Fermat-Catalan Conjecture

Author

Eric W. Weisstein
September 5, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DiophantineEquations/Fermat-CatalanConjecture.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Fermat-CatalanConjecture.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Known Solutions

Special case:

(1^x)+(2^3)==(3^2)

1^x+2^3==3^2

Solve[1/x+1/3+1/2==1,x]

Simplify[1/x+1/2+1/3<1,x>6]

l={
{{2,5},{7,2},{3,4}},
{{7,3},{13,2},{2,9}},
{{2,7},{17,3},{71,2}},
{{3,5},{11,4},{122,2}},{{17,7},{76271,3},{21063928,2}},{{1414,3},{2213459,2},{65,7}},{{9262,3},{15312283,2},{113,7}},{{43,8},{96222,3},{30042907,2}},{{33,8},{1549034,2},{15613,3}}
};

Plus@@Take[#,2]==Last[#]&/@Map[Power@@HoldForm/@#&,l,{2}]//ColumnForm

(Plus@@Take[#,2]==Last[#])&/@Map[Power@@#&,l,{2}]

Plus@@(1/Last[Transpose[#]])<1&/@l

GCD@@@(First[Transpose[#]]&/@l)

Gaussian Integers

l={
{{8+5I,2},{5+3I,3},{1+2I,7}},
{{20+9I,2},{1+8I,3},{1+I,15}}
};

Plus@@Take[#,2]==Last[#]&/@Map[Power@@HoldForm/@#&,l,{2}]//ColumnForm

(Plus@@Take[#,2]==Last[#])&/@Map[Power@@#&,l,{2}]

Plus@@(1/Last[Transpose[#]])<1&/@l

GCD@@@(First[Transpose[#]]&/@l)