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Extravagant number - mklabgr - 07-23-2026
Extravagant number
Summary
An extravagant number (also called a wasteful number) is a concept from number theory that describes numbers whose prime factorization requires more digits to write down than the number itself in a given numerical base. In base 10, for example, numbers such as 4, 6, 8, and 9 are extravagant because their factorizations ($4 = 2²$, $6 = 2 × 3$, $8 = 2³$, $9 = 3²$) use more written symbols when prime factors and exponents are counted than the original number.
This idea belongs to the broader study of “economical” representations of numbers, where mathematicians compare the efficiency of expressing a number directly versus describing it through its prime components. Extravagant numbers can be defined in any base, not only decimal, and there are infinitely many examples in every base.
The topic is interesting because it reveals hidden patterns in the structure of integers and shows how the way we write numbers influences mathematical properties. It connects arithmetic, prime factorization, and information efficiency, offering a different perspective on the simplicity or complexity of numbers.
Key takeaways: extravagant numbers are numbers whose prime factorizations are “longer” than the numbers themselves; they depend on the numerical base used; and they demonstrate that even ordinary integers can have surprising structural properties.
In conclusion, extravagant numbers provide a fascinating example of how mathematics explores not only what numbers are, but also how efficiently we represent them.
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