Generalized near-repdigit primes

[sweety439]

Are there any interest to search the smallest generalized near-repdigit 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.

[bur]

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 base-10, but it'd be nice to see at a glance how the number looks like in base-b.


To have an organized search someone would have to setup a website, there are so many possible near-repdigits in the larger bases a single text file gets hard to follow.

[sweety439]

Originally Posted by bur

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 base-10, but it'd be nice to see at a glance how the number looks like in base-b.


To have an organized search someone would have to setup a website, there are so many possible near-repdigits in the larger bases a single text file gets hard to follow.

You can copy the primes to factordb (I have already added all numbers in this list to factordb), and click "show", and choose the corresponding base, it will show the primes in base b (use digits a=10, b=11, c=12, ..., z=35)

Besides, these forms are proven to only contain composite numbers: (trivial cases: gcd(x,y) > 1 and/or gcd(y,b) > 1)

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)

But there are some families, such as 5{7} (i.e. 5777...777) in base 11 and 9{5} (i.e. 9555...555) in base 13, such that I cannot find a prime or PRP of such a form, neither can prove that there cannot be any prime of such a form, can you find the smallest prime or PRP of these two forms?

[bur]

To what limits did you search?

[sweety439]

Originally Posted by bur

To what limits did you search?

5000 base b digits