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.<p>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.<p>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:<p>1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37<p>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).<p>This explains why all but finitely many of the squares of primes lie in the two rays corresponding to these residue classes.<p>[1] <a href="https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions" rel="nofollow">https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arith...</a><p>[2] <a href="https://en.wikipedia.org/wiki/Prime_number_theorem#Prime_number_theorem_for_arithmetic_progressions" rel="nofollow">https://en.wikipedia.org/wiki/Prime_number_theorem#Prime_num...</a><p>[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.<p>(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.