Displaying similar documents to “Odd perfect polynomials over 𝔽 2 ”

On three questions concerning 0 , 1 -polynomials

Michael Filaseta, Carrie Finch, Charles Nicol (2006)

Journal de Théorie des Nombres de Bordeaux

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We answer three reducibility (or irreducibility) questions for 0 , 1 -polynomials, those polynomials which have every coefficient either 0 or 1 . The first concerns whether a naturally occurring sequence of reducible polynomials is finite. The second is whether every nonempty finite subset of an infinite set of positive integers can be the set of positive exponents of a reducible 0 , 1 -polynomial. The third is the analogous question for exponents of irreducible 0 , 1 -polynomials.

The Perfect Number Theorem and Wilson's Theorem

Marco Riccardi (2009)

Formalized Mathematics

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This article formalizes proofs of some elementary theorems of number theory (see [1, 26]): Wilson's theorem (that n is prime iff n > 1 and (n - 1)! ≅ -1 (mod n)), that all primes (1 mod 4) equal the sum of two squares, and two basic theorems of Euclid and Euler about perfect numbers. The article also formally defines Euler's sum of divisors function Φ, proves that Φ is multiplicative and that Σk|n Φ(k) = n.

Counting monic irreducible polynomials P in 𝔽 q [ X ] for which order of X ( mod P ) is odd

Christian Ballot (2007)

Journal de Théorie des Nombres de Bordeaux

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Hasse showed the existence and computed the Dirichlet density of the set of primes p for which the order of 2 ( mod p ) is odd; it is 7 / 24 . Here we mimic successfully Hasse’s method to compute the density δ q of monic irreducibles P in 𝔽 q [ X ] for which the order of X ( mod P ) is odd. But on the way, we are also led to a new and elementary proof of these densities. More observations are made, and averages are considered, in particular, an average of the δ p ’s as p varies through all rational primes.