How to formally prove global convergence to the 7-cycle for the digit-product map R(n) = P(n) + L(n)^2?

[NG. Randiv Singha]

Let P(n) be the product of the base-10 digits of n∈Z+, and let L(n) be the number of digits of n. Consider the map R:Z+→Z+ defined by:

R(n)=P(n)+L(n)2

Question

Can we formally prove that for every starting seed x0∈Z+, the iterated sequence xm+1=R(xm) always terminates in the periodic 7-cycle 6→7→8→9→10→4→5→6? Specifically, how can we analytically rule out alternative periodic cycles or orbits escaping to infinity?

[Allan Wechsler]

Randiv,

If I understand this procedure correctly, I think it can be proved fairly easily that R(n) < n for all n greater than some small threshold, which I'm pretty sure is 10. That alone prevents infinite escapes: the trajectory of all starting numbers eventually gets below that threshold. Exhaustive study of the numbers below the threshold then shows the uniqueness of the given limit cycle.

I think that all bases behave pretty much the same way. If this is the case, then any starting number n in any base will terminate in the corresponding limit cycle in a number of steps bounded above by some constant times log(n).

-- Allan

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[M F Hasler]

On July 13 2026, 11:21, NG. Randiv Singha:

Let P(n) be the product of the base-10 digits of n∈Z+, and let L(n) be the number of digits of n. Consider the map R:Z+→Z+ defined by: R(n)=P(n)+L(n)2 

> Can we formally prove that for every starting seed x_0 ∈ Z+, the iterated sequence x_m+1=R(x_m) always terminates in the periodic 7-cycle 6→7→8→9→10→4→5→6 ? Specifically, how can we analytically rule out alternative periodic cycles or orbits escaping to infinity?

 

If 10^(L-1) <= n < 10^L, then P(n) <= 9^L.

Therefore R(n) = P(n) + L² < n

(and more strictly < 10^(L-1)) for L large enough: 

P(n) + L² <= 9^L + L²  < 10^L  <=> L² < (10/9)^L

<=>  2 log L < L log(10/9)  <=> (log L)/L < log(10/9)/2 <=> L > 84.

So at least beyond 10^84, the map always brings down any integer to a smaller value.

But  P(n)=9^L  is an extreme case that happens only for n=10^L-1

With a little effort, you can certainly show that R² (twice applied) is *always* significantly smaller,

and bring down the limit within which a loop might occur to a much smaller value.

- Maximilian

[Gareth McCaughan]

As Brad Klee observed, the point is that unless n is rather small R(n) is smaller than n.

Here's the more detailed version.

Suppose the decimal digits of n, right to left, are d0, d1, ..., d(k-1), so k is the number of digits. Then P(n) <= d0 . 9^(k-1) while n >= d0 . 10^(k-1), so P(n) <= n - (10^(k-1) - 9^(k-1)), so R(n) <= n + k^2 - (10^(k-1) - 9^(k-1)).

Write f(k) := (10^(k-1) - 9^(k-1)) - k^2; then R(n) < n whenever n is a k-digit number and f(k) > 0. We have f(2) = (10-9)-4 = -3, f(3) = (100-81)-9 = 10 > 0, f(4) = (1000-729)-16 = 255 > 0, and in fact f(k) > 0 whenever k>2; I'll leave proving this as an exercise for you.

So if n >= 1000 we have R(n) < n. So iterating R(R(...(R(n))...)) must eventually produce a number <= 999, so every number ends up in _some_ cycle that includes a number <= 999.

We can then get a computer to use a few milliseconds of brute force to verify that starting with any number <= 999 you end up in the 4,5,6,7,8,9,10 cycle.

So anything <= 999 ends up in that cycle (proof: brute-force checking), and anything > 999 gets mapped to smaller and smaller numbers until it's <= 999, at which point it again must end up in that cycle. So everything does.

(I bet that with a bit more brainwork we could reduce the amount of brute force needed to a level that would be easy without computers, but it's easier not to :-).)

PS. I agree with Brad: don't name things after yourself. If you find something interesting enough then maybe someone else will name it after you, and that's much more worthwhile. (Because anyone can name anything after themselves, and all it shows is that they like their own name; but if _someone else_ thinks it's worth naming something after you, then that shows that someone else thought what you did was valuable.)

-- 
g