[seqfan] Re: Enumerating the rational numbers, supersingular primes and other monsters.
Jonathan Post
jvospost3 at gmail.com
Fri Feb 11 01:43:40 CET 2011
So, I wasn't being arbitrary...
A108764 Products of exactly two supersingular primes (A002267).
CROSSREFS
Cf. A001358, A002267.
KEYWORD
easy,fini,nonn
AUTHOR
Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 17 2005
On Thu, Feb 10, 2011 at 3:22 PM, peter.luschny
<peter.luschny at googlemail.com> wrote:
> 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
>
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