At 10:24 PM 2/16/2000 +0900, "Kotera Hiroshi"<[EMAIL PROTECTED]> wrote:
>Hi all
>Is a decimal 23-digit numbers 11111111111111111111111 prime ?
>Could you tell me the answer with proof?
>
>24-digit numbers 11111111111111111111 = 101*1100110011001100110011
>
>regards
For the smaller numbers get ubasic and run aprt-cle.ub. It shows it is
prime.
Also included with ubasic is ecm.ub and many other programs.
The numbers above are repunits. Repunits with nonprime number of digits
are provably nonprime. (This is why the mersenne search only considers
prime exponents. If a number has n identical digits r in some number base b,
it's a repunit, and if n is factorable into i and j, the number has at
least factors
f and g:
f=sum from k=0 to i-1 of r b^k
and similarly g=sum from k=0 to j-1 of r b^k)
since the nonprime number above has 24 digits, it's divisible by more factors
than you show. By inspection, we can see it's divisible by 3; its digit count
is divisible by 2, 3, 4, 6, and 12, predicting divisors of 11, 111, 1111,
111111,
and 111111111111. Note not all of these are prime either.
Prime Factorization by ECM
Input an integer =? 111111111111111111111111
3 * 7 * 11 * 13 * 37 * 73 * 101 * 137 * 9901 * 99990001
aprt-cle.ub
Finds first factor or indicates primality.
Accepts numbers of form 2^727-1 but apparently not x / y
"APRT-CLE" is the extended version of "APRT-CL"
Cohen-Lenstra version of Adleman-Pomerance-Rumely Test
programmed by Koichiro Akiyama
1988 2 12
modified for UBASIC version 7 by Yuji KIDA
1989 1 31 & 1989 3 30
amended by Frank O'Hara
1990 1 28
extended to 844 digits using extra memories by Yuji KIDA
1991 9 30
extended to 844 digits WITHOUT using extra memories by Frank O'Hara
1991 12 13
rearranged by Yuji KIDA
1992 1 24
Hmm. These numbers are in a sense the decimal analog of mersenne numbers.
I get primes for # of digits n=2,19, 23 and no others below 72,
and first factors for the nonprimes (of prime length) as follows:
n f
3 3
5 41
7 239
11 21649
13 53
17 2071723
29 3191
31 2791
37 2028119
41 83
43 173
47 35121409
53 107
59 2559647034361
61 733
67 493121
71 ?
Other than the special case 3, the factors above are all of form f=2kn+1, like
for mersenne numbers.
Ken
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