Problems & Puzzles: Puzzles

Puzzle 546.- Non-Pandigital primes

Anton Vrba sent the following nice puzzle.

We define non-pandigital expressed primes so that both the expression and the prime is non-pandigital, and all digits used in the expression are also used in the prime. We define the prime as unique if no digit in the expression is repeated.

Examples:

Unique non-pandigital prime:  A 37 digit prime that excludes the digit 3 and no digit is duplicated in the expression

6954 x 2^108 - 7 =  2256702022140699458050066967087284217

General non-pandigital prime: An 85 digit prime that excludes the digit 0 with duplication in the expression (both 6 and 7 are repeated)

27654 x 3^168 - 97 = 3963955326857335459158623686139952966832155324827848754999217785438166562392129386597

We define the quality of the prime as the ratio of the number of digits of the prime and number of digits used in the expression.

Thus the Q for above two examples are 37/9=4.11 and 85/11= 7.73 respectively. Solutions using powers of 10 or multiples of 10 are considered trivial

Q. Find other non trivial examples of higher quality for both types.

Contributions came from Seiji Tomita.

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Seiji wrote:

I searched the prime of the form k*2^n+1.
Condition:
k < 10^6.
n < 10^3.
Excludes the digit 0.
General non-pandigital prime.

Following prime is 102 digit and has the Q=102/11=9.27.

759486*2^319+1= 811126124981714157418932727765944228655574823433282392817
999421279233185327117784387369691779556179969

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