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)