sentences:  There are infinitely many extravagant numbers, no matter what base is used.Let formula_1 be a number base, and let formula_2 be the number of digits in a natural number formula_3 for base formula_4.A natural number formula_3 has the integer factorisation
  and is an extravagant number in base formula_4 if
  where formula_9 is the p-adic valuation of formula_3.