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%I #21 Jan 22 2021 10:02:48
%S 27,64,125,243,1000,1296,2187,50625,59049,194481,279841,456533,614125,
%T 3111696,6434856,22665187,25411681,38950081,62742241,96059601,
%U 131079601,418161601,506250000,741200625,796594176,1249198336,2136719872,2217342464,5554571841,5802782976
%N Lower of a pair of adjacent perfect powers, both with exponents > 2.
%C It is conjectured that the intersection of A340700 and A340701 is empty, i.e., that no 3 immediately consecutive perfect powers with all exponents > 2 (A076467) exist. No counterexample < 3.4*10^30 was found.
%H Hugo Pfoertner, <a href="/A340700/b340700.txt">Table of n, a(n) for n = 1..1670</a>
%H StackExchange MathOverflow, <a href="https://mathoverflow.net/questions/62444/are-there-ever-three-perfect-powers-between-consecutive-squares/62479">Are there ever three perfect powers between consecutive squares?</a> Answers by Gjergji Zaimi and Felipe Voloch (2011).
%H Michel Waldschmidt, <a href="https://arxiv.org/abs/0908.4031">Perfect Powers: Pillai's works and their developments</a>, arXiv:0908.4031 [math.NT], 27 Aug 2009.
%F a(n) = A340702(n)^A340704(n) = A340701(n) - A340706(n).
%e Initial terms of sequences A340700 .. A340706:
%e a(n) = x^p,
%e A340701(n) = A340703(n)^A340705(n) = y^q,
%e A340706(n) = A340701(n) - a(n) = y^q - x^p.
%e .
%e n a(n) x ^ p A340701 y ^ q A340706 adjacent squares
%e 1 27 = 3 ^ 3, 32 = 2 ^ 5, 5 5^2=25, 6^2=36
%e 2 64 = 2 ^ 6, 81 = 3 ^ 4, 17 8^2=64, 9^2=81
%e 3 125 = 5 ^ 3, 128 = 2 ^ 7, 3 11^2=121, 12^2=144
%e 4 243 = 3 ^ 5, 256 = 2 ^ 8, 13 15^2=225, 16^2=256
%e 5 1000 = 10 ^ 3, 1024 = 2 ^ 10, 24 31^2=961, 32^2=1024
%e 6 1296 = 6 ^ 4, 1331 = 11 ^ 3, 35 36^2=1296, 37^2=1369
%e 7 2187 = 3 ^ 7, 2197 = 13 ^ 3, 10 46^2=2116, 47^2=2209
%e 8 50625 = 15 ^ 4, 50653 = 37 ^ 3, 28 225^2=50625, 226^2=51076
%e 9 59049 = 3 ^ 10, 59319 = 39 ^ 3, 270 243^2=59049, 244^2=59536
%Y The corresponding upper members of the pairs are A340701.
%Y Cf. A001597, A076467, A097056, A340642, A340702, A340703, A340704, A340705, A340706.
%Y Cf. A117934 (excluding pairs where one of the members is a square).
%K nonn
%O 1,1
%A _Hugo Pfoertner_, Jan 16 2021