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**11 - 15**of**15**### EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS

, 2013

"... Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments invol ..."

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Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant. Contents

### A NEW ALGORITHM TO SEARCH FOR SMALL NONZERO |x 3 − y 2 | VALUES

"... Abstract. In relation to Hall’s conjecture, a new algorithm is presented to search for small nonzero k = |x 3 −y 2 | values. Seventeen new values of k<x 1/2 are reported. 1. Hall’s conjecture Dealing with natural numbers, the difference (1.1) k = x 3 − y 2 is zero when x = t 2 and y = t 3 but, in ..."

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Abstract. In relation to Hall’s conjecture, a new algorithm is presented to search for small nonzero k = |x 3 −y 2 | values. Seventeen new values of k<x 1/2 are reported. 1. Hall’s conjecture Dealing with natural numbers, the difference (1.1) k = x 3 − y 2 is zero when x = t 2 and y = t 3 but, in other cases, it seems difficult to achieve small absolute values. For a given k ̸ = 0, (1.1), known as Mordell’s equation, is an elliptic curve and has only finitely many solutions in integers by Siegel’s theorem. Therefore, for any nonzero k value, there are only finitely many solutions in x (which is hence bounded). There is a proven lower bound, due to A. Baker [1] and improved by H. M. Stark [14], that places the size of k above the order of log c (x) for any c<1. A bound concerning the minimal growth rate of |k | was found early by M. Hall [2, 7] by means of a parametric family of the form (1.2) f(t) = t 9 (t9 +6t 6 +15t 3 + 12), g(t) = t15 27 + t12 +4t9 +8t6 3 f 3 (t) − g2 (t) = − 3t6 +14t3+27

### Some Number-theoretic Conjectures and Their Relation to the Generation of Cryptographic Primes

, 1992

"... . The purpose of this paper is to justify the claim that a method for generating primes presented at EUROCRYPT'89 generates primes with virtually uniform distribution. Using convincing heuristic arguments, the conditional probability distributions of the size of the largest prime factor p 1 (n) ..."

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. The purpose of this paper is to justify the claim that a method for generating primes presented at EUROCRYPT'89 generates primes with virtually uniform distribution. Using convincing heuristic arguments, the conditional probability distributions of the size of the largest prime factor p 1 (n) of a number n on the order of N is derived, given that n satisfies one of the conditions 2n+1 is prime, 2an+1 is prime for a given a, or the d integers u 1 ; : : : ; u d , where u 1 = 2a 1 n + 1 and u t = 2a t u t\Gamma1 + 1 for 2 t d, are all primes for a given list of integers a 1 ; : : : ; a d . In particular, the conditional probabilities that n is itself a prime, or is of the form "k times a prime" for k = 2; 3; : : : ; is treated for the above conditions. It is shown that although for all k these probabilities strongly depend on the condition placed on n, the probability distribution of the relative size oe 1 (n) = log N p 1 (n) of the largest prime factor of n is virtually independent...

### Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 ARITHMETIC PROPERTIES OF POLYNOMIAL SPECIALIZATIONS OVER FINITE FIELDS

"... We present applications of some recent results that establish a partial finite field analogue of Schinzel’s Hypothesis H. For example, we prove that the distribution of gaps between degree n prime polynomials over Fp is close to Poisson for p large compared to n. We also estimate the number of polyn ..."

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We present applications of some recent results that establish a partial finite field analogue of Schinzel’s Hypothesis H. For example, we prove that the distribution of gaps between degree n prime polynomials over Fp is close to Poisson for p large compared to n. We also estimate the number of polynomial substitutions without prime factors of large degree (“smooth ” polynomial substitutions); this confirms a finite field analogue of a conjecture of Martin in certain ranges of the parameters. Other topics considered include an analogue of Brun’s constant for polynomials and “smooth ” values of neighboring polynomials. 1.

### Factorizations

, 1997

"... Many combinatorial structures decompose into components, with the list of component sizes car-rying substantial information. An integer factors into primes—this is a similar situation, but different ..."

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Many combinatorial structures decompose into components, with the list of component sizes car-rying substantial information. An integer factors into primes—this is a similar situation, but different