We now can prove that the primes are infinite in number, and we have no doubt sensed the feeling that their distribution is uneven (since so many appear in the first 10 natural numbers, and then start to get annihilated in the spiraling patterns of the sieve of Eratosthenes). But what can we say about the distribution of primes? Are there interesting patterns? This section seeks to sum up some of what we know.
Various famous interesting conundrums, mysteries, conjectures, etc. are discussed, including
- The Goldbach Conjecture (that every even n > 2 is the sum of two primes);
- Twin Primes, and other gaps between primes;
- Dirichlet's theorem about primes of the form a+kb, with gcd(a,b)=1; and
- Primes of various forms given by the division algorithm.
Twin primes: prime pairs of the form p, p+2.
Euler polynomial: 
(which produces primes for the
integers from 0 to 39). (Euler showed that it did not generate primes for all
n, which others apparently believed.)
Examples: other gaps in primes:
- Exercise #3, p. 59 - one minute!
- Exercise #9a, p. 59 - Use 6n+k
- Exercise #9b, p. 59 - prime-triplets - use the sieve, or Mathematica!
- Arbitrary long, primeless gaps.
Goldbach Conjecture (unproven yet heavily favored: hence, still a conjecture): Every even n > 2 is the sum of two primes.
Note: it's interesting that it's been shown (Vinogradov) that
where A(x) is the number of evens less than or equal to x and not expressible as the sum of two primes. (A(x) may be zero for all x!) This means that ``almost all'' integers satisfy the conjecture. What does that mean?! (``The Goldbach conjecture is false for at most 0% of all even integers; this at most 0% does not exclude, of course, the possibility that there are infinitely many exceptions.'' George Landau)
Twin Prime Conjecture: There are infinitely many twin primes.
Lemma: The product of two or more integers of the form 4n+1 is of the same form.
Theorem 3.6: There are infinitely many primes of the form 4n+3.
Example: Exercise #13, p. 60 asks us to show the same thing for integers of the form 6n+5.
Theorem 3.7 (Dirichlet): If a and b are relatively prime positive integers, then the arithmetic progression
contains infinitely many primes.
Theorem 3.8: If all the n>2 terms of the arithmetic progression
are prime numbers, then the common difference d is divisible by every prime q<n.