10=m^p-n^q?
ZAKIRS
zfseidov at ycariel.yosh.ac.il
Mon Oct 7 12:26:58 CEST 2002
Dear all,
i've just speculated that somel integers can't be represented as
"difference of two full powers >=2"
and the list of such (eligible) numbers <200 is:
10, 14, 22, 29, 31, 34, 42, 50, 52, 54, 58, 60, 62, 66, 68, 70, 72,
76, 78, 82, 84, 86, 88, 90, 92, 94, 96, 98, 102, 108, 110, 111,
112, 114, 118, 120, 122, 124, 126, 130, 132, 134, 136, 140, 142,
146, 150, 153, 156, 158, 160, 162, 164, 174, 176, 177, 178, 182,
186, 188, 190, 192, 193, 194
The reason is that the set of full powers is not so much dense
(but... - i say trembling - infinite).
If it's not OK then it'll be interesting to find the smallest cases for each
number in the list -
and for all integers as well!.
Otherwise one can prove or disprove that
"any number can be represented as difference of two full powers" or smth.
Or such theorem is well known to all save me. thanks, zak
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