4. Congruences, clocks, and calendars

‘Congruences, clocks, and calendars’ demonstrates how we might apply the idea of congruence, first introduced by Gauss in 1801, to problems such as testing which Mersenne numbers are primes and finding the day of the week on which a given date falls. Ancient Chinese puzzles depended on the solving of simultaneous linear congruences, inspiring mathematicians and giving rise to the Chinese Remainder Theorem. Exploring quadratic congruences leads towards the law of quadratic reciprocity, noted by Euler and Legendre and proved by Gauss. The problem, ‘Is 1066 a square or a non-square?’ can be solved by applying this law several times to reduce the numbers involved.