Pi (by a co-author of the Pi paper)
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As one of the two co-authors of the recent paper on pi, I thought it would be appropriate if I myself made a few comments:
1. Crandall and I did not prove that pi is normal. Ours is a partial proof, reducing this question to a conjecture of chaotic processes. But if nothing else it help explain why the digits of pi appear random -- they are generated by a chaotic sequence generator.
2. Our results deal strictly with base 2 and base 16 -- we can say nothing about decimal (base 10) digits of pi.
3. The "normality" property that we seek to prove is not the same as randomness in the general sense. The digits of pi are generated by a very simple, compact deterministic sequence which is thus not random in the Chaitin sense. Instead, we only claim that Pi is statistically normal -- its binary expansion contains every string of n binary digits, with limiting frequency 2^(-n).
4. The algorithm for computing individual binary or hex digits of pi is very simple. It is completely stated on my web site: http://www.nersc.gov/~dhbailey/pi-alg
5. The latest paper on the normality of pi, and the original paper giving the pi-digit-calcualting algorithm, are also available on my web site: http://www.nersc.gov/~dhbailey
Cheers, David H Bailey david@dhbailey.com
As one of the two co-authors of the recent paper on pi, I thought it would be appropriate if I myself made a few comments:
1. Crandall and I did not prove that pi is normal. Ours is a partial proof, reducing this question to a conjecture of chaotic processes. But if nothing else it help explain why the digits of pi appear random -- they are generated by a chaotic sequence generator.
2. Our results deal strictly with base 2 and base 16 -- we can say nothing about decimal (base 10) digits of pi.
3. The "normality" property that we seek to prove is not the same as randomness in the general sense. The digits of pi are generated by a very simple, compact deterministic sequence which is thus not random in the Chaitin sense. Instead, we only claim that Pi is statistically normal -- its binary expansion contains every string of n binary digits, with limiting frequency 2^(-n).
4. The algorithm for computing individual binary or hex digits of pi is very simple. It is completely stated on my web site: http://www.nersc.gov/~dhbailey/pi-alg
5. The latest paper on the normality of pi, and the original paper giving the pi-digit-calcualting algorithm, are also available on my web site:
http://www.nersc.gov/~dhbailey
Cheers,
David H Bailey
david@dhbailey.com
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As one of the two co-authors of the recent paper on pi, I thought it would be appropriate if I myself made a few comments:
1. Crandall and I did not prove that pi is normal. Ours is a partial proof, reducing this question to a conjecture of chaotic processes. But if nothing else it help explain why the digits of pi appear random -- they are generated by a chaotic sequence generator.
2. Our results deal strictly with base 2 and base 16 -- we can say nothing about decimal (base 10) digits of pi.
3. The "normality" property that we seek to prove is not the same as randomness in the general sense. The digits of pi are generated by a very simple, compact deterministic sequence which is thus not random in the Chaitin sense. Instead, we only claim that Pi is statistically normal -- its binary expansion contains every string of n binary digits, with limiting frequency 2^(-n).
4. The algorithm for computing individual binary or hex digits of pi is very simple. It is completely stated on my web site: http://www.nersc.gov/~dhbailey/pi-alg
5. The latest paper on the normality of pi, and the original paper giving the pi-digit-calcualting algorithm, are also available on my web site: http://www.nersc.gov/~dhbailey
Cheers, David H Bailey david@dhbailey.com
As one of the two co-authors of the recent paper on pi, I thought it would be appropriate if I myself made a few comments: 1. Crandall and I did not prove that pi is normal. Ours is a partial proof, reducing this question to a conjecture of chaotic processes. But if nothing else it help explain why the digits of pi appear random -- they are generated by a chaotic sequence generator. 2. Our results deal strictly with base 2 and base 16 -- we can say nothing about decimal (base 10) digits of pi. 3. The "normality" property that we seek to prove is not the same as randomness in the general sense. The digits of pi are generated by a very simple, compact deterministic sequence which is thus not random in the Chaitin sense. Instead, we only claim that Pi is statistically normal -- its binary expansion contains every string of n binary digits, with limiting frequency 2^(-n). 4. The algorithm for computing individual binary or hex digits of pi is very simple. It is completely stated on my web site: http://www.nersc.gov/~dhbailey/pi-alg 5. The latest paper on the normality of pi, and the original paper giving the pi-digit-calcualting algorithm, are also available on my web site: http://www.nersc.gov/~dhbailey Cheers, David H Bailey david@dhbailey.com