Follow by Email

Thursday, June 14, 2012

Transversal of primes


Choose a prime number p, draw a p×p array, and fill it with integers like so:

Now: Can we always find p cells that contain prime numbers such that no two occupy the same row or column? (This is somewhat like arranging rooks on a chessboard so that every rank and file is occupied but no rook attacks another.)

The example below shows one solution for p=11. Does a solution exist for every prime number? No one knows.

Wednesday, June 6, 2012

A problem from Litton Mathematical Recreations, which attributes it to Fermat circa 1635:



What is the remainder upon dividing 5999,999 by 7?

When successive powers of 5 are divided by 7, the remainders form a repeating series:
51 / 7 = 0 remainder 5
52 / 7 = 3 remainder 4
53 / 7 = 17 remainder 6
54 / 7 = 89 remainder 2
55 / 7 = 446 remainder 3
56 / 7 = 2232 remainder 1
57 / 7 = 11160 remainder 5
58 / 7 = 55803 remainder 4
59 / 7 = 279017 remainder 6
510 / 7 = 1395089 remainder 2

The 999,999th term of the series is 6.