Problems & Puzzles: Puzzles

Puzzle 401. Magnanimous primes

Perhaps you already know the sequence A089392, defined as:

"numbers n with following property. Let the digits of n be abcd. Then bcd+a, cd+ab, d+abc, abcd, etc. must all be primes. If n is a k-digit number then it must produce k such primes"

Amarnath Murthy conjectures that this sequence is infinite, but my own computations do not support clearly this statement. As  a matter of fact I have computed all the numbers of this type and my results are as follow:

 digits Quantity of numbers 2 10 (11, 89) 3 17 (101, 881) 4 15 (2221, 8821) 5 10 (20261, 80849) 6 15 (220021, 864883) 7 7 (2000221, 8608081) 8 2 (20266681, 48804809) 9 4 (228440489, 608844043 )

The largest number of this type I have found is 608844043.

Questions.

1. Can you argue about the Murthy's Conjecture?
2. Can you find a number of this type larger than 608844043?

Contributions came from J. K. Andersen, Farideh Firoozbakht & Giovanni Resta.

***

Jens Kruse Andersen wrote:

I searched the largest given number 608844043 in Google.
All given numbers are in http://www.research.att.com/~njas/sequences/A089393
and http://www.research.att.com/~njas/sequences/A089394.
David Wasserman comments "no members with 10, 11, or 12 digits.
It is unlikely that it has any with more than 12 digits".

http://primes.utm.edu/curios/page.php/608844043.html says:
"No larger prime of this type is known. [Rivera]"
Archive.org first shows the curio in January 2004 where it said:
"I conjecture that 608844043 is the largest prime of this type. [Rivera]"
http://web.archive.org/web/*/http://primes.utm.edu/curios/page.php/608844043.html

Based on some quick heuristics, I estimate roughly 1% chance of
a solution above 12 digits, and no chance of infinitely many solutions.

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Farideh wrote:

There is no 10-digit Magnanimous prime.

But for the 10-digit composite number 2240064227 all nine numbers 2+240064227,
22+40064227, 224+0064227, 2240+064227, 22400+64227, 224006+4227,
2240064+227, 22400642+27 & 224006422+7 are prime.
For the comosite numbers like 2240064227 we have the following similar table.

 digits Quantity of numbers 2 23 (12, 98) 3 62 (110, 998) 4 89 (1001, 9910) 5 102 (11116, 99998) 6 81 (111112, 999994) 7 64 (1115756, 9959374) 8 33 (11771992,95559998) 9 14 (117711170, 995955112 )

Note that for each k-digit composite such number it must produces k-1 primes.
Namely for n=abcd the three numbers a+bcd, ab+cd & abc+d must all
be primes.

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Giovanni wrote:

I've checked all the numbers up to 5*10^16 and I did not find any other
magnanimous prime.
I guess that there can be only a finite number of such primes.

The probability that a candidate of n digits can be magnanimous
can be upper bounded considering that we want to obtain n prime
numbers (with a number of digits ranging from about n/2 to n).
The probability that a number of n/2 digits is prime is very roughly
1/log(10^(n/2)), that is, 0.86/n. Since we want to have n such primes
formed by the digits of the candidate
this will happen with a probability less than
(0.86/n)^n. Multiplying this by the number of candidates (about 10^n) we
obtain
something like (8.6/n)^n which decreases very fast once n>8,
and even summing it up for every n up to infinite, we obtain a finite
number.
This very sketchy argument suggests that there are probably
only a finite number of magnanimous primes.

I have also investigated the "magnanimous numbers", which can be
defined as the magnanimous primes, but dropping the requirement
that the number itself is prime. That is, we only require that inserting
a "+" somewhere always gives a prime number. (Somehow I prefer them,
since the definition is more "symmetric").
I've checked all the numbers with up to 15 digits. The largest magnanimous
number I found is 97393713331910.
The stats are the following: (better viewed with a fixed width font)

d # min max
-------------------------------
2 32 11 98
3 78 110 998
4 103 1112 9910
5 112 11116 99998
6 96 111112 999994
7 71 1115756 9959374
8 35 11771992 95559998
9 18 117711170 995955112
10 6 1777137770 9151995592
11 5 22226226625 84448000009
12 0
13 1 5391391551358
14 1 97393713331910
15 0

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