Smooth solutions to the $abc$ equation: the $xyz$ Conjecture

Jeffrey C. Lagarias; Kannan Soundararajan

Displaying similar documents to “Smooth solutions to the $a b c$ equation: the $x y z$ Conjecture”

On an arithmetic function considered by Pillai

Florian Luca, Ravindranathan Thangadurai (2009)

Journal de Théorie des Nombres de Bordeaux

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For every positive integer $n$ let $p (n)$ be the largest prime number $p \leq n$ . Given a positive integer $n = n_{1}$ , we study the positive integer $r = R (n)$ such that if we define recursively $n_{i + 1} = n_{i} - p (n_{i})$ for $i \geq 1$ , then $n_{r}$ is a prime or $1$ . We obtain upper bounds for $R (n)$ as well as an estimate for the set of $n$ whose $R (n)$ takes on a fixed value $k$ .

Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves

Joseph H. Silverman (2012)

Journal de Théorie des Nombres de Bordeaux

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A number of authors have proven explicit versions of Lehmer’s conjecture for polynomials whose coefficients are all congruent to $1$ modulo $m$ . We prove a similar result for polynomials $f (X)$ that are divisible in $(ℤ / m ℤ) [X]$ by a polynomial of the form $1 + X + \dots + X^{n}$ for some $n \geq ϵ deg (f)$ . We also formulate and prove an analogous statement for elliptic curves.

An arithmetic function arising from Carmichael’s conjecture

Florian Luca, Paul Pollack (2011)

Journal de Théorie des Nombres de Bordeaux

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Let $φ$ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every $n$ , the equation $φ (n) = φ (m)$ has a solution $m \neq n$ . This suggests defining $F (n)$ as the number of solutions $m$ to the equation $φ (n) = φ (m)$ . (So Carmichael’s conjecture asserts that $F (n) \geq 2$ always.) Results on $F$ are scattered throughout the literature. For example, Sierpiński conjectured, and Ford proved, that the range of $F$ contains every natural number $k \geq 2$ . Also, the maximal order of $F$ has been investigated by Erdős and Pomerance....

On the ternary Goldbach problem with primes in arithmetic progressions having a common modulus

Karin Halupczok (2009)

Journal de Théorie des Nombres de Bordeaux

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For $ε > 0$ and any sufficiently large odd $n$ we show that for almost all $k \leq R : = n^{1 / 5 - ε}$ there exists a representation $n = p_{1} + p_{2} + p_{3}$ with primes $p_{i} \equiv b_{i}$ mod $k$ for almost all admissible triplets $b_{1}, b_{2}, b_{3}$ of reduced residues mod $k$ .

Oscillation of Mertens’ product formula

Harold G. Diamond, Janos Pintz (2009)

Journal de Théorie des Nombres de Bordeaux

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Mertens’ product formula asserts that $\prod_{p \leq x} (1 - \frac{1}{p}) log x \to e^{- γ}$ as $x \to \infty$ . Calculation shows that the right side of the formula exceeds the left side for $2 \leq x \leq 10^{8}$ . It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $π (x) - li x$ , this and a complementary inequality might change their sense for sufficiently large values of $x$ . We show this to be the case.

Displaying similar documents to “Smooth solutions to the a b c equation: the x y z Conjecture”

On an arithmetic function considered by Pillai

Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves

An arithmetic function arising from Carmichael’s conjecture

On the ternary Goldbach problem with primes in arithmetic progressions having a common modulus

Oscillation of Mertens’ product formula

Displaying similar documents to “Smooth solutions to the $a b c$ equation: the $x y z$ Conjecture”