13th root, 200-digit number, 77 seconds
From The Observer, a profile of Alexis Lemaire, possibly the world’s greatest mathlete: “On 24 July 2007, in controlled conditions at Oxford’s Museum of the History of Science, Lemaire broke the world record for finding the 13th root of a 200-digit number. This involved being presented with an enormously long random number (which, if reproduced here, would take up six lines of text) and working out in the space of 77.99 seconds that its 13th root was 2396232838850303.”
There’s a lovely piece in one of Oliver Sacks’ earlier books The Man Who Mistook His Wife For A Hat, about two brothers whose IQ scores were in the idiot range, who sat around reciting 7-digit numbers to each other. Dr. Sacks realized that the numbers they were tossing around were prime numbers.