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.

Attributes	Values
rdfs:label	Perfect digital invariant (en)
rdfs:comment	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)
dcterms:subject	Base-dependent integer sequences Arithmetic dynamics Diophantine equations
Wikipage page ID	61883319 (xsd:integer)
Wikipage revision ID	1104129448 (xsd:integer)
Link from a Wikipage to another Wikipage	Python (programming language) Quadratic equation Quartic equation Base-10 Cycle detection Signed-digit representation Perfect digit-to-digit invariant G. H. Hardy Base-dependent integer sequences Parity (mathematics) Persistence of a number Arithmetic dynamics Divisibility rule Divisor Dudeney number Cubic equation Euler's theorem Euler's totient function Exponentiation Factorion Singular point of a curve Balanced ternary Number theory Base-16 Base-20 Digit sum Happy number Kaprekar's routine Kaprekar number Iterated function Happy numbers Arithmetic dynamics A Mathematician's Apology Diophantine equations If and only if Integer Natural number Meertens number Narcissistic number Sum-product number Periodic point Periodic sequence Base-12 Base-3 Base-4 Base-5 Base-6 Base-8 Base-9 Number base
Link from a Wikipage to an external page	http://www.deimel.org/rec_math/DI_0.htm
sameAs	Perfect digital invariant Perfect digital invariant
dbp:wikiPageUsesTemplate	dbt:Main dbt:OEIS dbt:Reflist dbt:Quote_frame dbt:Math_proof dbt:Classes_of_natural_numbers
proof	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)
title	Proof (en)
drop	hidden (en)
has abstract	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)
prov:wasDerivedFrom	wikipedia-en:Perfect_digital_invariant?oldid=1104129448&ns=0
page length (characters) of wiki page	23697 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Perfect_digital_invariant
is rdfs:seeAlso of	Happy number
is Link from a Wikipage to another Wikipage of	List of number theory topics List of recreational number theory topics Perfect digit-to-digit invariant Dudeney number Factorion Digit sum Happy number Kaprekar's routine Kaprekar number Meertens number The Art of Computer Programming Narcissistic number Sum-product number Perfect Digital Invariant
is Wikipage redirect of	Perfect Digital Invariant
is foaf:primaryTopic of	wikipedia-en:Perfect_digital_invariant

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