Goldbach Variations: Problems with Prime Numbers

Praeludium We will call prime numbers the integers n ≥ 2 which are divisible only by 1 and themselves. Euclid (fourth century B. C.) first showed that there exist infinitely many prime numbers. His proof is an excellent example of a mathematical argument: if 2, 3, 5, . . . , p were the only prime numbers, we could construct the number 2 · 3 · 5 · · · p + 1, which has the property of leaving the remainder 1 when divided by any of our former prime numbers. Since every integer larger than 1 is either a prime number, or is divisible by at least a prime, our list can not be complete. According to the great British mathematician G. H. Hardy, this proof “is as fresh and significant as when it was discovered–two thousand years have not written a wrinkle on it.” The following is a short list of prime numbers.