| The article essentially looks at the distribution of primes modulo a fixed number m (here m=40). Each "ray" of the diagram corresponds to a residue class modulo 40. As the author observes, if x and m share a common divisor, then there can be at most one prime congruent x modulo m [3]. The interesting classes are those with gcd(x,m) = 1. For those Dirichlet's Theorem on primes in arithmetic progressions [1] tells us that we will in fact find infinitely many primes in these residue classes. More refined, the prime number theorem for arithmetic progressions [2] tells us something about their distribution. As for the squares of primes: If m = 40 then the residue classes modulo m which will contain infinitely many primes can be represented by: 1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37 Therefore the only residue classes in which infinitely many square of primes can lie are the squares of these residue classes, which turn out to be represented by 1 and 9 modulo 40 (just square the previous numbers and take remainders of dividing by 40). This explains why all but finitely many of the squares of primes lie in the two rays corresponding to these residue classes. [1] https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arith... [2] https://en.wikipedia.org/wiki/Prime_number_theorem#Prime_num... [3] Suppose d = gcd(x,m). If y is congruent x mod m, then m divides y-x. Since d therefore divides x and y-x, it divides y. If d is not 1, then y can only be a prime if y=d and d is a prime. (one more edit): We can also say which classes will contain a single prime and which ones will contain a unique prime square: If p is a prime that has a common divisor with m, then p mod m will contain a single prime (p itself). Then p^2 mod m will contain a single prime square (well, p^2).
In the case m=40, we have that 2 and 5 are the prime divisors of 40. Therefore the residue classes represented by 2 and 5 contain a single prime, and those represented by 4 and 25 contain a single prime square. |