That escalated quickly!

Number Theory Level 2

Find the number of positive integers n n , for which

2 n + 12 n + 159187 n 2^n+{12}^n+{159187}^n

is a square.


The answer is 1.

Bogdan Simeonov by Bogdan Simeonov , 21, Bulgaria

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2 solutions

Adarsh Kumar
Nov 28, 2014

2 n + 1 2 n + 15918 7 n = k 2 = o d d l e t u s t a k e t h a t n ≥ 2 t h e n 2 n + 1 2 n + 15918 7 n i s d i v i s i b l e b y 4 b u t 4 d o e s n ′ t d i v i d e \ a n y o d d i n t e g e r ⇒ 0 < n < 2 ⇒ n = 1 i s t h e o n l y s o l u t i o n w h i c h c a n b e t r u e . P u t t i n g 1 i n p l a c e o f n g i v e s t h a t t h e e x p r e s s i o n i s 39 9 2 . T h u s , n = 1 i s t h e o n l y s o l u t i o n . 2^{n}+12^{n}+159187^{n}=k^{2}=odd\\ let\ us\ take\ that\ n\geq2\\ then\ 2^{n}+12^{n}+159187^{n}\ is\ divisible\ by\ 4\ but\ 4\ doesn't\ divide\\\ any\ odd\ integer\\ \Rightarrow0<n<2\\ \Rightarrow\ n=1\ is\ the\ only\ solution\ which\ can\ be\ true.Putting\ 1\ in\\ place\ of\ n\ gives\ that\ the\ expression\ is\ 399^{2}.\\ Thus,n=1\ is\ the\ only\ solution.

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How is 2^n+12^n+159187^n divisible by 4?

Matthew Feng - 6 years, 5 months ago
Bogdan Simeonov
May 31, 2014

This is kind of a troll problem :D.You can substitute 159187 with any number, congruent to 3 mod 4 and 1 mod 3.

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