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.
Mp is shorthand for Mersenne Prime, a number having value one less than two raised to the power p where p is a prime, often written in ascii, 2^p-1 meaning (2^p) -1 and not 2^(p-1) because exponentiation is an operator of higher precedence than subtraction. 2^33219278-1 is not a Mersenne Prime since 33219278 is not prime, being even. The only exponent of the four above that is prime is 33219281. (33219279 mod 3 = 0 by eye; 33 21 9 27 9) M33219278 is the smallest Mersenne Number with at least 10^7 digits, but M33219281 is the smallest Mersenne Prime with at least 10^7 digits. This distinction is economically important because >I need a formula that will definitely give the exact number of decimal >digits in a Mp or Mersenne prime Mp. At 09:30 AM 10/25/2001 -0500, Tony Pryse <[EMAIL PROTECTED]> wrote: >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.) The rule in the FAQ answer misses a case. Consider some very small exponents: 2^0=1; log10(1)=0; # of digits =1 2^0-1=0; log10(Mn)= -infinity?; # of digits=1; rule yields 0 2^1=2; log10(2)=~.3010; 1 digit 2^2=4; log10(4)=~.6020; 1 digit 2^3=8; log10(8)=~.9030; 1 digit 2^4=16; log10(16)=~1.204; 2 digits ... 2^10=1024; log10(1024)=~3.0103; 4 digits 10^1=10; log10(10)=1; 2 digits 10^3=1000; log10(1000)=3; 4 digits The number of digits required is always at least one. The number of digits is rounded up, not down. I suggest the "or equal" is incorrect. Ken _________________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
