LOGical equation

Number Theory Level 4

F o r h o w m a n y d i f f e r e n t v a l u e s o f " w " c a n t h i s e q u a t i o n b e s a t i s f i e d ? log ⁡ x ( p w + q w ) = w A l l g i v e n v a r i a b l e s a r e n a t u r a l n u m b e r s . For\quad how\quad many\quad different\quad values\quad of\quad "w"\quad can\quad this\quad equation\quad be\quad satisfied?\\ \\ \log _{ x }{ \left( { p }^{ w }+{ q }^{ w } \right) } =w\\ \\ All\quad given\quad variables\quad are\quad natural\quad numbers.

1 2 15 13 Finite but >22 Infinitely many 6 22

Archit Boobna by Archit Boobna , 20, India

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1 solution

Anuj Mishra
Jun 22, 2015

Given equation is, log ⁡ x ( p w + q w ) = w \log_x \ (p^w+q^w) = w

This can also be written as p w + q w = x w p^w+q^w = x^w where p , q , x p,q,x & w w are all natural numbers.

By F e r m a t ′ s L a s t T h e o r e m Fermat's\ Last\ Theorem , for solution to exist for such an equation, w w can only be 1 and 2.

Hence, w w can have only 2 values i.e. 1 & 2

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