Editing Catalan's conjecture (section)

==Pillai's conjecture==
{{unsolved|mathematics|Does each positive integer occur only finitely many times as a difference of perfect powers?}}
'''Pillai's conjecture''' concerns a general difference of perfect powers {{OEIS|id=A001597}}: it is an [[open problem]] initially proposed by [[S. S. Pillai]], who conjectured that the gaps in the sequence of perfect powers tend to infinity.  This is equivalent to saying that each positive integer occurs only finitely many times as a difference of perfect powers: more generally, in 1931 Pillai conjectured that for fixed positive integers ''A'', ''B'', ''C'' the equation <math>Ax^n - By^m = C</math> has only finitely many solutions (''x'',&nbsp;''y'',&nbsp;''m'',&nbsp;''n'') with (''m'',&nbsp;''n'') ≠ (2,&nbsp;2).  Pillai proved that for fixed ''A'', ''B'', ''x'', ''y'', and for any λ less than 1, we have <math>|Ax^n - By^m| \gg x^{\lambda n}</math> uniformly in ''m'' and ''n''.<ref name=rnt>{{citation | pages=[https://archive.org/details/rationalnumberth00nark/page/n261 253]–254 | title=Rational Number Theory in the 20th Century: From PNT to FLT | url=https://archive.org/details/rationalnumberth00nark | url-access=limited | series=Springer Monographs in Mathematics | first=Wladyslaw | last=Narkiewicz | publisher=[[Springer-Verlag]] | year=2011 | isbn=978-0-857-29531-6}}</ref>

The general conjecture would follow from the [[ABC conjecture]].<ref name=rnt/><ref>{{citation | last=Schmidt | first=Wolfgang M. | author-link=Wolfgang M. Schmidt | title=Diophantine approximations and Diophantine equations | series=Lecture Notes in Mathematics | volume=1467 | publisher=[[Springer-Verlag]] | year=1996 | edition=2nd | isbn=3-540-54058-X | zbl=0754.11020 | page=207}}</ref>

Pillai's conjecture means that for every natural number ''n'', there are only finitely many pairs of perfect powers with difference ''n''. The list below shows, for ''n''&nbsp;≤&nbsp;64, all solutions for perfect powers less than 10<sup>18</sup>, such that the exponent of both powers is greater than 1. The number of such solutions for each ''n'' is listed at {{oeis|id=A076427}}. See also {{oeis|id=A103953}} for the smallest solution (&gt;&nbsp;0).

