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Old 2020-02-14, 13:06   #1
sweety439
 
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Nov 2016

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Default Always composite numbers?

Can a dozenal (duodecimal, base 12) number {111...111}21{111...111} (with the same number of 1s in the two brackets) (start with 21, 1211, 112111, 11121111, ...) be prime? I cannot find such prime with <=1000 (decimal 1728) digits, but find neither covering set nor algebra factors. (I have proved that dozenal (duodecimal, base 12) numbers 1{555...555}1 (start with 151, 1551, 15551, 155551, ...) cannot be primes, because of covering sets and algebra factors)

Also, can a dozenal (duodecimal, base 12) 414141...4141411 (start with 411, 41411, 4141411, 414141411, ...) be prime? I want to find the dozenal (duodecimal, base 12) analog of A086766, and I proved that there are no primes for n = 10 (decimal 12) and n = 33 (decimal 39), because of algebra factors, thus the conjecture in A086766 is not true in dozenal (duodecimal, base 12).

Besides, I want to find the dozenal (duodecimal, base 12) analog of many other OEIS sequences, such as A088782, A069568, A200065, A272232, A089776 (n%12 = 1, 5, 7, 11 instead of n%10 = 1, 3, 7, 9), A267720, A244424, A262300, A046035, A047777, A060421, A064118.

For A069568 case, I found the prime (12^1676*298-1)/11 for n = 23 (decimal 27) and proved that there are no primes for n = 34, 89, and 99 (decimal 40, 105, and 117), because of covering sets and algebra factors, but for the A089776 case, I cannot find prime for n = 65 (decimal 77) and n = EE (decimal 143).

Last fiddled with by sweety439 on 2020-02-14 at 13:30
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