[seqfan] Enumerating the rational numbers, supersingular primes and other monsters.

[peter.luschny]

Enumerating the rational numbers, supersingular primes and other monsters.

(a) Looking at the Calkin–Wilf tree (breadth-first traversal)
1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, ...

(b) Assuming the Schinzel-Sierpinski conjecture:
Every positive rational number can be represented as
a/b = (p+1)/(q+1) for primes p and q.

(c) Choosing such a pair by some minimality criterion,
plus prime-encoding gives

SchinzelSierpinski := proc(a,b) local p,q,r,P;
P := select(isprime,[$1..4000]):
for p in P  do for q in P  do
    if (p+1)/(q+1) = a/b then r := `if`(a<=b,p*q,-p*q);
        RETURN(r) fi;
od; od; 0 end:

(d) Applying (c) to (a) gives:

  4
-10,10
-33,15,-15,33
-22,6,-209,133,-133,209,-6,22
-57,667,-91,65,-14,589,-1189,39,-39,1189,-589,14,-65,91,-667,57

(e) Looking only at numbers > 0  gives
4, 6, 10, 14, 15, 22, 33, 39, 57, 65, 91, 133, 209, 589, 667, 1189

(f) Unexpected observation: All these integers are
products of exactly two supersingular primes.

The moon shines, good night.
Peter