To make payments, the Pythagoreans use coins in no more than three denominations. The three denominations are in whole Oboloi amounts, and the sum of the squares of the two smaller denominations equals the square of the largest denomination. The amount of $1,000$ Oboloi, and any larger whole Oboloi sum, can be obtained using the three distinct types of coins. However, the exact amount of $999$ Oboloi can not be obtained (and hence requires change).
What coin denominations do the Pythagoreans use?