Polynomial satisfying p(x)=3^{x} for x\in\mathbb{N}

haguemarineo6h 2022-04-25 Answered
Polynomial satisfying p(x)=3x for xN
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Answers (2)

Dexter Conner
Answered 2022-04-26 Author has 15 answers

Step 1
p(x+1)=3p(x) on
Np(x+1)=3p(x)p(1)=13 contra p(x)Z[x]
Or, continuing,  p(n)=3n:⇒ p(x):p(x)=1 on   p(x):p(x)=1 contra degree comparison. This proof works over much more general coefficient rings, e.g. Q

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bobthemightyafm
Answered 2022-04-27 Author has 16 answers
Suppose anxn+an1xn1++a0=3x for xN
Then
an(x+1)n+an1(x+1)n1++a0=3anxn+3an1xn1++3a0
But the coefficient of xn on the LHS is an and on the RHS it is 3an.
This contradicts the assumption.
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