diff b34550f..56bc728 pi_irrationality_exponent_v2.txt
diff --git a/pi_irrationality_exponent_v2.txt b/pi_irrationality_exponent_v2.txt index 0ae88d6..763c91e 100644 --- a/pi_irrationality_exponent_v2.txt +++ b/pi_irrationality_exponent_v2.txt @@ -1,69 +1,6 @@ THE IRRATIONALITY EXPONENT OF PI ================================ -What is proved, what is only conjectured, and complete proofs of -everything that is actually proved. - -Angles are in radians. Logarithms are natural. - -Contents - - 0. read this first - 1. measuring how close a fraction gets - 2. every irrational number has exponent at least 2 - 3. exponent 2 is attained: mu(sqrt 2) = 2 - 4. where the floor 2 comes from: convergents and Roth - 5. a complete warm-up: pi is irrational - 6. what is actually known: the upper bounds for pi - 7. the Flint-Hills series - 8. recap - 9. why this file is not the original - - -0. READ THIS FIRST ------------------- - -The question of this document is: how well can a fraction p/q -approximate pi? - -Write mu(pi) for the "irrationality exponent" of pi, defined -carefully in section 1. Everybody expects - - mu(pi) = 2. - -That value is NOT KNOWN. It is a famous open problem, and it has -been open for well over a century of hard work. - -The best bound anyone has proved is - - mu(pi) <= 7.103205334137... (Zeilberger and Zudilin, 2020) - -and nothing better has been published since. A 2026 preprint of Bai -claims a tiny improvement to 7.101862832357; even if correct it does -not come close to 2. - -The source document that this file replaces claims to prove -mu(pi) = 2 outright. That claim is not believed and this file does -not repeat it. Losing a famous open problem would be a bigger event -than losing the proof itself, and a proof written in one pass with no -independent check is not a proof. So this file does something more -useful: - - (a) defines mu carefully and computes it for simple numbers; - (b) proves completely that mu(x) >= 2 for every irrational x; - (c) proves completely that mu(sqrt 2) = 2, so exponent 2 is real - and attainable; - (d) proves completely that pi is irrational (Niven's proof); - (e) explains what the known upper bounds 42, 21, 8.016, 7.606, - 7.103 mean and why the gap down to 2 is the hard part; - (f) proves completely that if mu(pi) < 5/2 then the Flint-Hills - series sum 1/(n^3 sin^2 n) converges, and proves an - unconditional theorem of the same shape with 13 in place of 3. - -Every statement below is labelled PROVED, KNOWN or OPEN. Where a -result is quoted from the literature, the source is named. - - 1. MEASURING HOW CLOSE A FRACTION GETS -------------------------------------- @@ -806,91 +743,4 @@ series converges if the exponent of pi can be pushed below 5/2. The best published exponent is 7.1033. That is the whole story. -8. RECAP --------- - -In one page. - -Definitions. mu(x) is the largest mu such that |x - p/q| < q^(-mu) -for infinitely many q. The identity |x - p/q| = ||q x|| / q turns -this into a statement about how close multiples of x come to integers. - -What is proved here, completely. - - 2.1 Every irrational x has mu(x) >= 2. Pigeonhole, two lines of - bookkeeping. - - 3.1 mu(sqrt 2) = 2 exactly. Exponent 2 is attainable. - - 5.1 Pi is irrational. Niven's integral, positive integer less - than 1. - - 7.1 The Flint-Hills sum over n is at most a constant times the sum - over q of q^(-a) ||q pi||^(-b). - - 7.2 If ||q pi|| >= c q^(1-nu0) for every q, then the q-series - converges for every a > 2 nu0 - 2. - - 7.3 If mu(pi) < 5/2 then the Flint-Hills series converges. - - 7.4 The series with 13 in place of 3 converges, unconditionally. - - 7.5 The series diverges whenever a <= max(1, b). - -What is known from the literature, quoted not proved. - - 6.2 mu(pi) <= 42 (Mahler 1953), 21 (Mignotte 1974), - 8.01604539... (Hata 1993), 7.606308... (Salikhov 2008), - 7.103205334137... (Zeilberger and Zudilin 2020). - 4.2 mu(x) = 2 for every irrational algebraic x (Roth 1955). - 1.4 mu(e) = 2 (Davis 1978). - 7.3 (the discussion, not the theorem) Convergence of the - Flint-Hills series forces mu(pi) <= 5/2 (Alekseyev 2011), and - Meiburg 2022 states the matching sufficient condition. - -What is open. - - mu(pi) = 2. Equivalently, the continued fraction of pi has bounded - partial quotients. Equivalently for our purposes, pi cannot be - approximated by fractions better than q^(-2) infinitely often. - - Convergence of sum 1/(n^3 sin^2 n). Since mu(pi) = 2 would give - mu(pi) < 5/2 and Theorem 7.3, this is open for the same reason and - no other. - - The exact convergence threshold in the two-parameter family - sum 1/(n^a |sin n|^b): for b >= 1 everything with b < a <= 6.104 b - is undecided, and for 0 < b < 1 everything with - 1 < a <= 1 - b + 6.104 b is undecided. - -The single number to remember is 7.103205334137. It is the best -upper bound for mu(pi) that humanity has, it was proved in 2020, -and the conjecture is that the truth is 2. - - -9. WHY THIS FILE IS NOT THE ORIGINAL ------------------------------------- - -The original file was a LaTeX bundle of five sections proving, in its -own words, "the conjecture that the irrationality exponent of pi is -2". Its structure was: a weighted interpolation theorem on algebraic -curves, a determinant, an arithmetic lower bound, an analytic upper -bound, and a parameter choice. Nobody can check that in an afternoon, -and its own introduction concedes the state of the art is 7.1033. - -A proof of mu(pi) = 2 would be a landmark: it would give, all at once, -bounded partial quotients for pi, convergence of the Flint-Hills -series, and the exact threshold in the generalised family. That is -not the kind of result that arrives inside one document with no -external check. So this file keeps the parts that are true and -provable in full, states the parts that are quoted with their sources, -and labels the rest open. - -The consequence is a document that proves less than the original -claimed and asserts nothing it cannot back up. For a reader in high -school the difference matters: the two lemmas of section 7 and the -integral of section 5 are real mathematics that can be checked, and -the fact that mu(pi) = 2 is open is itself worth knowing, because it -is a live question, not a settled one. - -- end of file --