Polynomial problem involving divisibility, prime numbers, monotonyLet f be a polynomial function, with integer coefficients,...

Anton Huynh

Anton Huynh

Answered

2022-11-23

Polynomial problem involving divisibility, prime numbers, monotony
Let f be a polynomial function, with integer coefficients, strictly increasing on N such that f ( 0 ) = 1. Show that it doesn't exist any arithmetic progression of natural numbers with ratio r > 0 such that the value of function f in every term of the progression is a prime number. I believe that the solution includes a reductio ad absurdum, but I don't know how to solve it.

Answer & Explanation

cenjene9gw

cenjene9gw

Expert

2022-11-24Added 13 answers

Explanation:
If progression is a + n b, and f ( a ) = p, then f ( a + p n b ) 0 ( mod p ) and absolute value is more then p for large n.

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