##
**Die Lehre von den Kettenbrüchen. Band I. 3. erweiterte und verbesserte Aufl.**
*(German)*
Zbl 0056.05901

Stuttgart: B. G. Teubner Verlagsgesellschaft. vi, 168 S. (1954).

The first edition of this book appeared in 1913 see JFM 43.0283.04, the second with a few changes in 1929 (Leipzig) see JFM 55.0262.09.

This is volume I of the third edition, and has been considerably improved and brought up to date. It deals entirely with the elementary arithmetic portion, and it is still the only book ever published on this phase of continued fraction theory. Although the five chapters and most of the individual paragraphs have the same or almost the same titles as did those in the preceding editions, and the format is still along classical lines, many proofs have been simplified and a number of items which were formerly omitted and since have become important have been included in the appropriate places, as well as a number of new results. As before, the author has placed greatest value on ease of comprehension. He has carefully avoided complicated notation and has adhered to the classical presentation. Thus the reader is not unnecessarily weighed down with unimportant details and digressions and thereby does not lose sight of the main line of thought and the important ideas.

The book consists of the following topics:

Chapter I, definitions and general formulas: (1) notation, historical remark; (2) the transformation into an ordinary fraction; (3) the Euler-Minding formulas; (4) continuants; (5) the fundamental formulas, matrices; (6) extensions of the fundamental formulas; (7) irreducibility; (8) numerical continued fractions, convergence.

Chapter II, regular continued fractions: (9) finite regular continued fractions; (10) the diophantine linear equation; (11) inverse continued fractions, symmetric continued fractions; (12) infinite regular continued fractions; (13) the approximation theorem, criterion for a fraction to be an approximant; (14) approximation by rational fractions; (15) the theorem of best approximation; (16) in between approximants; (17) equivalent numbers; (18) two number-theoretic applications.

Chapter III, regular periodic continued fractions: (19) pure- and mixed-periodic continued fractions; (20) the Lagrange theorem on periodicity; (21) two additional proofs of the Lagrange theorem; (22) reduced numbers and pure periodicity; (23) inverse periods; (24) square roots of rational numbers; (25) square roots of whole numbers, necessary conditions; (26) sufficient conditions, tables; (27) the Pell equation; (28) number-theoretic applications; (29) culminating and almost-culminating periods; (30) the continued fraction expansion for \((\sqrt{1+4G}+ 1)/2\).

Chapter IV, Hurwitz continued fractions, transcendental numbers: (31) preparatory theorems; (32) definition of Hurwitz continued fractions; (33) the Hurwitz theorem; (34) the regular continued fractions for the numbers \(e\) and \(e^2\) and their \(m\)-th roots; (35) Liouville numbers; (36) quasi-periodic continued fractions.

Chapter V, semi-regular continued fractions: (37) the convergence theorem of Tietze; (38) definition of semi-regular continued fractions; (39) longest and shortest expansions; (40) transformation of semi-regular continued fractions into regular; (41) the approximation theorem; (42) periodicity; (43) continued fraction expansions according to nearest and more distant integers as well as reduced-regular continued fractions; (44) singular continued fractions; (45) semi-regular continued fractions; (46) continued fractions in the field of complex numbers.

The following improvements and additions are noted: In Chapter I, §2, the fundamental recurrence formulas are obtained from the point of view that a continued fraction is generated by a sequence of linear transformations. In §5, since matrix calculus is now widely used, the author has introduced a small part of matrix theory applied to the recurrence relations of continued fractions, which he uses later in the book.

In Chapter II, §14, the theorem of Hurwitz on the approximation of irrational numbers by rational is shown by the shorter and farther-reaching proof of Fujiwara. In §18, a new application consisting of two theorems, proved by means of continued fractions, in the theory of modular functions has been added.

In Chapter III, §20, a simple algorithm is inserted for the expansion of an irrational number into a periodic continued fraction. In §21 has been added a proof of the Lagrange theorem on periodicity due to Ballieu. The author has included it for its especial simplicity and for the fact that one does not at any time use the convergence of the continued fraction. Furthermore, the discussion on the continued fraction expansion for \((\sqrt{1+4G}+ 1)/2\) due to H. Schmidt and its connection as the solution of certain diophantine equations, §30, is new.

In Chapter IV, §31, the theory of Hurwitz continued fractions is expressed on a completely new basis in terms of matrices, according to Châtelet.

