Dan, The FAQ is correct. (The version I saw has "10,000,000 exponents" corrected to "10,000,000 digits", as you note it should.)
The number of digits, d, in a Mersenne number, 2^n-1, is "the least integer greater than or equal to n/log_2(10)." (The number of digits in an integer must itself be an integer.) Thus, in your examples: 33,219,278/3.321928094887 = 9,999,999.112..... and the least integer greater than or equal to 9,999,999.112..... is 10,000,000, which is then the number of digits in 2^33,219,278-1 . Likewise for 33,219,280/3.321928094887 = 9,999,999.71436166801 and for 33,219,281/3.321928094887 = 10,000,000.01539..... so the number of digits in 2^33,219,281-1 is 10,000,001, which is the least integer greater than or equal to 10,000,000.01539..... Since 33,219,277/3.321928094887 = 9,999,998.811...., the Mersenne number with exponent 33,219,277 has 9,999,999 digits (the least integer greater than or equal to 9,999,998.811...) and thus 33,219,278 is the first Mersenne number with 10,000,000 digits. Tony Pryse At 01:01 AM 10/25/2001 -0500, you wrote: >Forgive my ignorance but; > >In reading the Lucas Wiman Mersenne Prime FAQ I became confused at the Q5.3 >instruction. (see FAQ insert below). > >I want to know how many decimal digits are in a given MP. > >This part of the FAQ does not make sense to me. > >Specifically; > >First off this question seems to ask 10,000,000 exponents. It must mean >10,000,000 digits. > >The answer given below, M33219278, by my calculations, has less than >10,000,000 digits. The questions below ask "How many digits are in a given >Mp?" and "What is the smallest Mp with a given number of digits?" > >The explanation does not seem to answer that question. > >33,219,278/3.321928094887 = 9,999,999.11230167668 and > >33,219,279/3.321928094887 = 9,999,999.41333167235 and > >33,219,280/3.321928094887 = 9,999,999.71436166801 and > >33,219,281/3.321928094887 = 10,000,000.0153916636 > >This number 33,219,281 seems, from the explanation below, to be the first >Mp to have 10,000,000 decimal digits. Can I depend on this? This would seem >to make the answer 33,219,278 the third highest Mp with less than >10,000,000 digits. > >I need a formula that will definitely give the exact number of decimal >digits in a Mp or Mersenne prime Mp. > >Can you help? > >Thanks >Dan > ***************************************************** Kenneth M. (Tony) Pryse Department of Biochemistry and Molecular Biophysics Campus Box 8231 Washington University School of Medicine 660 South Euclid Avenue St. Louis, MO 63110-1093 Tel.: 314-362-3345 Fax: 314-362-7183 ***************************************************** _________________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
