41

This number is a prime.

Consider a "prime race" based on the values of p mod 100 for each prime p. Over all primes there are 42 possible values of p mod 100 (1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, or 99), but only 40 of these occur in the long term (since 2 and 5 are special cases). In this prime race we scan through the primes from the beginning, and keep track of how many primes there are having each of these 40 possible values for p mod 100, and we ask: when does each of the 40 racers "take the lead" (be the only one to have the largest count) for the first time? The first 39 first-time leaders that occur are, in order, 3, 7, 31, 57, 9, 83, 67, 19, 51, 23, 39, 37, 21, 87, 59, 73, 93, 97, 71, 33, 27, 53, 1, 91, 63, 17, 11, 69, 79, 49, 77, 47, 13, 81, 89, 61, 99, 29, 43 and they take the lead for the first time at primes 3, 307, 431, 857, 1109, 1583, 3967, 5519, 7451, 8423, 12739, 13337, 13921, 16087, 23059, 25873, 60793, 63997, 171671, 253433, 340127, 457553, 580201, 977791, 1267663, 1329217, 2059711, 2607469, 3032279, 6253549, 11761777, 34929547, 45740213, 51656281, 213350989, 227710961, 316867699, 1033090529, 11342219743, respectively. Every racer has been in the lead except for "Car #41" (i.e., p mod 100 = 41). At what prime does Car 41 finally takes the lead? The answer is known to be greater than 5 x 10¹³. [Keith]