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- Chris K. Caldwell, Yuanyou Cheng
- 2005

In 1947 Mills proved that there exists a constant A such that bA3nc is a prime for every positive integer n. Determining A requires determining an effective Hoheisel type result on the primes in short intervals—though most books ignore this difficulty. Under the Riemann Hypothesis, we show that there exists at least one prime between every pair of… (More)

- CHRIS K. CALDWELL, David Underbakke, Giovanni La Barbera, Sophie Germain
- 2003

2 dx (log x)2 ∼ 2C2N (log N)2 where C2, called the twin prime constant, is approximately 0.6601618. Using this we can estimate how many numbers we will need to try before we find a prime. In the case of Underbakke and La Barbera, they were both using the same sieving software (NewPGen by Paul Jobling) and the same primality proving software (Proth.exe by… (More)

- Chris K. Caldwell, Yves Gallot
- Math. Comput.
- 2002

- CHRIS K. CALDWELL, YVES GALLOT, John Schommer
- 2001

For each prime p, let p# be the product of the primes less than or equal to p. We have greatly extended the range for which the primality of n!± 1 and p#± 1 are known and have found two new primes of the first form (6380! + 1, 6917!− 1) and one of the second (42209# + 1). We supply heuristic estimates on the expected number of such primes and compare these… (More)

absolute prime: a prime that remains prime for all permutations of its digits. E.g., 199, 919 and 991 are all prime. Goal: definitions balanced with curio info. almost-all-even-digits prime: a prime with all even digits except for one odd right-most digit. E.g., 666 · 10 + 1. almost-equi-pandigital prime: a prime with all digits equal in number, except for… (More)

- AMY BRUNNER, CHRIS K. CALDWELL, DANIEL KRYWARUCZENKO, CHRIS LOWNSDALE
- 2008

Sierpiński proved that there are infinitely many odd integers k such that k ·2n+1 is composite for all n ≥ 0. These k are now called Sierpiński numbers. We define a Sierpiński number base b to be an integer k > 1 for which gcd(k+1, b−1) = 1, k is not a rational power of b, and k · bn+1 is composite for all n > 0. We discuss ways that these can arise, offer… (More)

What is the first prime? It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers. To find the first prime, we must also know what the first positive integer is. Surprisingly, with the definitions used at various… (More)

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