Prove that x^n_1+x^n_2 is an integer and is not divisible by 5

saucletbh 2022-09-18 Answered
Prove that x 1 n + x 2 n is an integer and is not divisible by 5
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Answers (1)

Miguel Shah
Answered 2022-09-19 Author has 8 answers
We have x 1 + x 2 =  Sum of roots  = 6. From the equation, we have
x 1 2 6 x 1 + 1 = 0  and  x 2 2 6 x 2 + 1 = 0
Adding both we get
x 1 2 + x 2 2 = 6 ( x 1 + x 2 ) Integer 2 = Integer
Now use strong induction and make use of the fact that
x 1 n + 2 6 x 1 n + 1 + x 1 n = 0  and  x 2 n + 2 6 x 2 n + 1 + x 2 n = 0
i.e.,
x 1 n + 2 + x 2 n + 2 = 6 ( x 1 n + 1 + x 2 n + 1 ) ( x 1 n + x 2 n )
Use the same idea to show that
x 1 n + x 2 n { 1 ( mod 5 ) n 1 ( mod 4 ) 4 ( mod 5 ) n 0 , 2 ( mod 4 ) 3 ( mod 5 ) n 3 ( mod 4 )

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