In Spring 2013, Villanova math major Stephanie Perez chose to study multiplicative persistence as her senior seminar project. The paper is: Stephanie Perez and Robert Styer, Persistence: A Digit Problem. Involve: a journal of mathematics Vol 8, Issue 3 (2015), pp. 439-446.

Here is a
version of the
paper; the abstract is:

We examine the persistence of a number, defined as the number of iterations of
the function which multiplies the digits of a number until one reaches a single
digit number. We give numerical evidence supporting Sloane's 1973 conjecture
that there exists a maximum persistence for every base. In particular, we give
evidence that the maximum persistence in each base 2 through 12 is 1, 3, 3, 6,
5, 8, 6, 7, 11, 13, 7, respectively.

Here are some supporting Maple calculations: base10, bases3to9highpowers, bases4to11lowpowers, base11high, base12 .

For bigger bases, we have not gone high enough to safely conjecture the true
value, but lower
bounds for the maximum persistence are:

15 for base 13;

13 for base 14;

11 for base 15;

8 for base 16.

In Fall 2016, Villanova math major Colin Lubner studied multiplicative persistence, focusing on the cases when the fixed point is nonzero. Surprisingly, in base 10 the only numbers collapsing to 1, 3, 7, or 9 have persistence at most 1. The same surprising result occurs with relatively prime digits in bases 4, 6 and 12.

The paper also gives the probable maximum persistence for other digits in
base 10:

Theorem.

Let n < 10^1000 be a positive integer.

If f^k(n) =
2, the persistence k is at most 6; only n with f(n) = 2^6*7^8 have persistence
6.

If f^k(n) = 4, the persistence k is at most 5; only n with f(n) =
2^23*3^7*7^1 have persistence 5.

If f^k(n) = 5, the persistence k is at most
5; only n with f(n) = 3^5*5^1*7^2 or 3^2*5^2*7^1 or 3^2*5^2*7^3 have

persistence 5.

If f^k(n) = 6, the persistence k is at most 8; only n with
f(n) = 2^6*3^27*7^1 have persistence 8.

If f^k(n) = 8, the persistence k is
at most 6; only n with f(n) = 2^20*3^5*7^1 or 2^39*3^3*7^2 have persistence 6.

Also, in base 4, our calculations suggest that the maximum persistence of any number collapsing to 2 is 3; in base 6, the maximum persistence of a number collapsing to the digit 2 or digit 4 is 5, to the digit 3 is 2; in base 12, the maximum persistence of a number collapsing to digits 2 or 10 is 3, to digit 3 is 4, to digit 4 is 6, to digits 6 or 8 is 5, to digit 9 is 4.

Here are some supporting Maple calculations: base 12 persistence 1, 5, 7, 11 digits (pdf, mw), base 10 digits 2, 4, 5, 6, 8 persistence bounds (pdf, mw)

Return to Robert Styer Home Page

02/21/2017: