Numbers Are Forever by Liz Strachan

Author:Liz Strachan
Language: eng
Format: epub
ISBN: 9781472111104
Publisher: Constable & Robinson

The Repunits

11 is a repunit. ‘Repunit’ means unity or the digit 1 repeated. Since the word does not appear in the Oxford English Dictionary, you won’t know how to pronounce it. You say, ‘re-pew-nit’. The ‘pun’ bit does not rhyme with ‘fun’ although the repunits are fun – well, mathematical fun, anyway.

At a board meeting of repunits, it was decided unanimously that 1 could not become a member of the exclusive repunit club as it only had one leg. However, 1 is a member of so many other clubs including the Happy Number Club, where it is the very important founder member who decides which other numbers may join the club. And in the famous Fibonacci sequence, 1 is not only the first number, it is also the second!

So ‘legs 11’ is the smallest repunit and the others, 111, 1111, 11111… go on forever. 111 and every 3rd one after that divides by 3; repunits with an even number of legs divide by 11. Some of them are chock-a-block with prime factors. The one with 30 legs, R30 = 3 × 7 × 11 × 13 × 31 × 37 × 41 × 211 × 241 × 271 × 2161 × 9091 × 2,906,161. (Don’t worry, that last factor is prime. Greater brains than ours have proved it.) Prime number repunits are very rare. Little legs 11 is a prime, of course, but the only other primes in the first 100 repunits (yes, that’s all the way up to a 100-legged monster) are R19 and R23. Beyond that, there is only one more known repunit prime and it has 317 legs. To write it out on A4 paper, with single spacing between the units, takes over six lines.

You don’t really need a formula for repunits but (10n – 1) ÷ 9 fits them all.

When squared, the repunits make pretty patterns:

11 × 11 = 121

111 × 111 = 12321

1111 × 1111 = 1234321

And up and up you can go to:

111111111 × 111111111 = 12345678987654321

It doesn’t work after that because a 10-legged repunit multiplied by another 10-legged repunit would have 10 in the middle of the answer, necessitating a carried 1, and the pattern becomes very messy. But 13 legs × 13 legs is nicely symmetrical and equals:

12345678900987654321

But that isn’t all. Continuing from the above pattern:

121 = 22 × 22 ÷ (1 + 2 + 1)

12321 = 333 × 333 ÷ (1 + 2 + 3 + 2 + 1)

1234321 = 4444 × 4444 ÷ (1 + 2 + 3 + 4 + 3 + 2 + 1)

… all the way to the monster:

12345678987654321

= 999999999 × 999999999 ÷ (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1)

Now, if you add the digits of the above squared numbers of repunits from 2 digits long to 9 digits long (121, 12321, etc.), you get 4, 9, 16, 25 … yes, the squares of 2 to 9.

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