Research and Advances

Some negative results concerning prime number generators

Programs attributed to Wirth and Misra for generating the prime numbers up to a specified limit are investigated. It is shown that Wirth's program is incorrect according to three increasingly weak criteria, and a composite number is exhibited that the program accepts as prime. This is the smallest known counterexample, and could not have been found by the usual method of program testing—the program would run for trillions of years on the fastest computer before reaching it! Closely related counterexamples are given to a conjecture of Misra concerning his program. An appendix gives a particularly simple algorithmic proof of the Chinese remainder theorem.


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