this is truly interesting...do you have a link to the original 1996 paper? do you know if anyone has incorporated this into a program? phillip > -----Original Message----- > From: [EMAIL PROTECTED] > [mailto:[EMAIL PROTECTED]]On Behalf Of Eric Cordian > Sent: Thursday, August 02, 2001 2:35 PM > To: [EMAIL PROTECTED] > Subject: Pi > > > > Interesting article recently posted on the Nature Web site about the > normality of Pi. > > http://www.nature.com/nsu/010802/010802-9.html > > "David Bailey of Lawrence Berkeley National Laboratory in California and > Richard Crandall of Reed College in Portland, Oregon, present evidence > that pi's decimal expansion contains every string of whole numbers. They > also suggest that all strings of the same length appear in pi with the > same frequency: 87,435 appears as often as 30,752, and 451 as often as > 862, a property known as normality." > > Of cryptographic interest. > > "While there may be no cosmic message lurking in pi's digits, if they are > random they could be used to encrypt other messages as follows: > > "Convert a message into zeros and ones, choose a string of digits > somewhere in the decimal expansion of pi, and encode the message by > adding the digits of pi to the digits of the message string, one after > another. Only a person who knows the chosen starting point in pi's > expansion will be able to decode the message." > > While there's presently no known formula which generates decimal digits of > Pi starting from a particular point, there's a clever formula which can be > used to generate HEX digits of Pi starting from anywhere, which Bailey et > al discovered in 1996, using the PSLQ linear relation algorithm. > > If you sum the following series for k=0 to k=infinity, its limit is Pi. > > 1/16^k[4/(8k+1) - 2/(8k+4) - 1/(8k+5) - 1/(8k+6)] > > (Exercise: Prove this series sums to Pi) > > Since this is an expression for Pi in inverse powers of 16, it is easy to > multiply this series by 16^d and take the fractional part, evaluating > terms where d>k by modular exponentiation, and evaluating a couple of > terms where d<k to get a digit's worth of precision, yielding the (d+1)th > hexadecimal digit of Pi. > > Presumedly, if one could express PI as a series in inverse powers of 10, > one could do the same trick to get decimal digits. Such a series has so > far eluded researchers. > > -- > Eric Michael Cordian 0+ > O:.T:.O:. Mathematical Munitions Division > "Do What Thou Wilt Shall Be The Whole Of The Law" >
