MathDB

Call a polynomial

P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_1x+a_0

with integer coefficients primitive if and only if

\gcd(a_n,a_{n-1},\dots a_1,a_0) =1

.
a)Let

P(x)

be a primitive polynomial with degree less than

1398

and

S

be a set of primes greater than

1398

.Prove that there is a positive integer

n

so that

P(n)

is not divisible by any prime in

S

.
b)Prove that there exist a primitive polynomial

P(x)

with degree less than

1398

so that for any set

S

of primes less than

1398

the polynomial

P(x)

is always divisible by product of elements of

S

Problem Statement