Also added is §39 in Chapter V on longest and shortest semi-regular continued fraction expansions for a rational number, and a part of the related §43 has been inserted. In §40 and §41, the theory of semi-regular continued fractions has been greatly altered on the basis of the work of Blumer. Finally, §46 is a new section on continued fractions with elements in the field of complex numbers.

There is a bibliography consisting of 116 references, only those which are cited in the book. In the text, much material is presented without references. This, however, the author specifically states is not necessarily his own.

As one reads this volume, one feels as though one were listening to the author speaking in person, perhaps auditing one of his lectures. It is written in that same clear, logical, simple language which is very easy to comprehend and for which Professor Perron is so well known. As was the case with the preceding editions, the book is exceedingly well coordinated, each step and each theorem a logical consequence of the preceding. This volume covers very well the theory of arithmetic continued fractions. It is with great anticipation that one looks forward to the publication of the second volume on the analytic function-theoretic portion of continued fraction theory. Volume I certainly is in every respect one of the classics of present-day mathematics.

This is volume I of the third edition, and has been considerably improved and brought up to date. It deals entirely with the elementary arithmetic portion, and it is still the only book ever published on this phase of continued fraction theory. Although the five chapters and most of the individual paragraphs have the same or almost the same titles as did those in the preceding editions, and the format is still along classical lines, many proofs have been simplified and a number of items which were formerly omitted and since have become important have been included in the appropriate places, as well as a number of new results. As before, the author has placed greatest value on ease of comprehension. He has carefully avoided complicated notation and has adhered to the classical presentation. Thus the reader is not unnecessarily weighed down with unimportant details and digressions and thereby does not lose sight of the main line of thought and the important ideas.

The book consists of the following topics:

Chapter I, definitions and general formulas: (1) notation, historical remark; (2) the transformation into an ordinary fraction; (3) the Euler-Minding formulas; (4) continuants; (5) the fundamental formulas, matrices; (6) extensions of the fundamental formulas; (7) irreducibility; (8) numerical continued fractions, convergence.

Chapter II, regular continued fractions: (9) finite regular continued fractions; (10) the diophantine linear equation; (11) inverse continued fractions, symmetric continued fractions; (12) infinite regular continued fractions; (13) the approximation theorem, criterion for a fraction to be an approximant; (14) approximation by rational fractions; (15) the theorem of best approximation; (16) in between approximants; (17) equivalent numbers; (18) two number-theoretic applications.

Chapter III, regular periodic continued fractions: (19) pure- and mixed-periodic continued fractions; (20) the Lagrange theorem on periodicity; (21) two additional proofs of the Lagrange theorem; (22) reduced numbers and pure periodicity; (23) inverse periods; (24) square roots of rational numbers; (25) square roots of whole numbers, necessary conditions; (26) sufficient conditions, tables; (27) the Pell equation; (28) number-theoretic applications; (29) culminating and almost-culminating periods; (30) the continued fraction expansion for \((\sqrt{1+4G}+ 1)/2\).

Chapter IV, Hurwitz continued fractions, transcendental numbers: (31) preparatory theorems; (32) definition of Hurwitz continued fractions; (33) the Hurwitz theorem; (34) the regular continued fractions for the numbers \(e\) and \(e^2\) and their \(m\)-th roots; (35) Liouville numbers; (36) quasi-periodic continued fractions.

Chapter V, semi-regular continued fractions: (37) the convergence theorem of Tietze; (38) definition of semi-regular continued fractions; (39) longest and shortest expansions; (40) transformation of semi-regular continued fractions into regular; (41) the approximation theorem; (42) periodicity; (43) continued fraction expansions according to nearest and more distant integers as well as reduced-regular continued fractions; (44) singular continued fractions; (45) semi-regular continued fractions; (46) continued fractions in the field of complex numbers.

The following improvements and additions are noted: In Chapter I, §2, the fundamental recurrence formulas are obtained from the point of view that a continued fraction is generated by a sequence of linear transformations. In §5, since matrix calculus is now widely used, the author has introduced a small part of matrix theory applied to the recurrence relations of continued fractions, which he uses later in the book.

In Chapter II, §14, the theorem of Hurwitz on the approximation of irrational numbers by rational is shown by the shorter and farther-reaching proof of Fujiwara. In §18, a new application consisting of two theorems, proved by means of continued fractions, in the theory of modular functions has been added.

In Chapter III, §20, a simple algorithm is inserted for the expansion of an irrational number into a periodic continued fraction. In §21 has been added a proof of the Lagrange theorem on periodicity due to Ballieu. The author has included it for its especial simplicity and for the fact that one does not at any time use the convergence of the continued fraction. Furthermore, the discussion on the continued fraction expansion for \((\sqrt{1+4G}+ 1)/2\) due to H. Schmidt and its connection as the solution of certain diophantine equations, §30, is new.

