Problem #PRU-64767

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?