Puzzle 1238 Puzzle 1236, revisited
Davide Rotond came up with this new observation related to
the decimal expression of the reciprocal of prime integers,
that have already posted in the
Puzzle 1236. Here is what he now added:
I analyzed the period of 1/7= 0,142857... that we
discovered follows a recursive rule with carry. But now
I want to see if the reverse of the digit of the period
follows a rule too
Take 7,5,8,2,4,1,7,5,8,2,4,1,...
I discovered that starting with the numbers
7,5,8 with the rule a(n)= a(n-2) + a(n-3) & obtain:
7, 5, 8, 12, 13, 10. 6, 5, 8, ...
2. 4 1. 7
Q1: Is it
possible to construct a rule for the reverse of the
digit of the period of every prime number?
Give a list with prime, the rule and the
starting integers needed.
I wrote to Mr. Rotondo that despite that his observation is
remarkable, I find it useless, because before using this
kind of in-reverse-rules all the decimal digits that are
going to be computed by these new in-reverse-recursive
equations, must be calculated in advance.
Nevertheless, here we are dealing with this puzzle. I only
add the following two perhaps interesting
questions:
Q2. For each prime p, the recursive rules-in-reverse
are defined and related in some way with the forward-rules?
Q3. Do you find useful for some purpose the
rules-in-reverse?
