4

Part of 2019 Bundeswettbewerb Mathematik

Problems(2)

Zero digits in the expansion of Sqrt(2)

Source: Bundeswettbewerb Mathematik 2019, Round 1 - Problem 4

In the decimal expansion of

\sqrt{2}=1.4142\dots

, Isabelle finds a sequence of

k

successive zeroes where

k

is a positive integer.
Show that the first zero of this sequence can occur no earlier than at the

k

-th position after the decimal point.

Digitssqrtnumber theory

between 10k and 10k + 100 there are not more than 23 primes

Source: Germany Federal - Bundeswettbewerb Mathematik 2019 round 2, p4

Prove that for no integer

k \ge 2

10k

10k + 100

there are more than

23

number theoryprimes