BackNumbers: givers of joy
When you play with numbers and find patterns, you learn something, you get stuff to think about
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Ponder the number 297. Square it (i.e., multiply it by itself). You get 88,209. Separate the digits to get two numbers, 88 and 209, and add them. Result: 297, your original number.
As you can imagine, not all numbers have this intriguing property.
Very few do. Below one thousand, there are these: 1, 9, 45, 55, 99, 297, 703 and 999. Though here is the billionth such number:
67939724934973926389828284012602181246021248515976590320719465368819692049657102676181412500787000603460107802882499378473201373667679461635359309773483851454476557937963244208934.
(Fear not. I will not ask you to square it).
These are called Kaprekar numbers, after Dattatreya Ramachandra Kaprekar, who first described them. A schoolteacher in Nashik until he retired in 1962, he had won the Wrangler Paranjpe mathematics prize while studying at Pune’s Fergusson College in 1927. He had no other university training in mathematics.
To wannabes like me, it’s an inspiration that he had his lifelong interest in numbers, studying them and publishing widely. For there’s something infinitely fascinating about amateur mathematicians who play with numbers and discover new things.
And if he found and described these Kaprekar numbers, that’s just a start. For example, there’s 6,174. Try this game:
Choose a four-digit number. Say 4,681. Rearrange its digits in descending order, thus 8,641. Subtract from this its reverse, 1,468, the number formed by rearranging the original digits in ascending order.
You get 7,173. Repeat the routine with this number: from 7,731, subtract 1,377 to get 6,354. Do it again: from 6,543, subtract 3,456 to get 3,087.
Again: 8,730-378=8,352. Again: 8,532-2,358=6,174. Again: 7,641-1,467 = 6,174...hmm, we have the same number, 6174, twice in a row, and clearly we will now get it every time. But mull over this curious thought: whatever four-digit number you start with—with certain exceptions, such as 8888—you’ll end up with 6174. Who first hit on this nugget, in 1949? Well, there’s a reason we call 6174 the Kaprekar constant.
Or try this: Is any given number divisible by the sum of its digits?
Yes, some are, and they’re not as infrequent as Kaprekar numbers. I imagine Kaprekar leaping around Nashik in delight every time he hit on another such number. That delicious vision, because he named them “Harshad", Sanskrit for “giver of joy".
The single digits, 1 through 9, are trivially Harshad numbers, and so is 10, 12, 133, 152 and 198 are Harshad numbers. Take 133:1+3+3=7, and 133/7=19. Or 198:1+9+8=18, and 198/18=11.
Number theorists have taken Harshad numbers and run off in all kinds of directions. For one example, notice that the first 10 Harshad numbers, 1 through 10, are consecutive. (Yes they are). Later in the sequence, 110, 111 and 112 are consecutive Harshad numbers, as are 152 and 153.
Mathematicians love patterns like these. So they wondered: are there other sequences of consecutive Harshad numbers? Are there sequences longer than 10? If so, how many are there? Is there a longest such sequence?
You’ll be overjoyed to know that the mathematicians Curtis Cooper and Robert Kennedy at Central Missouri State University supplied answers to these questions. In a 1994 paper in The Fibonacci Quarterly, they showed that while a sequence of more than 20 consecutive Harshad numbers is impossible, there are infinitely many 20-number sequences.
Though please don’t go searching for one such right now. The smallest begins with a number that has—I am not making this up—way over 44 billion digits.
(Which number may approach the sum of money another Harshad once made, but that’s another story).
All this, from what a Nashik schoolteacher found by simply playing around with numbers. For me, that’s the truly fascinating thing about Kaprekar. He was known to say of himself: “A drunkard wants to go on drinking wine to remain in that pleasurable state. The same is the case with me in so far as numbers are concerned." Perhaps this is why he is often called a “recreational" mathematician, though that is no kind of slight. I’d suggest that every passionate mathematician began with recreation: play with numbers, toss them at each other, look for sequences, patterns and oddities. Because when you play and find those things, you learn something about numbers, you get stuff to think about, you push the boundaries of knowledge that bit further.
And once you start looking, there’s no end to patterns and oddities.
For example, take a number and add its own digits to it to generate another number (e.g. 49 is generated from 38, because 38+3+8=49). Not very interesting? But you might wonder, are there numbers that cannot be generated like this? Like 20 and 97? Ah, you think, here’s something to pursue!
Except, they have already been pursued. These are called self numbers, though the man who first studied them called them Devlali numbers.
His name? That’s right: D.R. Kaprekar.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences.
Comments are welcome at dilip@livemint.com. To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza
Follow Mint Opinion on Twitter at https://twitter.com/Mint_Opinion
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