 Son of SPECIAL NUMBERS 99066

We begin this second page of special numbers with one that has two interesting facts. For the first, just rotate the number 180°. What you see is what you had before, right?

But the second is even more outstanding. This time we will ask you to merely "square it"; that is,

990662 = ?

If done carefully, you will obtain a product that contains each of the ten digits exactly once. Numbers that contain each of the ten digits exactly once are called pandigital.

 23

In the first edition of Special Numbers you saw the number 24. Now how about the number 1 less than 24, namely 23. Everybody knows that it is a prime number and is composed of two consecutive digits. In fact, it's the only prime whose digits are two consecutive primes.

But the really outstanding characteristic of 23 lies in its role as a number's divisor. [Recall, that was 24's tour de force also.]

If we write the first six prime numbers, forming one single number, like so

## 23571113

then 23 just happens to be a divisor of it.

Next, if we start with 4, then write the primes from 2 to 23, as we did above, we obtain this number:

## 423571113171923

We'll bet you can guess what's next, right? Yes, 23 is a divisor of this monster as well. 'Way to go, 23!

 58

This time we will teach you a new category of numbers, called Smith numbers. A Smith number is an integer the sum of whose digits is equal to the sum of the digits in its prime factorization. Got that? Well, 58 will help us to get a better grasp on that definition.

Observe:

and

## 5 + 8 = 13          2 + 2 + 9 = 13

bAnd that's all there is to it! I suppose we might say that 58's last name is Smith. [Nah! Better not.]

There are several more two-digit Smith numbers; can you find them?

 26

A few years ago the famous junior high math program MATHCOUNTS gave this problem to the competitors:

Last year my age was a perfect square.
Next year it will be a perfect cube.
How old am I?

You have probably assumed by now that the answer must be 26. Good, but that's not all the story. I thought that the idea of a number coming between a square and a cube was certainly interesting, but with all those jillions of numbers out there to choose from, wasn't there another number like 26, tucked between a square and a cube? Or maybe even a cube and a square? I felt that with a little patience and the aid of my calculator --or my computer's spreadsheet-- surely I could find a brother (or sister) for 26.

Algebra would write this situation as: find a solution (x, y), where x and y are integers for this equation:

## y3 = x2 + 2.

Much to my chagrin, I was not successful with either tool. So while becoming suspicious that it might not be possible, I was not willing to throw in the towel yet. I wrote to the good folks at MATHCOUNTS for help. They weren't sure either, so they put the question to two of their volunteer experts, a computer official and a university math professor. The computer person chose to attack the matter by, you guessed it, computer. After checking up to some very high values, he found nothing. (Of course, you know that this is not a proper proof in the eyes of a mathematician.)

The math professor was successful, but in a different way. He proved that it was impossible! He researched the idea and found a proof over 200 years old, using algebra, showing that 26 was the only solution. Who was responsible for that proof? The great Leonhard Euler! Well, if Euler says so, that's good enough for me.

Every Number Is Special. Sure enough, Mr. Henry also mentions this. So I guess it's decided now: 26 is the only number that is one more than a square AND one less than a cube.

[But I can't help but wonder about the reverse case. Algebra would write this as

## x2 = y3 + 2.

Mr. Euler! Where are you?]

 64 & 125

This time we are going to use two numbers, because they work together so well. First, we might point out that 64 is a square, a cube, and a sixth power all at once. And 125 is a cube as well. Those facts by themselves should make them special enough for most purposes.

bBut there's more (as shown to us by Boyd Henry).

He says and we quote:

Select any odd square and subtract 1. Divide the result by 125 (= 53).

Ignore the decimal point and treat the entire quotient as if it were a whole number. This number will always be divisible by 64 (= 43).

Example: Select 37.

372 - 1 = 1368

1368 ũ 125 = 10.944

10,944 ũ 64 = 171

 14

This page of special numbers would not be complete without some mention of one that is important in my family; it's my son age. But since he will some day turn 15, the number 14 won't be so important, right? We must, therefore, look elsewhere to give 14 its rightful place in the family of special numbers. Once again Boyd Henry's book comes to our aid.

He points out an interesting relationship between it and three consecutive powers of 2. Watch.

First, let's find the first 6 digits of the reciprocal of 14, namely 071428. Now separate those digits into three pairs. [Recall how we did this sort of thing in Special Numbers.] This gives us

07 14 28

Now, get ready. Here come those consecutive powers of 2.

07 = 14 Ũ 2-1     14 = 14 Ũ 20     28 = 14 Ũ 21

P.S. You can see a photo of my son at 14 years of age by clicking here.

 1,152,921,504,606,846,976

By way of paying one last tribute to Mr. Boyd and his delightful little book, we will present the final number that he gives and tell you why it is so special, not to mention long.

"Except for 1, the number 1,152,921,504,606,846,976 (1 quintillion, 152 quadrillion, 921 trillion, 504 billion, 606 million, 846 thousand, 976) is the smallest number that is a perfect square, cube, fourth, fifth, and sixth power. It is also a 10th, 12th, 15th, 20th, 30th, and 60th power."

Ah, Boyd, I thought everybody knew THAT!!!

P.S. Check out another big number that stretches up into the quintillions by going to my page called 2 Big Numbers.