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John Derbyshire

There are innumerable fascinating questions here. I’ll just offer a glimpse of one.

Think of a number, not necessarily a whole number. (And if it’s negative, that’s fine, though there is not much point.) Square it (see?), then add 1. Take your answer, square it, and add 1. Take the answer to *that*, square it, and add 1. Continue forever.

A bit more formally: Think of a number *u*_{0}. Form the new number *u*_{1} =*u*_{0}² + 1. Then *u*_{2} =*u*_{1}² + 1, and so on.

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However formally it’s posed, the process gives you an infinite sequence of numbers *u*_{0}, *u*_{1}, *u*_{2}, *u*_{3}, *u*_{4}, *u*_{5}, *u*_{6}, . . . , each one of which is bigger by 1 than the square of the previous one. The question is: Do the leading digits of the*u*’s follow Benford’s Law?

The answer depends on your initial choice of *u*_{0}. Taking the set R of all possible real numbers as our domain of interest for *u*_{0}, it divides into two subsets: one — call it*B _{Q}* — of all the

It can be proved that both *B _{Q }*and

The loose way mathematicians use to describe a situation like this is: *Almost all *real numbers are in*B _{Q}*. An alternative, even looser way: Though

If you have a decent math/stats package, or are serving a long prison sentence, you might want to try that out for some particular real number.

π is an obvious candidate. Then *u*_{0 }is 3.141592 . . . Square and add one: That gives *u*_{1} = 10.869604 . . . Squaring that and adding one, *u*_{2} = 119.148299 . . . Proceeding, you get *u*_{3} = 14197.317353 . . . , *u*_{4} = 201563821.046002 . . . , *u*_{5} = 40627973954664757.854260 . . . and so on. Leading digits are 3, 1, 1, 1, 2, 4 . . . Hey, it looks Benford-compliant already! (It is.)

That’s all well and good, you may say, but how do I *know*, without experimentation, whether some particular value of *u*_{0 }will generate a Benford-compliant sequence under the rule *Q*? (With, or course, an infinity of corresponding questions for an infinity of other rules you might think up.)

So far as I know, there is no general method for determining this, though you can of course always take the experimental-math approach: Generate the sequence out to *u _{bazillion }*or so and run the stats on its leading digits, as I just did for the primes. With

It is possible to figure out particular numbers that*don’t*generate Benford-compliant sequences — numbers, that is to say, that are members of the set *B′ _{Q}*, even though the chance of any given number being in

**u**_{0}= 9.94962308959395941218332124109326 . . .

as your starting value for the rule *Q*, then *every single number in the sequence *has leading digit 9. How noncompliant is *that!*

Can you figure out why? Or find a closed form for this number?