Numbers narcissistic

Dik T. Winter from Amsterdam is an expert on a somewhat related class of numbers that are the sums of powers of their digits. In other words, these -digit numbers are equal to the sum of the nth powers of their digits. For example, 153 = P -I- 5 -i- 3. Variously called narcissistic numbers, numbers in love with themselves, Armstrong numbers, or perfect digital variants, these kinds of numbers have fascinated number theorists for decades. The English mathematician Godfrey Hardy (1877-1947) noted that, There are just four numbers, after unity, which are the sums of the... 

Base Total Dig Largest Narcissistic Number (number of digits)... 

The largest narcissistic number discovered to date is the incredible 39-digit number... 

The frequency of occurrence of narcissistic numbers varies according to the base of the number system in which one conducts a search. For example, in our standard (base 10) system are 88 known numbers of this type, whereas in base 4 there are only 11. Table 22.1 lists numbers from Dik T. Winter. The number of digits for the largest narcissistic number known is in parentheses. 

As one searches for larger and larger narcissistic numbers, will they eventually run out, as in the case of the lonely factorions If they are proved to die out in one number system, does this mean they are finite in another (Martin Gardner wrote to me recently and indicated that the number of narcissistic numbers has been proved finite. They cannot have more than 58 digits in our standard base 10 number system.)... 

The following allows you to search for narcissistic numbers, as described in the text. 

Borderline disorders have been viewed both from a behavioral-symptomatic perspective (such as that of DSM-IV) and a theoretical perspective (for example, borderline personality organization a la Kemberg and others). As Grinker and many subsequent researchers and clinicians agree, borderline patients are not all alike they present in many different sizes and shapes. You can see borderline characteristics in the context of a number of different personality styles schizoid-detached, obsessional, paranoid, narcissistic, and histrionic. However, all varieties share a few common characteristics ... 

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