Displaying similar documents to “On a limit point associated with the abc-conjecture”

Power-free values, large deviations, and integer points on irrational curves

Harald A. Helfgott (2007)

Journal de Théorie des Nombres de Bordeaux

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Let f ∈ ℤ [ x ] be a polynomial of degree d ≥ 3 without roots of multiplicity d or ( d - 1 ) . Erdős conjectured that, if f satisfies the necessary local conditions, then f ( p ) is free of ( d - 1 ) th powers for infinitely many primes p . This is proved here for all f with sufficiently high entropy. The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov’s theorem from the theory of large deviations. ...

Variants of the Brocard-Ramanujan equation

Omar Kihel, Florian Luca (2008)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we discuss variations on the Brocard-Ramanujan Diophantine equation.

The divisor problem for binary cubic forms

Tim Browning (2011)

Journal de Théorie des Nombres de Bordeaux

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We investigate the average order of the divisor function at values of binary cubic forms that are reducible over ℚ and discuss some applications.

Some solved and unsolved problems in combinatorial number theory, ii

P. Erdős, A. Sárközy (1993)

Colloquium Mathematicae

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In an earlier paper [9], the authors discussed some solved and unsolved problems in combinatorial number theory. First we will give an update of some of these problems. In the remaining part of this paper we will discuss some further problems of the two authors.

.121221222... is not quadratic.

Florian Luca (2005)

Revista Matemática Complutense

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In this note, we show that if b > 1 is an integer, f(X) ∈ Q[X] is an integer valued quadratic polynomial and K > 0 is any constant, then the b-adic number ∑ (a / b), where a ∈ Z and 1 ≤ |a| ≤ K for all n ≥ 0, is neither rational nor quadratic.

A consequence of an effective form of the abc-conjecture

Jerzy Browkin (1999)

Colloquium Mathematicae

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T. Cochrane and R. E. Dressler [CD] proved that the abc-conjecture implies that, for every > 0, the gap between two consecutive numbers A A 0 . 4 with two exceptions given in Table 2.