20210804, 06:33  #1 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
6474_{8} Posts 
Generalized nearrepdigit primes
Are there any interest to search the smallest generalized nearrepdigit prime in base b of a given form xyyy...yyy or xxx...xxxy (where x,y are base b digits)? I found there is already this search for the form 444...4445 in base 6, see the OEIS sequence A248613.
I have searched all possible forms in bases up to 36, here is my data, "b, x, {y}" means xyyy...yyy form in base b, "b, {x}, y" means xxx...xxxy in base b, and the prime given is the smallest prime of this form (including the repunit case, i.e. 111...111, which have x=y=1) (note: we allow the prime xy to be the prime of the form xyyy...yyy and xxx...xxxy, but we do not allow x to be the prime of the form xyyy...yyy, neither allow y to be the prime of the form xxx...xxxy), my list skips the forms which have NUMERICAL covering set (e.g. gcd(x,y) > 1, which are all divisible by gcd(x,y), or gcd(y,b) > 1, which are all divisible by gcd(y,b), or 5111...111, which is always divisible by either 2 or 5), but does not skip the forms which have a full covering set with all or partial ALGEBRAIC factors (e.g. 3111...111 in base 9, which factored as difference of squares, or 8DDD...DDD in base 14, which is divisible by 5 if the number of D's is odd, and factored as difference of squares if the number of D's is even), I know that exactly which forms can be proven composite, and the forms which I cannot find a prime nor can prove as only contain composites includes 5777...777 in base 11 and 9555...555 in base 13, can someone find them? Two interesting cases are the form 777...7771 in base 13, whose smallest prime has 1504 7's, and the form 999...9995 in base 13, whose smallest prime has 1362 9's. Last fiddled with by sweety439 on 20210804 at 06:55 
20210811, 17:56  #2 
Aug 2020
79*6581e4;3*2539e3
503 Posts 
That's a nice list, but it would be good if you indicated how of {n} is repeated. I know you give the prime in base10, but it'd be nice to see at a glance how the number looks like in baseb.
To have an organized search someone would have to setup a website, there are so many possible nearrepdigits in the larger bases a single text file gets hard to follow. 
20210811, 18:34  #3  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
6474_{8} Posts 
Quote:
Besides, these forms are proven to only contain composite numbers: (trivial cases: gcd(x,y) > 1 and/or gcd(y,b) > 1) Code:
5, {1}, 3 (i.e. 111...1113 in base 5): all numbers are divisible by either 2 or 3 (thus not shown in my list) 5, {1}, 4 (i.e. 111...1114 in base 5): all numbers are divisible by either 2 or 3 (thus not shown in my list) 5, 3, {1} (i.e. 3111...111 in base 5): all numbers are divisible by either 2 or 3 (thus not shown in my list) 5, 4, {1} (i.e. 4111...111 in base 5): all numbers are divisible by either 2 or 3 (thus not shown in my list) 9, {1}, 1 or 9, 1, {1} (i.e. 111...111 in base 9): factored as difference of squares (since my list does not check algebra factorization, thus still shown in my list) 9, {1}, 5 (i.e. 111...1115 in base 9): all numbers are divisible by either 2 or 5 (thus not shown in my list) 9, 2, {7} (i.e. 2777...777 in base 9): all numbers are divisible by either 2 or 5 (thus not shown in my list) 9, 3, {1} (i.e. 3111...111 in base 9): factored as difference of squares (since my list does not check algebra factorization, thus still shown in my list) 9, {3}, 5 (i.e. 333...3335 in base 9): all numbers are divisible by either 2 or 5 (thus not shown in my list) 9, {3}, 8 (i.e. 333...3338 in base 9): all numbers are divisible by either 2 or 5 (thus not shown in my list) 9, 3, {8} (i.e. 3888...888 in base 9): factored as difference of squares (since my list does not check algebra factorization, thus still shown in my list) 9, 5, {1} (i.e. 5111...111 in base 9): all numbers are divisible by either 2 or 5 (thus not shown in my list) 9, 5, {7} (i.e. 5777...777 in base 9): all numbers are divisible by either 2 or 5 (thus not shown in my list) 9, 6, {1} (i.e. 6111...111 in base 9): all numbers are divisible by either 2 or 5 (thus not shown in my list) 9, {7}, 2 (i.e. 777...7772 in base 9): all numbers are divisible by either 2 or 5 (thus not shown in my list) 9, {7}, 5 (i.e. 777...7775 in base 9): all numbers are divisible by either 2 or 5 (thus not shown in my list) 9, {8}, 5 (i.e. 888...8885 in base 9): factored as difference of squares (since my list does not check algebra factorization, thus still shown in my list) 11, {1}, 3 (i.e. 111...1113 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, {1}, 4 (i.e. 111...1114 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, {1}, 9 (i.e. 111...1119 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, {1}, 10 (i.e. 111...111A in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, 2, {5} (i.e. 2555...555 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, 3, {1} (i.e. 3111...111 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, 3, {5} (i.e. 3555...555 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, 3, {7} (i.e. 3777...777 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, 4, {1} (i.e. 4111...111 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, 4, {7} (i.e. 4777...777 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, {5}, 2 (i.e. 555...5552 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, {5}, 3 (i.e. 555...5553 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, {5}, 8 (i.e. 555...5558 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, {5}, 9 (i.e. 555...5559 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, {7}, 3 (i.e. 777...7773 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, {7}, 4 (i.e. 777...7774 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, {7}, 9 (i.e. 777...7779 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, {7}, 10 (i.e. 777...777A in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, 8, {5} (i.e. 8555...555 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, 9, {1} (i.e. 9111...111 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, 9, {5} (i.e. 9555...555 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, 9, {7} (i.e. 9777...777 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, 10, {7} (i.e. A111...111 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 11, 10, {7} (i.e. A777...777 in base 11): all numbers are divisible by either 2 or 3 (thus not shown in my list) 

20210812, 04:51  #4 
Aug 2020
79*6581e4;3*2539e3
503 Posts 
To what limits did you search?

20210812, 09:28  #5 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2^{2}×7×11^{2} Posts 

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