Problem #PRU-64767

Problems Methods Examples and counterexamples Calculus Real numbers Rational and irrational numbers

Problem

In the Republic of mathematicians, the number \(\alpha > 2\) was chosen and coins were issued with denominations of 1 pound, as well as in \(\alpha^k\) pounds for every natural \(k\). In this case \(\alpha\) was chosen so that the value of all the coins, except for the smallest, was irrational. Could it be that any amount of a natural number of pounds can be made with these coins, using coins of each denomination no more than 6 times?