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.


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.


Seiji wrote:

I searched the prime of the form k*2^n+1.
 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



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