The author discusses some promising new random number generators, as well as formulates the mathematical basis that makes them random variables in the same sense as more familiar ones in probability and statistics, emphasizing his view that randomness exists only in the sense of mathematics. He discusses the need for adequate seeds that provide the axioms for that mathematical basis, and gives examples from Law and Gaming, where inadequacies have led to difficulties. He also describes new versions of the widely used Diehard Battery of Tests of Randomness.Key words: Random number generator, Diehard Test

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