ABSTRACT
Prime numbers and divisibility form the basis of very many problems in number theory. Some are as easy as, “What positive integer less than 100 has exactly 7 distinct divisors (including 1 and itself)? Others are more difficult and require some work, but some such as the Goldbach conjecture that every even number greater than 2 is the sum of two prime numbers, though simple to state, and very probably true, have defied the efforts of some of the greatest mathematicians of all time. But who knows, perhaps there is a lateral solution lurking somewhere that some reader of this book may discover!
Many problems on primes and divisibility can double as party tricks to amaze one's friends. An example, using a pocket calculator, is to show that every number of the form abcabc (such as 317317) is exactly divisible by the primes 3, 7 and 11, and when one divides these primes into the original number, one is left with abc! Lateral magic!
Problem 2.4, looking for positive integers n such that n 5 + n + 1 is a prime number, involves the incredibly lateral principle that problems involving real whole numbers are sometimes best solved using complex or so-called imaginary numbers. This is the notion of thinking not just outside the box but in a different box altogether!
