Problems & Puzzles: Puzzles

Puzzle 193. The Andersen's theorem & primes

Very recently, 9 April 2002, Joseph L. Pe (1, 2, 3) defined the "picture-perfect numbers (ppn)" the following way:

n is a ppn if the reversal of n is equal to the sum of the reversals of the proper divisors of n

Example

• The proper divisors of n = 10311 are 1, 3, 7, 21, 491, 1473 & 3437
• 11301 = 1+3+7+12+194+3741+7343
• So, 10311 is a ppn

Here are the known earliest seven ppn numbers, as consigned in the EIS, A069942:

6, 10311, 21661371, 1460501511, 7980062073, 79862699373, 798006269373

The ppn definition seems so fancy that one would hardly expect to find an interesting theorem associated to the ppn numbers. Nevertheless, this is exactly what was achieved by Jens Kruse Andersen:

Andersen's Theorem

If p=140{(0)z10(9)n89}k is prime, then 3*19*p is a ppn , and conversely. Here:

• (0)z is a string of z=>0 "zeros"
• (9)n is a string of n=>0 "nines"
• k is the number of repetitions of the part {(0)z10(9)n89} with varying numbers of zeros and nines in each repetition.

Honoring the author of this theorem, any prime of this type p=140{(0)z10(9)n89}k is named Andersen's prime. See the earliest Andersen primes at A075130 and the corresponding ppn numbers at A075131

Using this form Andersen has told me by email that he has produced many large primes - and the ppns associated - being the largest prime a 2461 digits one, 140*10^2458+1089, verified by PRIMO). Moreover, Andersen has some empirical evidence that suggests that there are infinite Andersen's primes

On the other hand, a fast examination of the known earliest seven ppn numbers shows that the first four ppn numbers are out of the Andersen's rule. This may be indicative that it could exists some other theorems similar, to the Andersen's one, waiting to be discovered.

In any case, all known ppn numbers (other than 6) are odd and divided by 3. As a matter of fact Mark Ganson conjectures that every ppn is divided by 3.

The logical questions are:

1. Are odd all the ppn numbers (other than 6), or can you find an even one?
2. Are divided by 3 all the ppn, or can you find a counterexample of this?
3. Can you find one single ppn out of the Andersen's theorem scheme, other than the first four of the sequence
4. Can you devise another theorem alike to the Andersen's one?
5. Are there an infinite quantity of Andersen's primes (p=140{(0)z10(9)n89}k)?

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