I'm not expecting this to lead anywhere, but I don't think
I've seen these ideas approached from this particular angle
before. I suspect almost everything is trivial and well-known.

It's long and rambling - print it out and take it with you
to the thinking room next time you go there!

I'm looking at vanishing values of n!%q+/-1 for q prime, n<q.

(Can you tell I've been sieving for factorial primes?)

So let's fix q, and work in the ring of integers modulo q.
The process of evaluating n! for each n is a simple iterative
process, highly regular, and with clearly defined starting
points and ending points.

The sequence always starts:
1!%q == +1

And it always ends:
(q-2)! == +1
(q-1)! == -1
(c.f. Wilson's theorem)

I find the 'gentle landing' most appealing, I synaesthetically picture the
residues as behaving like a quantum packet starting at, and fading to,
nothing, but having wild perturbations in the middle:

vvVVVVVvv
--^|||||||||||||_--
^^WWWWW^^

Very _unlike_ traditional stochastic behaviour due to the gentle landing
at the end. So perhaps the behaviour of the residues within the superficially
chaotic area in the middle will have some interesting patterns.

The first thing to notice is that the pattern of the residues has a
symmetry to it.

i! * (q-1-i)! == +/-1

Therefore if i!+/-1 vanishes modulo q, then so does (q-1-i)!+/-1.

If one is like me, one is then immediately led to wonder if there are
primes q for which the exact middle point, ((q-1)/2)!+/-1, vanishes.
In fact, they aren't rare at all:

ptest(p)={
local(pr=1);
print1("P = "p" :");
for(i=2,(p-1)/2,
pr=pr*i%p;
if(pr==1,print1(" "i"!-1%"p));
if(pr==p-1,print1(" "i"!+1%"p))
);
print(if(pr^2%p==1,"=middle","")
)
forprime(pt=5,100,ptest(pt))

P = 5 :
P = 7 : 3!+1%7=middle
P = 11 : 5!+1%11=middle
P = 13 :
P = 17 : 5!-1%17
P = 19 : 9!+1%19=middle
P = 23 : 4!-1%23 8!-1%23 11!-1%23=middle
P = 29 : 10!-1%29
P = 31 : 15!-1%31=middle
P = 37 :
P = 41 :
P = 43 : 21!+1%43=middle
P = 47 : 23!+1%47=middle
P = 53 : 15!-1%53
P = 59 : 15!+1%59 18!-1%59 29!-1%59=middle
P = 61 : 8!+1%61 16!+1%61 18!+1%61
P = 67 : 18!+1%67 33!+1%67=middle
P = 71 : 7!+1%71 9!+1%71 19!+1%71 35!-1%71=middle
P = 73 : 17!-1%73
P = 79 : 23!+1%79 39!+1%79=middle
P = 83 : 13!+1%83 36!+1%83 41!-1%83=middle
P = 89 : 21!-1%89
P = 97 : 43!-1%97

Summarising:
a) Primes with +1 at the middle: 3,23,31,59,71,83,...
b) Primes with -1 at the middle: 7,11,19,43,47,67,79,...
c) Primes without +/-1 at the middle: 5,13,17,29,37,41,53,61,73,89,97,...

The pattern behind the dichotomy "+/-1 or not" should have been
detected after only a few terms. Obviously the families q=4n+1 and
q=4n+3 have different behaviour.

That might ring 'jacobi(-1,q)' bells, and one is compelled to
investigate whether square roots of -1 are in any way relevant.

Changing the above GP script's final print statement to
if(pr^2%p==1,print("=middle"),print(" ("pr":"(pr^2+1)%p-1")")
the investigation leads to an instant conclusion:

P = 5 : (2:-1)
P = 13 : (5:-1)
P = 17 : 5!-1%17 (13:-1)
P = 29 : 10!-1%29 (12:-1)
P = 37 : (31:-1)
P = 41 : (9:-1)
P = 53 : 15!-1%53 (23:-1)
P = 61 : 8!+1%61 16!+1%61 18!+1%61 (11:-1)
P = 73 : 17!-1%73 (27:-1)
P = 89 : 21!-1%89 (34:-1)
P = 97 : 43!-1%97 (22:-1)

Quite simply - if a square root of -1 modulo q exists, it is ((q-1)/2)!

So apparently we've completely tamed the very centre of that quantum
packet above. It's either +/-1, or sqrt(-1), depeinding on q%4.

Curiously, this gives us a deterministic way of uniquely specifying
a 4th root of unity modulo q. That's something we can't do in C,
as +/-i are indistinguishable due to the field automorphism that
exists.

Of course, these numerical curiosities are nothing more than
observation as presentled. I don't believe proofs that they are
not just a coincidence should be too hard. In textbook style,
I should leave them as an exercise for the reader (and of course
the writer).

And once that's been done, the open questions remain -

1) What's the difference between primes in sequences (a) and (b) above?
They are already on OEIS, but with no explanation:
http://www.research.att.com/~njas/sequences/A058302
http://www.research.att.com/~njas/sequences/A055939

2) Are there other points apart from the very middle and the ends
where the sequence can be so simply tamed?

3) Do higher roots of unity occur with any regularity?
E.g. for sequence (c), where the primes q do have sqrt(-1), does the
sequence p!%q+/-sqrtmod(-1,q) vanish at any predictable points?
P = 37 : 3!+/-i%37
P = 61 : 21!+/-i%61
P = 89 : 40!+/-i%89
P = 101 : 7!+/-i%101 12!+/-i%101
P = 109 : 14!+/-i%109
P = 113 : 27!+/-i%113
P = 149 : 16!+/-i%149
P = 157 : 21!+/-i%157
P = 173 : 51!+/-i%173
P = 181 : 58!+/-i%181
P = 193 : 34!+/-i%193 69!+/-i%193 79!+/-i%193
P = 197 : 82!+/-i%197
I don't see a pattern.

4) Do double factorials (or higher) have similar properties?
(I suspect that the double might have, but I've not checked any
values at all.)

Does anyone else have any insights into these matters?

Phil

() ASCII ribbon campaign () Hopeless ribbon campaign
/\ against HTML mail /\ against gratuitous bloodshed

[stolen with permission from Daniel B. Cristofani]

__________________________________________________
Do You Yahoo!?
Tired of spam? Yahoo! Mail has the best spam protection around
http://mail.yahoo.com