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.

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

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

Mathematical definition

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

 

where   is the p-adic valuation of  , and   is an equidigital number in base   if

 

Properties

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  • Every prime number is equidigital. This also proves that there are infinitely many equidigital numbers.

See also

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Notes

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  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

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