Download PDF

perfect power∗
pahio†
2013-03-22 3:35:32
The power mn is called a perfect power if m and n are integers both greater
than 1. The perfect powers form the ascending order sequence (cf. Sloane’s
A001597)
4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, . . . ,
(1)
i.e.
22 , 23 , 32 , 24 = 42 , 52 , 33 , 25 , 62 , 72 , 26 = 43 = 82 , 34 = 92 , 102 , 112 , 53 , . . .
S. S. Pillai has conjectured in 1945, that if the ith member of the sequence (1)
is denoted by ai , then
lim inf (ai+1 −ai ) = ∞.
i→∞
(2)
This does not necessarily mean that one had limi→∞ (ai+1 − ai ) = ∞, since
there may always exist little differences ai+1 −ai arbitrarily far from the begin
of the sequence (1).
The equation (2) is equivalent to the
Pillai’s conjecture. For any fixed positive integer k, the Diophantine
equation
xm −y n = k
has only a finite number of solutions (x, y, m, n) where the integers x, y, m, n
all are greater than 1.
Pillai’s conjecture generalises the Catalan’s conjecture (k = 1) in which the
number of solutions is 1.
∗ hPerfectPoweri created: h2013-03-2i by: hpahioi version: h42194i Privacy setting: h1i
hDefinitioni h11D61i h11D45i h11B83i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1
The series formed by the inverse numbers of the perfect powers converges
absolutely, and its sum may be calculated easily:
∞ X
∞
X
1
1
=
n
n
m
m
m=2 n=2
m,n=2
∞
X
=
∞ X
∞
X
1
1
· n−2
2
m m
m=2 n=2
∞
∞
X
1 X 1
m2 n=2 mn−2
m=2
n
∞
∞ X
1 X 1
=
m2 n=0 m
m=2
=
=
∞
X
1
1
·
1
2
m 1− m
m=2
∞
X
1
m(m−1)
m=2
∞ X
1
1
=
−
m−1 m
m=2
=
The sum of this telescoping series is equal to 1.
References
[1] S. S. Pillai: On ax −by = c. – J. Indian math. Soc. 2 (1936).
2