Displaying similar documents to “Infinite families of noncototients”

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.

Restriction theory of the Selberg sieve, with applications

Ben Green, Terence Tao (2006)

Journal de Théorie des Nombres de Bordeaux

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The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L 2 – L p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k -tuples. Let a 1 , ⋯ , a k and b 1 , ⋯ , b k be positive integers. Write h ( θ ) : = ∑ n ∈ X e ( n θ ) , where X is the set of all n ≤ N such that the numbers a 1 n + b 1 , ⋯ , a k n + b k are all prime. We obtain upper bounds for ∥ h ∥ L p ( 𝕋 ) , p > 2 , which are (conditionally on the Hardy-Littlewood prime tuple conjecture) of the correct...

A note on a conjecture of Jeśmanowicz

Moujie Deng, G. Cohen (2000)

Colloquium Mathematicae

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Let a, b, c be relatively prime positive integers such that a 2 + b 2 = c 2 . Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of ( a n ) x + ( b n ) y = ( c n ) z in positive integers is x=y=z=2. If n=1, then, equivalently, the equation ( u 2 - v 2 ) x + ( 2 u v ) y = ( u 2 + v 2 ) z , for integers u>v>0, has only the solution x=y=z=2. We prove that this is the case when one of u, v has no prime factor of the form 4l+1 and certain congruence and inequality conditions on u, v are satisfied.

Powers and alternative laws

Nicholas Ormes, Petr Vojtěchovský (2007)

Commentationes Mathematicae Universitatis Carolinae

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A groupoid is alternative if it satisfies the alternative laws x ( x y ) = ( x x ) y and x ( y y ) = ( x y ) y . These laws induce four partial maps on ℕ + × ℕ + ( r , s ) ↦ ( 2 r , s - r ) , ( r - s , 2 s ) , ( r / 2 , s + r / 2 ) , ( r + s / 2 , s / 2 ) , that taken together form a dynamical system. We describe the orbits of this dynamical system, which allows us to show that n th powers in a free alternative groupoid on one generator are well-defined if and only if n ≤ 5 . We then discuss some number theoretical properties of the orbits, and the existence of alternative loops without two-sided inverses. ...

Wilson’s theorem

Chandan Singh Dalawat (2009)

Journal de Théorie des Nombres de Bordeaux

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We show how K. Hensel could have extended Wilson’s theorem from Z to the ring of integers 𝔬 in a number field, to find the product of all invertible elements of a finite quotient of 𝔬 .