{|class="wikitable" style="border:none;text-align:right;"
! ''n''
! solution<br />count
! numbers ''k'' such that ''k'' and ''k'' + ''n''<br />are both perfect powers
|rowspan="33" style="padding:2px;background:white;border:none;"|
! ''n''
! solution<br />count
! numbers ''k'' such that ''k'' and ''k'' + ''n''<br />are both perfect powers
|-
| 1 || 1 ||style="text-align:left"| 8
| 33 || 2 ||style="text-align:left"| 16, 256
|-
| 2 || 1 ||style="text-align:left"| 25
| 34 || 0 ||style="text-align:left"| ''none''
|-
| 3 || 2 ||style="text-align:left"| 1, 125
| 35 || 3 ||style="text-align:left"| 1, 289, 1296
|-
| 4 || 3 ||style="text-align:left"| 4, 32, 121
| 36 || 2 ||style="text-align:left"| 64, 1728
|-
| 5 || 2 ||style="text-align:left"| 4, 27
| 37 || 3 ||style="text-align:left"| 27, 324, {{val|14348907}}
|-
| 6 || 0 ||style="text-align:left"| ''none''
| 38 || 1 ||style="text-align:left"| 1331
|-
| 7 || 5 ||style="text-align:left"| 1, 9, 25, 121, {{val|32761}}
| 39 || 4 ||style="text-align:left"| 25, 361, 961, {{val|10609}}
|-
| 8 || 3 ||style="text-align:left"| 1, 8, {{val|97336}}
| 40 || 4 ||style="text-align:left"| 9, 81, 216, 2704
|-
| 9 || 4 ||style="text-align:left"| 16, 27, 216, {{val|64000}}
| 41 || 3 ||style="text-align:left"| 8, 128, 400
|-
| 10 || 1 ||style="text-align:left"| 2187
| 42 || 0 ||style="text-align:left"| ''none''
|-
| 11 || 4 ||style="text-align:left"| 16, 25, 3125, 3364
| 43 || 1 ||style="text-align:left"| 441
|-
| 12 || 2 ||style="text-align:left"| 4, 2197
| 44 || 3 ||style="text-align:left"| 81, 100, 125
|-
| 13 || 3 ||style="text-align:left"| 36, 243, 4900
| 45 || 4 ||style="text-align:left"| 4, 36, 484, 9216
|-
| 14 || 0 ||style="text-align:left"| ''none''
| 46 || 1 ||style="text-align:left"| 243
|-
| 15 || 3 ||style="text-align:left"| 1, 49, {{val|1295029}}
| 47 || 6 ||style="text-align:left"| 81, 169, 196, 529, 1681, {{val|250000}}
|-
| 16 || 3 ||style="text-align:left"| 9, 16, 128
| 48 || 4 ||style="text-align:left"| 1, 16, 121, 21904
|-
| 17 || 7 ||style="text-align:left"| 8, 32, 64, 512, {{val|79507}}, {{val|140608}}, {{val|143384152904}}
| 49 || 3 ||style="text-align:left"| 32, 576, {{val|274576}}
|-
| 18 || 3 ||style="text-align:left"| 9, 225, 343
| 50 || 0 ||style="text-align:left"| ''none''
|-
| 19 || 5 ||style="text-align:left"| 8, 81, 125, 324, {{val|503284356}}
| 51 || 2 ||style="text-align:left"| 49, 625
|-
| 20 || 2 ||style="text-align:left"| 16, 196
| 52 || 1 ||style="text-align:left"| 144
|-
| 21 || 2 ||style="text-align:left"| 4, 100
| 53 || 2 ||style="text-align:left"| 676, {{val|24336}}
|-
| 22 || 2 ||style="text-align:left"| 27, 2187
| 54 || 2 ||style="text-align:left"| 27, 289
|-
| 23 || 4 ||style="text-align:left"| 4, 9, 121, 2025
| 55 || 3 ||style="text-align:left"| 9, 729, {{val|175561}}
|-
| 24 || 5 ||style="text-align:left"| 1, 8, 25, 1000, {{val|542939080312}}
| 56 || 4 ||style="text-align:left"| 8, 25, 169, 5776
|-
| 25 || 2 ||style="text-align:left"| 100, 144
| 57 || 3 ||style="text-align:left"| 64, 343, 784
|-
| 26 || 3 ||style="text-align:left"| 1, {{val|42849}}, {{val|6436343}}
| 58 || 0 ||style="text-align:left"| ''none''
|-
| 27 || 3 ||style="text-align:left"| 9, 169, 216
| 59 || 1 ||style="text-align:left"| 841
|-
| 28 || 7 ||style="text-align:left"| 4, 8, 36, 100, 484, {{val|50625}}, {{val|131044}}
| 60 || 4 ||style="text-align:left"| 4, 196, {{val|2515396}}, {{val|2535525316}}
|-
| 29 || 1 ||style="text-align:left"| 196
| 61 || 2 ||style="text-align:left"| 64, 900
|-
| 30 || 1 ||style="text-align:left"| 6859
| 62 || 0 ||style="text-align:left"| ''none''
|-
| 31 || 2 ||style="text-align:left"| 1, 225
| 63 || 4 ||style="text-align:left"| 1, 81, 961, {{val|183250369}}
|-
| 32 || 4 ||style="text-align:left"| 4, 32, 49, 7744
| 64 || 4 ||style="text-align:left"| 36, 64, 225, 512
|}