Number Theory Archives - Math Research Problems https://mathresearchproblems.com/category/number-theory/ from fun to challenging Thu, 22 Feb 2024 20:41:09 +0000 en-US hourly 1 https://wordpress.org/?v=6.9.4 https://mathresearchproblems.com/wp-content/uploads/2023/06/cropped-favicon-3-32x32.png Number Theory Archives - Math Research Problems https://mathresearchproblems.com/category/number-theory/ 32 32 Continued fraction representations of the Archimedes number π https://mathresearchproblems.com/continued-fractions-of-pi/ https://mathresearchproblems.com/continued-fractions-of-pi/#comments Wed, 14 Jun 2023 18:04:03 +0000 https://mathresearchproblems.com/?p=356 [latex] \pi=3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+\cdots}}}}=3.14159265 \cdots[/latex]

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Illustration by Al Mamun

We present a collection of continued fraction (cf) representations of fractional multiples of \pi. Where possible we add references and year of first discovery. At the end of the article you will find ideas about open research problems.

Why Archimedes number?

Because before Archimedes time, \pi, was only known up to 1, decimal digit. Archimedes not only improved the approximation to, \pi=3.1418\cdots, which is accurate to, \frac{1}{12155}, but did so inventing a beautiful method to bound the perimeter of the circle between inscribed and circumscribed regular polygons. This is called the method of exhaustion and is a precursor to integral calculus. His method was invariably used to appx \pi, for almost the next 2000 years! He was so much ahead of his time and contributed so much to the importance of \pi, that in our opinion it is just to be called the Archimedes number.

Presentation

We will represent fractional multiples of \pi, as continued fractions (cf). Since only the fractional part of a real number  matters for the continued fractions, we only represent fractional multiples bounded between 0, and 1:

0\lt\frac{a+b \hspace{0.5 mm}\pi}{c+d \hspace{0.5 mm}\pi} \lt 1

where, a,b,c,d\in \mathbb{Z}. We will represent the continued fractions in traditional, compact or Gauss operator notation \large\mathcal{K},  where in all cases, a_0=0. Then

\frac{a+b \hspace{0.5 mm}\pi}{c+d \hspace{0.5 mm}\pi}= \frac{b_1}{a_1+\frac{b_2}{a_2+\frac{b_3}{a_3+..}}}= \large\mathcal{K}_{k=1}^\infty\normalsize \frac{b\left( k \right)}{a\left( k \right)}

where, a_k=a\left( k \right),b_k=b\left( k \right), k\in \mathbb{N^{*}}, and, \mathbb{N}^{*}=1,2,3,\cdots . We call b_k, the partial numerators (PN) and a_k, the partial quotients (PQ). Notice that in this article we chose the index k, of the Gauss operator to start with 1. This facilitates the comparison between cf.

Simple continued fraction of π

The partial numerators (PN) of simple (also called regular) continued fractions (scf) are, b_k=1, whereas the partial quotients (PQ) are positive integers, a_k\in \mathbb{N^{*}}. The scf of \pi, is

\pi=3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+\frac{1}{1+…}}}}}

\pi=\{3,7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,2,1,1,15,3,13,1,4,2,6,6,99,1,\cdots \}

where the sequence represents the compact notation of the PQ. In this case we have no Gauss representation of the scf, since there is no known pattern for the PQ. As for most irrational numbers the partial quotients do not follow any predictable pattern. L. J. Lange [1] puts it in a very elegant way saying in 1999, “There is no known regularity to the partial denominators (of the regular continued fraction of π) and the only known means to obtain them is to compute them one by one from a known decimal appx of \pi“.

General continued fractions of π

All general continued fraction formulas are transformed to comply with the presentation rules described above.

1. Gottfried Wilhelm Leibniz (1673) [1]

\frac{4-\pi}{\pi}=\frac{{1^{2}}}{2+\frac{3^{2}}{2+\frac{5^{2}}{2+\frac{7^{2}}{2+\cdots }}}}=\large\mathcal{K}_{k=1}^\infty\normalsize \frac{\left( 2k-1 \right)^{2}}{2}

The formula Leibniz discovered was actually the infinite sum below. The cf then follows from the infinite sum.

\frac{\pi}{4} =\sum_{k=0}^{\infty }\frac{\left( -1 \right)^{k}}{2k+1}

2. L. J. Lange (1999) [2]

\pi-3=\frac{{1^{2}}}{6+\frac{3^{2}}{6+\frac{5^{2}}{6+\frac{7^{2}}{6+\cdots }}}}

The continued fraction discovered by Lange was previously discovered by Nilakantha Somayaji around 1500 in the form of an infinite sum, from which the cf can be derived.

\pi-3=\sum_{k=1}^{\infty }\frac{\left( -1 \right)^{k+1}}{k\left( k+1 \right)\left( 2k+1 \right)}

Suggested research problems

  1. EASY – Find new general continued fractions of fractional multiples of \pi, with integer PN/PQ that have not been published. The PN/PQ should follow a predictable pattern, which ideally follows from a polynomial function. More generally the PN/PQ may be entries of the Online Encyclopedia of Integer Sequences (OEIS).
    Predictable here means that the nth PN/PQ can be computed without prior knowledge of all previous PN/PQ.
  2. VERY DIFFICULT – It has not been proved that the PQ of the simple continued fraction of \pi, follow Khinchin’s statement that their geometric mean is Khinchin’s constant. Prove or disprove that this is indeed the case.
  3. VERY DIFFICULT – It has not been proved that the PQ of the simple continued fraction of \pi, follow a random order (within the constraint of Khinchin’s statement for the geometric mean). Prove that this is indeed the case.

Comments

  1. EASY – There are many cf of \pi, with integer PN/PQ scattered in the literature and in books. There are also general methods to find cf of irrational numbers from series expansions or integrals. Sometimes it is more difficult to be sure that a certain cf has not been published than to actually find one.
  2. VERY DIFFICULT – Although Khinchin proved that the geometric mean of the PQ of the scf of almost all real numbers (except a set of measure zero) is Khinchin’s constant, it has not been proved that any particular number not specifically constructed for this purpose satisfies Khinchin’s statement. This makes the problem more or less intractable.
  3. VERY DIFFICULT – This problem is similar, but not equivalent as far as we know, to the statement that \pi, is a normal number, which implies that its decimal digits follow are random. Also intractable.

References

  1. Roy, Ranjan (1990). “The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha” (PDF).
  2. Lange, L. J.. “An Elegant Continued Fraction for π.” American Mathematical Monthly 106 (1999): 456-458.

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