Remarks on the sequence of 'perfect' powers (cf. Sequences A001597 and A023057)
From: alf(AT)ics.mq.edu.au (Alf van der Poorten)
Date: Thu, 9 May 2002 13:59:59 +1000
The sequence of powers x^a is popularly spoken of as the sequence
of `perfect' powers (though I have been archly asked just what
constitutes the sequence of `imperfect' powers. For that matter, it
follows from Euclid that no square is a perfect number, but Carl
Pomerance reminds me that it is not known whether any cube is a
perfect number). It is notorious that, other than for gaps 1 (or
-1) in the sequence --- thus Catalan's Conjecture, now, finally, a
theorem --- nothing is proved.
By the way, it seems that the gap 1 is `easy' exactly because 1
is the unique power perfect enough to be any power whatsoever.
In the following remarks, the word 'gap' may refer to the difference
of any pair of powers x^a rather than just to the difference of
consecutive elements in the sequence of powers.
It is morally obvious (and experimentally plain; that is, it is
certainly not counter indicated by experiment) that the sequence of
gaps has the following properties: (a) each gap k occurs at most
finitely many times, and --- this is just a p-adic analogue of the
preceding remark --- (b) given primes p_1, ..., p_k, gaps
p_1^{e_1} ... p_k^{e_k} also occur just finitely many times; for
example, gaps 2^a 3^b occur just finitely many times. Certainly one
feels confident that \liminf |x^a-y^b|, as at least one of the
different powers goes to infinity, is infinite. Accordingly, if a
gap, say 6, does not occur `early on' it is safe to bet that it
never occurs.
Running slightly skew to these remarks it also seems morally clear
that gaps of a `given shape' occur only finitely often unless they
obviously occur infinitely often: thus it is reasonable to hold the
view that a gap that is itself a perfect power occurs only finitely
many times (an 'obvious exclusion' here includes squares occurring as
differences of squares or, for that matter, as the difference of a
cube and a square, or of a fifth power and a cube --- in such cases
there are infinitely many solutions parametrised by polynomials in
several variables); however, I have in mind something rather more
general than `facts' (such as the generalised Fermat conjecture, aka
the Fermat-Catalan conjecture) immediately implied by the
abc-conjecture.
Nonetheless, the bottom line is that none of these things remotely
seem accessible to proof by present techniques. Thus, for all we can
prove, the sequence A023057 of non-gaps could well be empty.
The rather more interesting A023055 gives the sequence of (apparent) gaps.
However, it does seem worthwhile to add that the sum of the
reciprocals of the perfect powers is well known (exercise for the
reader), namely
\sum_{a=2}^{\infty}\sum_{b=2} ^{\infty} 1/a^b =1\,.
Just so, for further exercise, one can sum the the reciprocals of the
odd powers of the integers that are 3 \mod 4, namely
\sum_{n=0}^{\infty}\sum_{m=1} ^{\infty} 1/(4n+3)^{2m+1}.
Eight times the sum is \pi-4\log 2.
------------------------
alf(AT)math.mq.edu.au (Alf van der Poorten)
Department of Mathematics
Macquarie University, Sydney 2109 Australia
phone: +61 2 9850 8947 fax: +61 2 9850 8114
home: +61 2 9416 6026 mobile: +61 4 1826 3129 (from MQ: #6335)