My neighbor, who looks exactly like Ernest Hemingway, celebrated his 80th birthday last year and there was much ado. This year he’s 81, a number of a relatively rare kind — at least as far as birthdays go — and it was no big deal.

Within the life spans of human beings, we tend to emphasize those that are powers of ten. But there are birthday numbers that are much more exciting.

Multiples of ten are mundane artifacts of the decimal system. Numbers geeks look past them to numbers with deeper qualities such as power, perfection, or abundance, the way a little kid on December 25 ignores the religious significance in favor of opening presents.

For the purpose of this discussion I will restrict our perusal of numbers to the length in years of a human life, arbitrarily set at 85 on the average and 125 at its maximum.

## The Powers That Be

What’s so interesting about my erstwhile Hemingway’s 81st birthday? Well, he’s only had two others of the type, the first when he was a year old and couldn’t really comprehend the significance, and the second when he was sixteen and more interested in cars and girls than numbers.

Get it yet? 81 is a perfect fourth power — the fourth power of three. It’s been 65 years since he was two to the fourth power. And this will be his last fourth power birthday because four to the fourth power is 256.

There are other powers of course, but my earnest Ernest has already had four birthdays that were cubes, and a person who challenges the longevity maximum could squeeze out a fifth. Squares are hardly worth mentioning: nine within a normal lifetime and 11 in a maximum one.

## The Perfect Birthday

A ‘perfect’ number is one which is the sum of all its factors save itself (known as proper factors). Six is the smallest perfect number (0ne plus two plus three). Can you find the next one?

The fact is that perfect numbers are extremely rare, and Perfect Birthdays should be celebrated accordingly, possibly by issuing coins, but more practically by having an extra glass of Barrow’s Intense (or wine if you don’t like ginger). There are only four perfect numbers with fewer than eight digits, and yet two occur within a human lifetime.

And the second perfect number is — 28. Strangely, there are no known odd perfect numbers, though it has not been proven that there cannot be any.

## Abundant Birthdays

Very few numbers, the perfect ones, have proper factors that add up to the number itself. This invites comparison with those whose sum of proper factors is more or less than the number itself: the former are called abundant and the latter deficient. As it happens, though there are more deficient numbers than abundant ones, there are quite a few abundant multiples of ten: 20, 30, 40, 60, 70, 80, 90, 100, and 120. So my neighbor’s 80th was not quite as mundane as it seemed.

The amount by which the sum of proper factors of an abundant number exceeds the number is called its abundance. A 40th enjoys an abundance of six; my neighbor’s 80th sported an abundance of 22. Life gets more abundant as you get older.

## A Unique Birthday

The twenties are an exciting time for birthamathophiles. As we saw, there is a Perfect Birthday in the twenties. There is also a Unique Birthday at age 26. This is the only number that comes directly between a square and a cube (of integers of course). It’s my favorite number among those within the current human life span.

## Palindromic Birthdays

A palindrome is a group of words or numbers that read the same backwards and forwards. With words, the definition of ‘the same’ is rather loose, as punctuation and capitalization are ignored. The most famous word palindrome is Napoleon’s lament, “*Able was I, ere I saw Elba*.”

Number palindromes don’t have the deep mathematical meaning of perfect or abundant numbers, but they still have a certain cache. Aside from the unedifying single-digit palindromes, there are seven in a normal lifetime and 12 within the maximum life span. Those who reach the age of centenarian plus one can experience the thrill of observing two palindromic birthdays within three years. Talk about exciting!

## Stretching A Human Life To A Really Great Number

Modern medical miracles have raised the average life expectancy considerably, while the maximum isn’t budging. Future advances might stretch the maximum age to other fascinating numbers. I can’t help mentioning my favorite: 153.

One hundred fifty-three is the first non-trivial Armstrong number, also known as a Perfect Digital Invariant, or for those with psychological insight, a narcissistic number. An Armstrong number is equal to the sum of its digits, each raised to the power of the number of digits. The power is called the order of the Armstrong number. There are 88 Armstrong numbers in the universe (excluding the trivial single digits) of which four are of order three. The smallest is 153 (1^{3} + 5^{3} + 3^{3} = 153).

Interestingly, there are no Armstrong numbers of order two. Interestingly again, the next two Armstrong numbers of order three (after 153) are consecutive integers. Do you think you can find them before your next birthday?

LORNE MOUNTAIN says

Speaking of math???

What is the maximum age someone could be on their first birthday?

Any assistance would be appreciated.

Thanx

Frank Parker says

Hi Jon, Since “A palindrome is a group of … numbers …” how can there be a “single digit palindrome”, unedifying or not! I have been fascinated by palindromic dates (when expressed in the European way dd/m/yy) ever since I realised my 18th wedding anniversary would/did occur on 18/9/81..

Jon Plotkin says

OK, Frank, to me one qualifies as a group. But since we’re talking math, here, perhaps I should have used the word ‘set.’Congratulations on your wedding palindrome, but don’t get me started on palindromic dates. I have written about birthday palindromes, and it is rare for a person to be alive on his birthday palindrome in the mm/dd/yyyy format. My grandson will celebrate his palindromic birthday on 12/02/2021.