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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)
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)
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)
This brief glossary is written to explain some of the definitions we use at the Prime Curios! website: http://primes.utm.edu/curios/. A term used only once or twice can be defined in the curio itself, so those we have here are for terms that are used repeatedly. Please let us know of any errors or missing terms.