# bunching up of perfect powers

Neil Fernandez primeness at borve.demon.co.uk
Thu Oct 10 23:47:50 CEST 2002

```Continuing the 'differences between perfect powers' theme, I've just
submitted these:

..........................................

perfect powers for which the two closest perfect powers are smaller
(e.g. 144 is closer to both 128 (diff 16) and 125 (diff 19) than it is
to 169 (diff 25)):

9, 16, 36, 128, 144, 256, 361, 529, 1024, 1369, 1764, 2209, 2809, 3136,
3481, 5041, 5929, 6889, 8281, 9409, 10816, 12321, 13924, 16900, 17689,
19881, 22201, 24649, 27225, 29929, 33124, 36100, 39601, 43264, 51076,
55225, 59536, 64009, 69169, 74529, 78400, 79524, 85264, 91204, 97344,...

(128 is the only number here that is not a perfect square)
..........................................

perfect powers for which the three closest perfect powers are smaller:
36, 144, 2209 (no others <10^5)
..........................................

perfect powers for which the two closest perfect powers are greater:
25, 121, 2187 (no others <10^5)
..........................................

As for:

perfect powers for which the four closest perfect powers are smaller:
(no terms <10^5)

perfect powers for which the three closest perfect powers are greater:
(no terms <10^5).

Neil
--
Neil Fernandez

```