Problems & Puzzles: Puzzles Puzzle 24.- Primes in several bases Lets convert a prime base 10 (P10) to its corresponding numbers Mb in bases b, such that 2<=b<=9 : P10 à {M2 , M3, M4, M5 , M6, M7, M8, M9 } Now lets forget that all these Mb numbers are in a bases b other than 10 and lets ask how many of them keep being primes base 10. I have found 3 primes base 10 (210, 310 & 37908110) 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 its composite only in his representation base 2 : Prime =37908110 à Primes{6370019, 13443118, 31361237, 120430016, 441123115, 11302030214, 2010210000013}, Composite={10111001000110010012 } 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 (1-2 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 prime-gem 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): I would like to submit 3 new numbers (as requested!)
that are solution of puzzle 24. They are: 727533146383, 2250332130313,
2651541199513, and my search was interrupted around n = 3,103,417,510,867. Well done! Does anybody wants to add some more entries? *** Giovanni Resta wrote (29/07/18):
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