AN AMAZING PRIME HEURISTIC

2 dx (log x)2 ∼ 2C2N (log N)2 where C2, called the twin prime constant, is approximately 0.6601618. Using this we can estimate how many numbers we will need to try before we find a prime. In the case of Underbakke and La Barbera, they were both using the same sieving software (NewPGen by Paul Jobling) and the same primality proving software (Proth.exe by Yves Gallot) on similar hardware–so of course they choose similar ranges to search. But where does this conjecture come from? In this chapter we will discuss a general method to form conjectures similar to the twin prime conjecture above. We will then apply it to a number of different forms of primes such as Sophie Germain primes, primes in arithmetic progressions, primorial primes and even the Goldbach conjecture. In each case we will compute the relevant constants (e.g., the twin prime constant), then compare the conjectures to the results of computer searches. A few of these results are new–but our main goal is to illustrate an important technique in heuristic prime number theory and apply it in a consistent way to a wide variety of problems.