4

Part of 1979 Bundeswettbewerb Mathematik

Problems(2)

Almost Divisible!

Source:

Prove that the polynomial

P(x) = x^5-x+a

is irreducible over

\mathbb{Z}

5 \nmid a

algebrapolynomial

Bundeswettbewerb Mathematik 1979 Problem 2.4

Source: Bundeswettbewerb Mathematik 1979 Round 2

An infinite sequence

p_1, p_2, p_3, \ldots

of natural numbers in the decimal system has the following property: For every

i \in \mathbb{N}

the last digit of

p_{i+1}

is different from

9

, and by omitting this digit one obtains number

p_i

. Prove that this sequence contains infinitely many composite numbers.

SequenceLast digitinfinitely many solutionsnumber theory