On 7/22/07, David Wilson <davidwwilson@comcast.net> wrote:
Well, Wilson's Theorem Jr. is out the window.
And of course, Bell pseudoprime is infinitely preferable to Wilson pseudoprime.
So I remain in obscurity.
At least I've discovered a primality test that is at once computationally difficult and unreliable.
But not as unreliable as my typing: some gremlin introduces invisible typos into text as I copy and paste it, which only become visible after the send button is pressed ... Dobinski's sum should actually have read: B(n) = (1/e) \sum_{i >= 0} i^n / i! A Maple script implementing this method is (hopefully) appended below: it finds B(n) for n = 173 in 1/4 sec, for n = 2999 in 20 sec on a Mac G4. [The Maple 10 library function took 102 sec for n = 170, was aborted after 480 sec for n = 173; Maple 9 aborted after similar time for n = 2999.]
Would you be the discoverer of the first Bell pseudoprime, [MD]r. Lunnon?
As far as I know --- tho' I don't plan to have it engraved on my tombstone --- nor the 66,000 or so digits of the corresponding Bell number! Note that there may be a smaller one --- my program searched in order of smallest prime factor p, and found only that there are no more pseudoprimes for p < 60. I can prove that B(p^2)mod p = 3 for (odd) prime p; so the next candidate would be n = 61*67 = 4087 ... Fred Lunnon # Bell number B(n) via Dobinsky sum bell := proc (n) local i,i_0,i_1,digo,digs,beln; digo := Digits; Digits := 10; i_0 := fsolve(i*log(i) = n, i); # index max term i_1 := n + i_0; # number of terms required digs := n^2/i_0/log(10.0); # precision > log_10 of max term Digits := max(10, round(digs)); beln := add(evalf(i)^n/factorial(evalf(i)), i = 0..i_1)/exp(1.0); # floating-point evaluation! Digits := digo; # restore precision round(beln) end; tim := time(); # test n := 173; bell(n)mod n; # 2 for n prime tim := time() - tim;