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- Chris K. Caldwell, Yuanyou Cheng
- 2005

In 1947 Mills proved that there exists a constant A such that A 3 n 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

For each prime p let p# be the product of the primes less than or equal to p. Using a new type of microcomputer coprocessor, we have found five new primes of the form n\-1 , two new primes of the form p# + 1 , seven new primes of the form p#-1 , and greatly extended the search bounds for primes of the form n\ ± 1 and p# ± 1. For each prime p let p# be the… (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)

- 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… (More)

The way mathematicians have viewed the number one (unity, the monad) has changed throughout the years. Most of the early Greeks did not view one as a number, but rather as the origin, or generator, of number. Around the time of Simon Stevin (1548–1620), one (and zero) were first widely viewed as numbers. This created a period of confusion about whether or… (More)

The first author is a professor of mathematics at UT Martin. He lives on a small " farm " in rural northwest Tennessee with his wife, five children, two cats, and numerous chickens. The second author is a schoolteacher and amateur number theorist. He is an avid chess player. The prime counting function, π(x), counts exactly how many primes there are less… (More)

- DAN KRYWARUCZENKO, Chris K. Caldwell
- 2008

A generalized Sierpi´nski number base b is an integer k > 1 for which gcd(k +1, b−1) = 1, k is not a rational power of b, and k ·b n +1 is composite for all n > 0. Given an integer k > 0, we will seek a base b for which k is a generalized Sierpi´nski number base b. We will show that this is not possible if k is a Mersenne number. We will give an algorithm… (More)

Have you ever seen the great stone pyramids of ancient Egypt or Central America? For over 5000 years, mankind has been building, visiting, and even sleeping in pyramids. When Memphis, Tennessee, decided to build a new arena in 1991, they chose the shape of a pyramid— this time constructed of steel and glass rather than rock and rubble. In this paper we also… (More)

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

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