Problems & Puzzles: Puzzles Puzzle 24. Primes in several bases Let’s convert a prime base 10 (P_{10}) to it’s corresponding numbers M_{b} in bases b, such that 2<=b<=9 : P_{10} à {M_{2} , M_{3}, M_{4}, M_{5} , M_{6}, M_{7}, M_{8}, M_{9} } Now let’s forget that all these M_{b} numbers are in a bases b other than 10 and let’s ask how many of them keep being primes base 10. I have found 3 primes base 10 (2_{10}, 3_{10} & 379081_{10}) that remain primes in other seven bases of the eight available. For example, 379081 is a prime base 10 that remains prime for his representation in bases 9, 8, 7, 6, 5, 4 & 3 and that it’s composite only in his representation base 2 : Prime =379081_{10} à Primes{637001_{9}, 1344311_{8}, 3136123_{7}, 12043001_{6}, 44112311_{5}, 1130203021_{4}, 201021000001_{3}}, Composite={1011100100011001001_{2 }} a)Can you find 3 more primes base 10 that remain primes in 7 bases b of the other 8 available 2<=b<=9 ? b) Can you find a prime that remains prime in all the 9 bases 2<=b<=10, or give a theoretical reason why this can not be possible ? Solution Question a) Jack Brennen, found (12 October, 1998) not 3 but 4 primes base 10 that remain primes in other 7 bases less than 10: 59771671 is prime in all bases <=10 except in base 4. *** Jack Brennen (who else?) found the 13/7/2001 the least example of prime numbers in base 10 that remains prime when it's expressed in all the bases from 2 to 9 supposing they are in base 10. This the primegem 50006393431 that forces to say: "My goodness!... they exist": In base 10: 50006393431: There are no other solutions <= 280000000000 Proved as it has been that they exist the next challenge is to find 3 more in order to be published in the Nei'ls sequences database. Who says I will try? As a follow up of this result, can somebody calculate how fortunate was Jack in finding this prime number, that is to say the probability of finding a prime like 50006393431 with these properties? *** Giovanni Resta wrote (5/5/03):
Well done! Does anybody wants to add some more entries? *** Giovanni Resta wrote (29/07/18):
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