Problem 11594
(American Mathematical Monthly, Vol.118, October 2011) Proposed by Harm Derksen and Jeffrey Lagarias (USA).
Let
Gn=
n
Y
k=1
k−1
Y
j=1
j k
,
and let Gn= 1/Gn.
(a) Show that if n is an integer greater than 1, then Gn is an integer.
(b) Show that for each prime p, there are infinitely many n greater than 1 such that p does not divide Gn.
Solution proposed by Roberto Tauraso, Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.
The number Gn is an integer because
Gn=
n
Y
k=1
k−1
Y
j=1
k j
=
n
Y
k=1
kk k! =
n−1
Y
k=1
n k
.
Moreover, if p is a prime then by Lucas’ Theorem
n k
≡ns−1 ks−1
ns−2 ks−2
· · · ·n0 k0
(mod p)
where n = ns−1ps−1+ ns−2ps−2+ · · · + n0 and k = ks−1ps−1+ ks−2ps−2+ · · · + k0 are the base p expansions of n and k respectively. Hence, by letting n = ps− 1 with s > 0 then for k = 1, . . . , n − 1
n k
≡p − 1 ks−1
p − 1 ks−2
· · · ·p − 1 k0
≡ (−1)ks−1· (−1)ks−2· · · (−1)k0 6≡ 0 (mod p).
Therefore p does not divide Gn =Qn−1 k=1
n
k.
