RIS TY - JOUR T1 - MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS AU - Luis H.Gallardo Y1 - 2020 PY - 2020 N1 - doi: 10.24330/ieja.768086 DO - 10.24330/ieja.768086 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 1 EP - 8 VL - 28 IS - 28 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.768086 UR - https://doi.org/10.24330/ieja.768086 Y2 - 2022 ER -
EndNote %0 International Electronic Journal of Algebra MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS %A Luis H. Gallardo %T MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS %D 2020 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 28 %N 28 %R doi: 10.24330/ieja.768086 %U 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
AMA Gallardo L. H. MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS. IEJA. 2020; 28(28): 1-8.
Vancouver Gallardo L. H. MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS. International Electronic Journal of Algebra. 2020; 28(28): 1-8.
IEEE L. H. Gallardo , "MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS", International Electronic Journal of Algebra, vol. 28, no. 28, pp. 1-8, Jul. 2020, doi:10.24330/ieja.768086