MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS

Year 2020, , 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.