Turning Numbers Inside Out

On one of my visits to Princeton, I stopped by the math department and, as usual, asked John H. Conway what he was up to. He told me that he was turning numbers inside out. He explained that to perform this procedure on a number you need to reverse every prime factor, multiply the reversed factors back and reverse the result. For example, for 34, which is the product of 2 and 17, we need to reverse 2 and 17 (turning inside), changing them into 2 and 71, multiply back, getting 142, and reversing again (turning outside), leading to the resulting number 241.

He started with a number, turned it inside out, then turned the result inside out, and so on, thus getting an infinite sequence for any number. Every sequence he had calculated up to this point ended with a cycle.

Before I had interrupted him, he was calculating the sequence starting with 78 and it was growing. I suggested that Mathematica could do this calculation faster than John could do in his head. Although that was very rude considering his reputation for speed, John agreed, and we moved to a computer. The computer confirmed that the sequence starting with 78 was growing wildly.

While playing around with this, I became very interested in numbers that are fixed under this turning inside-out operation. First, prime numbers do not change — you just reverse them twice. Second, palindromes with palindromic primes do not change, as every reversal encounters a palindrome to apply itself to. I started to wonder if there are palindromes that are fixed under the turning inside-out procedure, but are not products of palindromic primes.

Here is where John had his revenge. He told me that he would be able to find such a number faster than I could write a program to find it. And he won! He found such a number while I was still trying to debug my program. The number he found was 1226221.

Here is how he beat me. If you have two not-too-big primes that consist of zeroes and ones and that are reversals of each other, their product will be a palindrome. And John is really fast in checking primes for primality. See his lesson in my essay Remember Your Primes.

The next day, when I stumbled on John again, he was doing something else. I asked him about the numbers and he told me that he was no longer interested. Initially he had hoped that every sequence would end in a cycle. The turning inside-out operation doesn’t produce much growth in a number. On top of that, prime numbers are stable. That means that if the turning inside-out operation was a random operation with a similar growth pattern, there would have been a very high probability of every sequence eventually hitting a prime. But the operation is not random, as it doesn’t change remainders modulo 9. In particular, sequences that start with a composite number divisible by 3 would never hit a prime. Our experiment with 78 discouraged him by showing no hope for a cycle.

I asked him, “Why not do it in binary?” He answered that he had sinned enough playing with a base 10 sequence.

A year later when I next visited Princeton and saw John again, I asked him if he had published or done something with the operation. He had not. He agreed to submit the sequence to the online database, but only if we came up with a name he liked. And we did. We now call this operation TITO (turning inside, turning outside). Please welcome TITO.

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3 Comments

  1. Leo:

    Congrats!

    Number gossip for 78 should be updated, then.

  2. Martijn:

    Very cool. 🙂

    I think it’s interesting how mathematicians (and I count myself as one) can become so interested in one thing, hoping to find some regularity or proving something fun or useful.

  3. Tanya Khovanova’s Math Blog » Blog Archive » Destinies of Numbers:

    […] 5. TITO. The next interesting example is the TITO operation. TITO is an abbreviation of “Turning Inside, Turning Outside”. By definition, […]

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