Determine all monic polynomial with integer coefficient $P$ such that for every integer $a,b$ there exists integer $c$ so that \[P(a)P(b)=P(c)\]
Problem
Source: Swiss TST 2024
Tags: algebra, polynomial
13.01.2025 15:42
Bump this
13.01.2025 16:46
Only xⁿ and x+C will work
13.01.2025 16:53
Rohit-2006 wrote: all odd degree polynomials are the solutions Uhh ? Considering $P(x)=x^3+1$, off odd degree and $a=b=1$, could you kindly give us the value of integer $c$ such that $P(a)P(b)=P(c)$ ?
13.01.2025 17:29
Oh sorry I didn't noticed a,b,c integers
14.01.2025 18:21
bump thisone
14.01.2025 19:36
Rohit-2006 wrote: Only xⁿ and x+C will work What about $x^3+6x^2+12x+8$ ?
14.01.2025 20:48
hint: if f is monic and take infinitely many power of his degree then it is (x+c)^deg
14.01.2025 20:49
so P=(x+c)^n or P=1 or P=0
17.01.2025 18:28
Bump please
21.01.2025 01:38
EtacticToe wrote: Determine all monic polynomial with integer coefficient $P$ such that for every integer $a,b$ there exists integer $c$ so that \[P(a)P(b)=P(c)\] Solved with emotional aupport from sebasl195 and jgxry.