About: Perfect digital invariant

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In number theory, a perfect digital invariant (PDI) is a number in a given number base that is the sum of its own digits each raised to a given power.

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  • In number theory, a perfect digital invariant (PDI) is a number in a given number base that is the sum of its own digits each raised to a given power. (en)
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  • Let the digits of be , , and . Then : Thus is a perfect digital invariant for for all . (en)
  • Let : be a natural number with digits, where , and , where is a natural number greater than 1. According to the divisibility rules of base , if , then if , then the digit sum : If a digit , then . According to Euler's theorem, if , . Thus, if the digit sum , then . Therefore, for any natural number , if , and , then for every natural number , if , then . (en)
  • Let the digits of be , , and . Then : Thus is a perfect digital invariant for for all . (en)
  • Let the digits of be , , and . Then : :: :: :: :: :: :: :: :: :: :: :: :: Thus is a perfect digital invariant for for all . (en)
  • Let the digits of be , , and . Then : :: :: :: :: :: :: :: :: :: :: :: :: :: :: :: Thus is a perfect digital invariant for for all . (en)
  • If , then the bound can be reduced. Let be the number for which the sum of squares of digits is largest among the numbers less than . : : because Let be the number for which the sum of squares of digits is largest among the numbers less than . : : because Let be the number for which the sum of squares of digits is largest among the numbers less than . : : Let be the number for which the sum of squares of digits is largest among the numbers less than . : : . Thus, numbers in base lead to cycles or fixed points of numbers . (en)
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  • Proof (en)
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  • In number theory, a perfect digital invariant (PDI) is a number in a given number base that is the sum of its own digits each raised to a given power. (en)
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  • Perfect digital invariant (en)
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