View Single Post
Old 2020-07-14, 19:33   #8
sweety439
 
Nov 2016

22·691 Posts
Default

Quote:
Originally Posted by kriesel View Post
What formula?

The alleged prime 2^283243137 - 1 can easily be shown to be composite without using a computer or calculator.
Sum the decimal digits of the exponent: 33. (Sometimes called "casting out nines") The digit sum is obviously divisible by 3. Any number 2^n-1 where n= a x b is composite, is composite, and is a repdigit (number with repeating digits), with factors 2^a-1 and 2^b-1 easily visible when expressed in base 2^a and 2^b respectively.

Consider 2^8-1 = 255 = 2^(2*4)-1
2^2-1 = 3 = 255/85.
2^4-1 = 15 = 255/17.
2^4-1 = 15 = 2^2-1 * cofactor 5.

For an exponent with 4 distinct prime factors, for example from the OP, 283243137:
https://www.alpertron.com.ar/ECM.HTM: 283243137 = 3 × 17 × 23 × 241469
a=3 (repdigit 2^3 - 1 = 7's in base 2^3 = 8)
b=17 (repdigit 2^17 - 1 = 131071's in base 2^17 = 131072)
c=23 (repdigit 2^23 - 1 = 8388607's in base 2^23 = 8388608)
d=241469 (repdigit 2^241469 - 1 in base 2^241469)

The number has numerous factors (at least 14, as shown below), each of which corresponds to being able to express the number as a repdigit in some base 2^B where 2^B=factor+1. For an exponent with four distinct prime factors, a, b, c, d, there are unique factors as follows

prime factors
2^a-1
2^b-1
2^c-1
2^d-1

composite factors
2^(ab)-1
2^(ac)-1
2^(ad)-1
2^(bc)-1
2^(bd)-1
2^(cd)-1

2^(abc)-1
2^(abd)-1
2^(bcd)-1

There's also a cofactor, whatever 2^(abcd)-1 / (2^a-1) / (2^b-1) / (2^c-1) / (2^d-1) is. Which may be prime or composite.
The general number is Phi_n(2), where Phi is the cyclotomic polynomial, which may be prime or composite, the value of Phi_n(2) for n = 1, 2, 3, ... are 1, 3, 7, 5, 31, 3, 127, 17, 73, 11, 2047, 13, 8191, 43, 151, 257, 131071, 57, 524287, 205, 2359, 683, 8388607, 241, 1082401, 2731, 262657, 3277, 536870911, 331, 2147483647, 65537, 599479, 43691, 8727391, 4033, 137438953471, 174763, 9588151, 61681, 2199023255551, 5419, 8796093022207, 838861, 14709241, 2796203, 140737488355327, 65281, 4432676798593, 1016801, ...

Phi_n(2) is prime for n = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208, 234, 254, 261, 280, 296, 312, 322, 334, 345, 366, 374, 382, 398, 410, 414, 425, 447, 471, 507, 521, 550, 567, 579, 590, 600, 607, 626, 690, 694, 712, 745, 795, 816, 897, 909, 954, 990, 1106, 1192, 1224, 1230, 1279, 1384, 1386, 1402, 1464, 1512, 1554, 1562, 1600, 1670, 1683, 1727, 1781, 1834, 1904, 1990, 1992, 2008, 2037, 2203, 2281, 2298, 2353, 2406, 2456, 2499, 2536, ...

Last fiddled with by sweety439 on 2020-07-14 at 19:34
sweety439 is offline