
nsequences
Erdös, in a letter dated 79=10=31 observed that: 3*4 = 5*6*7 = 1 mod 11
and asked for the least prime p such that 3 consecutive products of
integers = 1 mod p.
Landon Noll and Chuck Simmons solved Erdös'
problem by solving a more general problem of:
q1! = q2! = ... = qn! mod p
where q1 < q2 < ... < qn.
The following table shows that least prime p for which there is a solition:
n p q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11
1 2 0
2 2 0 1
3 5 0 1 3
4 17 0 1 5 11
5 17 0 1 5 11 15
6 23 0 1 4 8 11 21
7 71 8 10 20 52 62 64 71
8 599 29 51 123 184 251 290 501 540
9 599 29 51 123 184 251 290 501 540 556
10 3011 0 1 611 723 749 805 2205 2261 2287 2399
11 3011 0 1 611 723 749 805 2205 2261 2287 2399 3009
For more information see
Unsolved Problems in Number Theory, 1st edition, problem
A15.
FYI: The 1st edition is no longer in print, but the
2nd
edition is available.
