The Fascinating World of Number Persistence

Every so often, a number comes along that makes mathematicians stop and wonder. One such number is 277777788888899, which has captured attention because of its remarkable multiplicative persistence. But what does that actually mean?

What Is Multiplicative Persistence?

The persistence of a number is the number of times you apply an operation to an integer until that operation can no longer change it due to its reduction. This persistence can be additive or multiplicative. Today, we're focusing on multiplicative persistence.

A key goal in studying persistence is finding the smallest numbers with the greatest number of steps. The prevailing hypothesis suggests that no numbers exist with a persistence greater than 11. That's where the strange 277777788888899 comes into play. What makes it so special?

Let's run a quick test. Pick any four-digit integer. I'll choose, say, 7793. Calculating its multiplicative persistence is quite simple: you multiply its digits, take the result, and repeat the process until it's no longer possible. Let's see:

  • 7 × 7 × 9 × 3 = 1323
  • 1 × 3 × 2 × 3 = 18
  • 1 × 8 = 8

The multiplicative persistence of 7793 is just 3 steps. Now, let's drop the four-digit rule and think of any number. Your mission, should you choose to accept it, is to find one that exceeds 11 steps of persistence. But beware: it's much harder than it looks. How about a demonstration?

The Persistence of 277777788888899

In a recent video published by the Numberphile channel, they explore the peculiar condition of 277777788888899. An interesting variant of the challenge is finding the smallest numbers for a specific persistence, and 277777788888899 is the smallest with persistence 11.

Initially, they spend a couple of minutes on the traditional calculation, but those with some programming knowledge might be more interested in the code they write for automatic computation. The manual reduction of 2 × 7 × 7 × 7 × 7 × 7 × 7 × 8 × 8 × 8 × 8 × 8 × 8 × 9 × 9 follows this pattern:

  • 4996238671872
  • 438939648
  • 4478976
  • 338688
  • 27648
  • 2688
  • 768
  • 336
  • 54
  • 20
  • 0

The big question is whether a number with persistence greater than 11 truly exists. Numberphile recommends not searching below 10^233 (they haven't found one), avoiding the digit 5 altogether, and prioritizing the use of 7, 8, and 9. If you can write code or think you can improve on Numberphile's, you're invited to try.

https://old.neoteo.com/ramanujan-bot-extension-para-resolver-ecuaciones-y-problemas-matematicos/

(From the NeoTeo Archive, article originally published on March 29, 2019)