In Chapter IV, §31, the theory of Hurwitz continued fractions is expressed on a completely new basis in terms of matrices, according to Châtelet.

Also added is §39 in Chapter V on longest and shortest semi-regular continued fraction expansions for a rational number, and a part of the related §43 has been inserted. In §40 and §41, the theory of semi-regular continued fractions has been greatly altered on the basis of the work of Blumer. Finally, §46 is a new section on continued fractions with elements in the field of complex numbers.

There is a bibliography consisting of 116 references, only those which are cited in the book. In the text, much material is presented without references. This, however, the author specifically states is not necessarily his own.

As one reads this volume, one feels as though one were listening to the author speaking in person, perhaps auditing one of his lectures. It is written in that same clear, logical, simple language which is very easy to comprehend and for which Professor Perron is so well known. As was the case with the preceding editions, the book is exceedingly well coordinated, each step and each theorem a logical consequence of the preceding. This volume covers very well the theory of arithmetic continued fractions. It is with great anticipation that one looks forward to the publication of the second volume on the analytic function-theoretic portion of continued fraction theory. Volume I certainly is in every respect one of the classics of present-day mathematics.

Reviewer: Evelyn Frank (Chicago)

### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11A55 | Continued fractions |

11J70 | Continued fractions and generalizations |

40A15 | Convergence and divergence of continued fractions |

### Keywords:

continued fractions; regular continued fractions; approximation by rational fractions; theorem of best approximation; regular periodic continued fractions; Lagrange theorem on periodicity; Pell equation; Hurwitz continued fractions; transcendental numbers; Liouville numbers; semi-regular continued fractions; semi-regular continued fractions### Online Encyclopedia of Integer Sequences:

Numbers k such that the continued fraction for sqrt(k) has odd period length.Numbers such that no smaller positive integer has the same maximal palindrome in the periodic part of the simple continued fraction for its square root.

a(n) = Fibonacci(n)*Fibonacci(n+2).

Minimal positive solution a(n) of Diophantine equation a(n)^2 - a(n)*b(n) - G(n)*b(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077058(n).

Minimal positive solution a(n) of Diophantine equation b(n)^2 - b(n)*a(n) - G(n)*a(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077057(n).

a(n) is smallest natural number satisfying Pell equation a^2 - d(n)*b^2= +1 or = -1, with d(n)=A000037(n) (a nonsquare). Corresponding smallest b(n)=A077233(n).

a(n) is smallest natural number satisfying Pell equation b^2- d(n)*a^2= +1 or = -1, with d(n)=A000037(n) (nonsquare). Corresponding smallest b(n)=A077232(n).

Nonsquare integers n such that the continued fraction (sqrt(n)+1)/2 has odd period length.

Primitive period length of (regular) continued fraction of (sqrt(D(n))+1)/2 for D(n)=A077425(n).

Minimal (positive) solution a(n) of Pell equation a(n)^2 - D(n)*b(n)^2 = +4 with D(n)= A077425(n). The companion sequence is b(n)=A078355(n).

Minimal (positive) solution a(n) of Pell equation b(n)^2 - D(n)*a(n)^2 = +4 with D(n)= A077425(n). The companion sequence is a(n)=A077428(n).

Minimal positive solution z of Pell equation z^2 - A077426(n)*t^2 = -4.

Minimal positive solution x of Pell equation y^2 - A077426(n)*x^2 = -4.

Non-oblong numbers: Complement of A002378.

Minimal positive solution a(n) of Pell equation a(n)^2 - D(n)*b(n)^2 = +4 or -4 with D(n)=A077425(n). The companion sequence is b(n)=A077058(n).

A Chebyshev T-sequence with Diophantine property.

A Chebyshev T-sequence with Diophantine property.

A Chebyshev T-sequence with Diophantine property.

A Chebyshev T-sequence with Diophantine property.

a(n) = 27a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 27.

a(n) =29a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 29.

a(n) = 21a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 21.

a(n) = 23a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 23.

a(n) = 25*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 25.

The Diophantine equation x^2 - x*y - G*y^2 = -1, G a positive integer, D = 4*G + 1 not a perfect square, has no solution precisely for G = a(n).

Pair deficit of the most equal partition of n into two parts using standard rounding of the expectations of n, floor(n/2) and n-floor(n/2), assuming equal likelihood of states defined by the number of 2-cycles.