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.

[1]  Richard P. Brent,et al.  The Distribution of Small Gaps Between Successive Primes , 1974 .

[2]  Twenty-two primes in arithmetic progression , 1995 .

[3]  Michael Schroeder.  Number theory in science and communication : With applications in cryptog-raphy , 1997 .

[4]  Eric Bach,et al.  The Complexity of Number-Theoretic Constants , 1997, Inf. Process. Lett..

[5]  Daniel Shanks,et al.  On the conjecture of Hardy & Littlewood concerning the number of primes of the form ²+ , 1960 .

[6]  E. Grosswald.  Arithmetic progressions that consist only of primes , 1982 .

[7]  M. R. Schroeder,et al.  Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity (Springer Series in Information Sciences) , 1986 .

[8]  H. Iwaniec.  Almost-primes represented by quadratic polynomials , 1978 .

[9]  Donald E. Knuth,et al.  The Art of Computer Programming, Volume I: Fundamental Algorithms, 2nd Edition , 1997 .

[10]  Where is the next mersenne prime hiding? , 1983 .

[11]  Pieter Moree,et al.  Approximation of singular series and automata , 2000 .

[12]  Wieb Bosma,et al.  Algorithmic Number Theory , 2000, Lecture Notes in Computer Science.

[13]  R. Horn,et al.  A heuristic asymptotic formula concerning the distribution of prime numbers , 1962 .

[14]  Günter Löh.  Long chains of nearly doubled primes , 1989 .

[15]  Anders Björn,et al.  Factors of generalized Fermat numbers , 1998, Math. Comput..

[16]  M. D. MacLaren.  The Art of Computer Programming—Volume 1: Fundamental Algorithms (Donald E. Knuth) , 1969 .

[17]  Jonathan M. Borwein,et al.  Computational strategies for the Riemann zeta function , 2000 .

[18]  G. Pólya.  Heuristic Reasoning in the Theory of Numbers , 1959 .

[19]  J. Littlewood,et al.  Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes , 1923 .

[20]  Harvey Dubner,et al.  Distribution of generalized Fermat prime numbers , 2002, Math. Comput..

[21]  A. Schinzel,et al.  Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers" , 1961 .

[22]  J. E. Littlewood,et al.  Some problems of diophantine approximation , 1914 .

[23]  H. Riesel.  Prime numbers and computer methods for factorization , 1985 .

[24]  J. Wrench,et al.  Evaluation of Artin's Constant and the Twin-Prime Constant , 1961 .

[25]  Roger A. Horn,et al.  Primes represented by irreducible polynomials in one variable , 1965 .

[26]  Pierre Dusart,et al.  The kth prime is greater than k(ln k + ln ln k - 1) for k >= 2 , 1999, Math. Comput..

[27]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[28]  B. Mayoh.  The second goldbach conjecture revisited , 1968 .

[29]  A. Harles.  Sieve Methods , 2001 .

[30]  S. Wagstaff.  Divisors of Mersenne numbers , 1983 .

[31]  Jeffrey Shallit,et al.  Algorithmic Number Theory , 1996, Lecture Notes in Computer Science.

[32]  S. Kravitz,et al.  On the distribution of Mersenne divisors , 1967 .

[33]  Samuel Yates.  SOPHIE GERMAIN PRIMES , 1991 .

[34]  Ian Richards.  On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem , 1974 .

[35]  Emil Grosswald,et al.  Arithmetic progressions consisting only of primes , 1979 .

[36]  E. Wright,et al.  THE FREQUENCY OF PRIME-PATTERNS , 1960 .

[37]  Karl Dilcher,et al.  A search for Wieferich and Wilson primes , 1997, Math. Comput..

[38]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[39]  D. Shanks.  Solved and Unsolved Problems in Number Theory , 1964 .

[40]  P. Ribenboim.  The new book of prime number records , 1996 .

[41]  Lord Cherwell.  NOTE ON THE DISTRIBUTION OF THE INTERVALS BETWEEN PRIME NUMBERS , 1946 .

[42]  P. Stäckel.  Die Darstellung der geraden Zahlen als Summen von zwei Primzahlen , 1916 .