[oavjlpdx] Mersenne number factorizations, binary metric prefixes

Mersenne numbers (prime exponent) have "weird" (non-algebraic) factorizations.

2¹¹ - 1 = 2047 = "2 K - 1" = 23 * 89

2¹³ - 1 = 8191 = "8 K - 1" is prime
2¹⁷ - 1 = 131,071 = "128 K - 1" is prime
2¹⁹ - 1 = 524,287 = "512 K - 1" is prime

2²³ - 1 = 8,388,607 = "8 meg - 1" = 47 * 178,481

2²⁹ - 1 = "512 meg - 1" = 233 * 1103 * 2089

2³¹ - 1 = "2 gig - 1" is prime

2³⁷ - 1 = "128 gig - 1" = 223 * 616,318,177

2⁴¹ - 1 = "2 tera - 1" = 13,367 * 164,511,353

2⁴³ - 1 = "8 tera - 1" = 431 * 9719 * 2,099,863

2⁴⁷ - 1 = "128 tera - 1" = 2351 * 4513 * 13,264,529

2⁵³ - 1 = "8 peta - 1" = 6361 * 69,431 * 20,394,401

2⁵⁹ - 1 = "512 peta - 1" = 179,951 * 3,203,431,780,337 ("~ 3 tera")

2⁶¹ - 1 = "2 exa - 1" is prime

2⁶⁷ - 1 = "128 exa - 1" = 193,707,721 * 761,838,257,287 ("~ 761 gig")

What is the largest completely factored composite prime-exponent Mersenne number? It appears to be 2⁶⁸⁴¹²⁷-1 = 23765203727 * 205933-digit PRP (4/2009 Jens Kruse Andersen).

Fermat number factorizations have a bit more structure.

2³² + 1 = "4 gig + 1" = 641 ( = 5 * 2⁷ + 1 ) * 6,700,417

2⁶⁴ + 1 = "16 exa + 1" = 274,177 ( = 1071 * 2⁸ + 1 ) * 67,280,421,310,721 ("~ 67 tera")

The largest completely factored Fermat number is 2²⁰⁴⁸+1.