Ever since school I’ve loved the pure truth of mathematics. The fact that you can be just right or wrong, with no middle ground or ambiguity, always appealed to me. As a toolset, it sits neatly in one’s arse pocket, ready to be used throughout life.

Recreational Mathematics can introduce some fun into the subject. I was listening to a podcast of The Infinite Monkey Cage a few days ago and a guest introduced the concept of Narcissistic Numbers – numbers that are in love with themselves. Here’s an example:

The four-digit number 8208.

Take each digit, raise it to the fourth power (because there are four digits), added the numbers and you get … 8208.

To work it out:

(8^4)+(2^4)+(0^4)+(8^4)

= (8x8x8x8)+(2x2x2x2)+(0x0x0x0)+(8x8x8x8)

= 4096+16+0+4096

= 8208

The number loved itself so much, it turned back into itself.

There are only three four-digit numbers that are Narcissistic Numbers. Same with five digit numbers. I’m busy writing some Java code to calculate them for everything up to 39 digits. There is a 39-digit Narcissistic Number, the largest possible:

115,132,219,018,763,992,565,095,597,973,971,522,401

The amazing thing is that it can be proven fairly easily that there are no numbers above 39 digits that are Narcissistic. The proof is this:

Take the largest 39-digit number, which is 999,999,999,999,999,999,999,999,999,999,999,999,999

Perform the calculation on this i.e. add 9 to the 39th power to itself 39 times, or

(9^39) + (9^39) + (9^39) + (9^39) + … + (9^39) + (9^39) + (9^39)

*the above 39 times*

= 39 x (9^39)

= 640,504,927,462,165,500,000,000,000,000,000,000,000 (approx.)

Note that my calculator only has a limited precision to a certain number of digits (around 16). But it is sufficient to demonstrate the proof: the number, itself 39 digits, is considerably less than the largest 39-digit number i.e. it is **deficient** in that it cannot produce a number large enough to equal the originating number. A 40-digit number is going to have the same problem, in fact even more so, as is a 41-digit number, and so on.

I dunno about you, but I love the idea that (a) this is possible and (b) someone thought of it. Also, because this is an exploration of numbers in Base 10 (i.e. a man-made condition) it would be worth exploring in other bases for comparison.

Numbers for numbers’ sake. Love it.

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Louis Barbe

said:You should check out a Ronald Graham!

glennwestmore

said:There are only 88 narcissistic numbers, as proven by D. Winter in 1985 and verified by D. Hoey. 153 is the smallest narcissistic number, and there is only one narcissistic number whose digits are all raised to the power of 6. Read more: http://www.glennwestmore.com.au/narcissistic-numbers/