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.

Contributions came from Claudio Meller, Maximilian Hasler, Carlos Rivera & Emmanuel Vantieghem.

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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

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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?

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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).

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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 ...