Problems & Puzzles: Puzzles

 Puzzle 1050. A follow-up to Puzzle 1049 While solving the Puzzle 1049, Emmanuel Vantieghem found an infinite chain of composite terms starting in 3345, that obviously is not a solution to Puzzle 1049, but... Q. Is there a prime that generates an infinite chain solution to Puzzle 1049, as the composite 3345 almost does?

Contributions during the week 14-19 of August came from Giorgos Kalogeropoulos, Oscar Volpatti and Emmanuel Vantieghem

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Giorgos wrote:
3345 produces the following infinite chain 3345->3435->3435->3435->3435...
So, 3435 repeats forever. This happens because 3435 has a unique property: 3435 = 3^3 + 4^4 + 3^3 + 5^5.

3435 is called  "Munchausen Number" and no  other integer in base 10 has this property (except 1).

Here is a paper on this number: https://arxiv.org/pdf/0911.3038.pdf
The first prime number that produces the same infinite chain is 1122224353->3453->3435->3435->3435...

So, a prime number that repeats itself forever in an infinite chain does not exist.

On the other hand, in order to have an infinite chain, there may exist a set of primes that create a loop which repeats forever. I believe that finding such a loop is a very hard task...

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Oscar wrote:

About puzzle 1050, no prime generates an infinite chain of primes.

Let's drop zeroless constraint by defining 0^0 = 1.
Now every integer generates an infinite chain of integers.
I'll prove that any such chain eventually enters some loop of finite period, involving only numbers below 10^10.

Inequality 1: if x >= 10^10, then f(x) < x.
If x has m digits, we have the bounds  x >= 10^(m-1)  and  f(x) <= m*9^9;
inequality m*9^9 < 10^(m-1) also holds for every m >= 11.

Inequality 2: if x < 10^10, then f(x) < 10^10.

As x has 10 digits or less, we have the bound  f(x) <= 10*9^9 < 10*10^9 = 10^10.

If the starting value is above 10^10, then next values are strictly decreasing, until a value below 10^10 is reached.
As the following values remain below 10^10, some value is eventually repeated, starting a loop of finite period.
QED.

In particular, an infinite chain of primes must enter a loop involving only primes below 10^10.

But no such loop exists: even after dropping zeroless constraint, every prime below 10^10 is followed by at most three more primes before reaching a composite.

However, there are exactly five loops involving only composites below 10^10; the first of such loops is also zeroless.

Loop 1, with period 1:
3435, 3435, ...

Loop 2, with period 2:
16780890, 421845123, 16780890, ...

Loop 3, with period 3:
3418, 16777500, 2520413, 3418, ...

Loop 4, with period 11:
824599, 791621579, 776537851, 19300779, 776488094, 422669176, 388384265, 50381743, 17604196, 388337603, 34424740, 824599, ...

Loop 5, with period 40:
3388, 33554486, 16830688, 50424989, 791621836, 405114593, 387427281, 35201810, 16780376, 18517643, 17650825, 17653671, 1743552, 830081, 33554462, 53476, 873607, 18470986, 421845378, 34381644, 16824695, 404294403, 387421546, 17651084, 17650799, 776537847, 20121452, 3396, 387467199, 793312220, 388244100, 33554978, 405027808, 34381363, 16824237, 17647707, 3341086, 16824184, 33601606, 140025, 3388, ...

What about a prime followed by an infinite chain of composites?

I found the smallest prime for each loop.

Loop 1 reached after 2 steps from prime p = 1122224353:
1122224353, 3453, 3435.

Loop 2 reached after 4 steps from prime p = 2081:
2081, 16777222, 2517298, 405024382, 16780890.

Loop 3 reached after one step from prime p = 1483:

1483, 16777500.

Loop 4 reached after 10 steps from prime p = 881:
881, 33554433, 6870, 17647416, 1740912, 388244295, 420978593, 792445150, 388250800, 50334807, 17604196.

Loop 5 reached after 19 steps from prime p = 2:
2, 4, 256, 49785, 405024629, 387470792, 406668622, 16964105, 387517185, 35207797, 389894275, 809222365, 404247526, 874101, 17601018, 17647420, 1694260, 387514063, 17650852, 17653671.

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Emmanuel wrote:

Let's define  F(m)  as the sum of the digts of  m  each powered to themselves.
There is no other number than  3435  that is mapped on itself.
I examined the absolute value of the difference  m - F(m)  for all zerofree numbers with  n  digits.
This is what I found :

n     Min | m - F(m) |
2          1          for  m = 32, F(m) = 31
3          20        for  m = 241, F(m) = 261
4          0          for  m = 3435 = F(m)
5          123    for  m = 56155, F(m) = 56032
6          1440    for  m = 141666, F(m) = 140226
7          266      for  m = 2473775, F(m) = 2474041
8          1          for  m = 34378338, F(m) = 34378339
9          7          for  m =  792445398, F(m) = 792445305
10         4          for  m = 1179958669, F(m) = 1179958665
No need to search for  n > 10 since the value of  F(m)  then is at most  n*9^9
while  m  is at least  (10^n - 1)/9.

I made similar computations for  | m - F(F(m)) |  and found  0  only when  m  was  3435.

So, no chain of the form  a -> b -> a  exist.

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