Does Pi Conspire?

Gödel’s Lost Letter and P=NP 2024-03-26

With integers like 222, perhaps

James Franklin is a mathematician and philosopher at the University of New South Wales in Sydney. He wrote an article in 2016 for the Mathematical Intelligencer titled, “Logical probability and the strength of mathematical conjectures.” One example in his article concerns statistical properties of the decimal digits of ${\pi}$ .

Today, Pi Day, we ask whether ${\pi}$ ‘s failure of certain statistical tests could ever manifest in practice.

Franklin’s essay begins with a volley of questions:

What are probabilities, really? Actual or long-run relative frequencies? Measures satisfying Kolmogorov’s axioms? Physical propensities? Degrees of belief? All those views and more have their defenders, and the differences have an impact on what to believe and do on the basis of data.

He then goes on to his main question:

What sense can be made of the notion of the probability that an unproved mathematical conjecture is true?

Extreme Risk and Doom

In our previous post, we considered whether P=NP can be assigned, not a probability, but a present financial value. Franklin works on this wavelength too, under the heading of extreme risk theory. He says on his webpage:

My research on extreme risks looks at the problems of combining very small amounts of data with expert opinion to reach reasonable estimates of the chance of very rare events with large negative consequences.

P=NP counts as an extreme risk because equality—or anything denting confidence in the practical most-case intractability of factoring or related problems—would collapse Bitcoin and similar cryptocurrencies. Our query of whether there is a present value of hedging such risk drew a private parallel from Ron Rivest, who pointed to a new manuscript on his website titled “Betting on Doom.” It derives a present value on a bet even when one side brings the end of the world. Whether a fully effective form of P=NP would cause human extinction—perhaps by supercharging AI—is up for debate.

Franklin’s 2008 paper on evaluating extreme risk events emphasizes how to assess their probability, when one cannot infer from frequency of occurrence because the event may occur once or never. Note that P=NP is a one-shot event by definition: it cannot “occur” twice. That’s the tack that he picks up in his 2016 essay, where the conjectures need not involve risk.

Can the digits of ${\pi}$ pose any risk? Would anything collapse if the normality conjecture failed—that is, if some string of ${b}$ digits occurs with density different from ${10^{-b}}$ ? Could the failure of some other simple statistical property of truly random sequences have unforeseen consequences?

Testing Pi

The digits of ${\pi}$ are not only efficient to generate, one can compute a stretch of digits at any desired place ${n}$ without having to compute any of the digits up to ${n}$ . We covered the BBP algorithm by David Bailey, Peter Borwein, and Simon Plouffe that does this. This stands in contrast to known pseudorandom generators that are conjectured to pass important statistical tests of randomness and that appear to be more inherently sequential.

Reinhard Ganz discovered a relatively simple test that the first ${10^{13}}$ (ten trillion) digits of ${\pi}$ fail according to common standards of statistical hypothesis testing. This was in a 2014 paper titled, “The Decimal Expansion of ${\pi}$ Is Not Statistically Random.”

The first step in his test broke the digits into two-hundred twenty-two blocks, each partitioned into five billion 9-tuples of digits. Note that ${222 \times 5\cdot 10^9 \times 9 = 999 \times 10^{10}}$ , which uses up almost all the digits. Then define a binary string ${x}$ of length 222 by

$\displaystyle x_i = 1 \text{ if block } i \text{ has more than 2,300,000 tuples in which some digit repeats at least five times}.$

The further steps test whether ${x}$ behaves as a random binary string. Ganz showed that a quantity based on lengths of runs of the same bit within ${x}$ diverges from the symmetric Laplace distribution that holds when ${x}$ is truly random.

Bailey and Borwein, with Richard Brent and Mohsen Reisi, reviewed Ganz’s experiment. They reproduced his findings, but found an unsuspected loophole:

The number 222 is a “magic number.” Replace it by neighboring numbers of blocks, for instance ${n =}$ 213, 217, 226, 231, 236, 241, 246, 252, or 258 blocks, and the final statistical tests of the resulting strings ${x'}$ of length ${n}$ all pass.

Put another way, there was an unseen thumb on the scale: The number 222 was not merely a numerical convenience to get up to a number 99,900,000,000,000 that is real close to ${10^{13}}$ , it exploited a degree of freedom to use for ${p}$ -hacking. Bailey et al. compare this to issues in the so-called replication crisis in general.

Conspiracy?

Still, there remains an unsettling element in this explanation. The number 222 still represents a “pothole” for ${\pi}$ . (Note also this 2016 rebuttal by Ganz.)

Could ${\pi}$ have such exceptional linkages to innocent-looking parameter choices in other applications? This feeds into our question stated more generally:

Is there a natural, non-contrived application that takes auxiliary input ${A}$ in the form of an unlimited sequence of digits, such that ${A}$ behaves differently when given the digits of ${\pi}$ versus the digits of ${e}$ or Euler’s constant, not to mention truly random digits?

Our notion of “natural” allows ${A}$ to have some hand-tuned parameters, but an exceptional value akin to 222 above would need to have some distinctive justification. It also rules out applications that of themselves expressly reference that ${\pi}$ has small specifications via efficient algorithms like BBP.

Open Problems

Putting our question another way, might the digits of ${\pi}$ be “malign” for certain applications in a sense described here, and perhaps connected to ideas and results here?

[fixed links to BBBR paper and Calude et al.]