We begin this page of special numbers with one of my all-time favorites: 153. The reason I like it so much, I guess, is that it was the first one I learned about many years ago from a number category called narcissistic. Its unique quality, and what qualifies it as narcissistic, can be stated in these words: it is equal to the sum of the cubes of its digits. Observe:
153 = 13 + 53 + 33 = 1 + 125 + 27 = 153
Now, how'd you like that? Clever, huh! Well, there are only three other numbers that share this same quality. They are also three-place numbers. Can you find them?
Now, here are two more reasons why 153 is special.
If you add up all the whole numbers from 1 to 17, you get 153. This can be expressed mathematically in this way:
This is the same as saying that 153 is the 17th triangular number. (For more about triangular numbers, click here.)
The other reason that 153 is interesting uses the concept of factorial. Recall that n! means "n(n-1)(n-2)...2×1." So look at this expression:
What do you think it equals? Right! 153.
In an email just received from a WTM reader, the following extra curious fact about 153 can be shown...
where SoD(n) means the "Sum of the Digits of n".
Continuing with the theme of 153, with but one slight and rather important change, I will show why 1634 is rather special too. The change this time is that we will use the fourth powers of the digits. This gives this equation:
1634 = 14 + 64 + 34 + 44 = 1 + 1296 + 81 + 256 = 1634
Will numbers ever cease to amaze me? I hope not. There are two more cases like this one. Again we challenge you to find them.
Mathematicians call such numbers as those we've just discussed above as PDI's (Perfect Digital Invariants). A PDI is a number that can be expressed as some combination of its digits and various operations, powers, or whatnot. The two numbers discussed so far used the cubes and 4th powers of the digits. Numbers that use higher powers also exist; some examples include:
54,748 and fifth powers, 548,834 and sixth powers, 1,741,725 and seventh powers.
Do you care to verify these cases, or do you trust me by now? ;)
This special number, which involves cubes in a different way than described above, has a special story to go along with it. Some time ago in the early part of this century, a famous young mathematician from India was visiting the noted English mathematician Hardy in London. Ramanujan, the Indian, had become sick and Hardy had gone to see him. Hardy later wrote of the incident, saying: "I had ridden in a taxi-cab No. 1729, and remarked to my guest that the number seemed to me a rather dull one, and that I hoped it was not an unfavorable omen. 'No', he remarked, 'it is a very interesting number; it is the smallest number expressable as the sum of two cubes in two different ways.'"
Here is what Ramanujan meant:
1729 = 13 + 123 = 93 + 103 = 1 + 1728 = 729 + 1000
By way of asking an easier question so that you may better appreciate this concept:
Three final notes about 1729 that I think are equally interesting:
1st: If you factor it into primes, you get
and those primes form an arithmetical sequence with a common difference of 6, making them "sexy" primes (see Sexy Primes).
2nd: Then by combining 7 and 13 into one factor and rearranging the order, we have
That's sort of a palindromic arrangement of the digits, right?
3rd: If we note that the sum of the digits of 1729 is 19, then this is sufficient to declare that "1729 is a Niven number!" (See Trotter Dates and Niven Numbers.)
Wow! 1729 is some fantastic number, for sure.
I can hear you saying on this one: "Now how can that monster of a number be special?" It's really quite simple if I tell you to use Ramanujan's idea just explained above. Believe it or not, but it is the smallest number that is expressable as the sum of two 4th powers in two different ways.
If I present this in symbols of algebra, it would look like this:
where A, B, C, and D are distinct whole numbers.
Of course, you'd like to know what those letters stand for, wouldn't you? To give them out to you directly is not the way of "Trotter Math"; but I will give you some clues, okay?
Clue #1: One pair of numbers are consecutive, in the 130-140 range.
Clue #2: One number of the second pair is 59.
Now you can find the numbers.
An historical fact about this number is that it was discovered by the brilliant Swiss mathematician Leonhard Euler in the 18th century.
This is a very special number, eventhough at first glance there seems to be nothing unique about it. In fact, it's another narcissistic number but in a way different than what was discussed earlier. To understand why, we need to know the meaning of a special mathematical symbol: "!" That's it; the ordinary exclamation point.
But in math, it is called the factorial symbol. It is used in this way:
We read that as "5-factorial". So it means nothing more nor less than the product of the given whole number with all the smaller ones down to 1.
So what does this have to do with 145, you ask? Well, just observe this neat little statement:
145 = 1! + 4! + 5! = 1 + 24 + 120 = 145See? We are back to 145 again. Now that's special, wouldn't you agree?
In addition to being two dozen and 4! (4-factorial), the number 24 has another interesting characteristic. If you select any prime number, greater than 3, square it, then diminish that by 1, then 24 is always a divisor (factor) of the result.
Here is an example:
1. Choose the prime 17.
2. Its square is 289.
3. Subtracting 1 gives 288.
4. Then 288 divided by 24 gives exactly 12.
See? It works. I kid you not. Now you might wish to test a few more primes of your own choosing, just so you believe it more firmly. (Then later you can amaze your friends with your new found knowledge.)
A good activity here for students of Algebra is to PROVE that it always works. Just using many examples is not considered as proof in mathematics. (However, if many examples do work out, it is, I suppose, an indication that something "might" be true.)
Probably you recognize this number; it's the 6-digit period of the rational number 1/7.
1 --- = 0.142857142857142857... 7It is a favorite of math buffs because it has some unusual characteristics when multiplied by other numbers. But one that seems to be overlooked in most books is the fact that the missing digits are 0, 3, 6, and 9, which are the multiples of 3. (This is an observation that becomes more important when one investigates the periods of other fractions.)
But back to our multiplying idea... Note these products.
142857 × 2 = 285714 142857 × 3 = 428571 142857 × 4 = 571428 142857 × 5 = 714285 142857 × 6 = 857142When you multiply by 7, you get a surprise. Do it!
Here is another trick to show how special the number is. Separate it into two halves, and then add.
142 + 857 = 999
Next split it into thirds before adding.
14 + 28 + 57 = 99
Finally, let's square the number before splitting and adding.
What did you find, my friend?
The year 1992 has been gone for some time now, but the short form of the year '92 is a number that produces a lot of 9's (remember what happened to the previous entry just above?). (I found this idea in a book titled "Every Number Is Special" by Boyd Henry.) It is done in this unusual manner:
Multiply 92 by 8, then that product by 8, then that product by 8, etc. List the products one under the other, shifting the digits two places to the right as shown below. Continue indefinitely. Add. The sum converges to a string of 9s.
92 736 5888 47104 376832 3014656 24117248 9999999...Not to let the full year number be left out, WTM takes pleasure in pointing out that 1992 is sorta nice by itself. Look:
WTM now hopes that you will take a new interest in the world of numbers and try to uncover their inner personalities. They're a little like we people, each unique in some way, yet at the same time, sharing common properties with others.
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