Demonstration, with Cuisenaire rods, that the composite number 10 is equidigital: 10 has two digits, and 2 · 5 has two digits (1 is excluded)
Demonstration, with Cuisenaire rods, that the composite number 10 is equidigital: 10 has two digits, and 2 · 5 has two digits (1 is excluded)

In number theory, an equidigital number is a natural number in a given number base that has the same number of digits as the number of digits in its prime factorization in the given number base, including exponents but excluding exponents equal to 1.[1] For example, in base 10, 1, 2, 3, 5, 7, and 10 (2 · 5) are equidigital numbers (sequence A046758 in the OEIS). All prime numbers are equidigital numbers in any base.

A number that is either equidigital or frugal is said to be economical.

Mathematical definition

Let be a number base, and let be the number of digits in a natural number for base . A natural number has the integer factorisation

and is an equidigital number in base if

where is the p-adic valuation of .

Properties

See also

Notes

  1. ^ Darling, David J. (2004). The universal book of mathematics: from Abracadabra to Zeno's paradoxes. John Wiley & Sons. p. 102. ISBN 978-0-471-27047-8.

References