Problems & Puzzles:
Puzzles
Puzzle 607.
A zeroless Prime power
Shyam Sunder Gupta in his interesting site "Number
Recreations", published the
CYF40 asking
for: A zeroless Prime power p^n�(n >1) as large as possible.
The best solution published there is:
6161527^21 =
383226448283881161158925442286272657358177586728919276714197
431918815293137131182142282513485878467995834596155592319882
82189668125135144387927 (143 digits), solution by SSG, Feb. 2011.
This week I made a little search using a code in Ubasic,
looking for "d-less prime powers", for d equal any decimal digit
from 0 to 9.
As a matter of fact I found six larger solutions than
6161527^21, being the largest of these the following
one:
92364991^22, zeroless, 176 digits.
Q1. Send only your largest d-less prime
power.
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|
� |
Contributions came from Claudio Meller, Maximilian Hasler, Carlos Rivera & Emmanuel
Vantieghem.
***
Claudio wrote:
Here are my best results:
|
d |
p^n |
digits |
|
1 |
� 20019953^22 � |
� �161� |
|
2 |
� 501570439^16 |
� �140 |
|
3 |
� 27793541^23 � |
� �172 |
|
4 |
� 29536943^23 � |
� �172 |
|
5 |
� 18655619^20 � |
� �146 |
|
6 |
� 25240057^21 � |
� �156 |
|
7 |
� 23663407^18 � |
� �133 |
***
Maximilian found (empirically?) that for a fixed n value, (10^n)\3+k or
(10^n)\6+k is at the same time prime and this prime squared is zeroless
after trying a few k values in a short range.
His three examples are valid solutions for this puzzle, nut they are not
only zeroless because frequently they lack digits other than zero. Nevertheless his method is a good guide in order to seek
for only zeroless squares, it just need to test a few more k cases.
Examples:�
- The square of the prime (10^89)\6+1773 is a
zeroless (& 4-less) solution,177 digits
- The square of
the prime
(10^89)\3+130� is a zeroless (& 4 & 8-less) solution, 178 digits.
- The square of
the prime
�(10^500)\3+10210
is a zeroless (&3-less) solution,
1000 digits.
He proposed to "add some "merit" function to the puzzle (giving a
better score to smaller primes and larger exponents) to make it more
interesting..."
Q2. Any Merit Index proposal?
***
Carlos Rivera tested the Maximilian approach (to the limit of his Ubasic code) and got
two more results, confirming the effectiveness of his solution by counting
the primes of the form 10^n/3+k or
10^n/6+k needed to test for a given fixed n value. As a matter of fact the
primes needed to test are very few, never more than 8, in my runs.
Examples:
-
The square of the prime (10^1300)\6+5767 is a
zeroless (& 5-less) solution, 2599 digits.
-
The square of the prime (10^1300)\3+15380 is a only
zeroless solution, 2600 digits.
BTW, due that the Maximilian's approach focus on zeroless squares,
he has solved at the same time the question Q2 of an old puzzle of these
pages ( Puzzle 195,
Primes such that their squares are free of the digit D ).
***
Emmanuel wrote:
To find a big prime power with a missing digit, we can proceed as
follows:
choose a small��numbers��a, b��and find a prime number��p��of the
form��a10^n +b.��Then, there may exist very small��k��such
that��p^k�misses one or more digits.��If��n��is very big,
then��p^k��is also very big.��My biggest example is this : p =
19x10^1169+1��(is prime, proved with PRIMO in about 30
minutes)��and��p^6��has��7022��digits, none of them being 2.
However, this will not work if you want to find �0-less numbers ...
Later he added:
�I found�the
prime�p = 10^1470 + (10^1470 - 1)/3 - 2 = 133...331.� Its square has
2941 digits among which no zero (neither 2,3,4,8 & 9)
***
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