Figurate numbers. (English) Zbl 1258.11001

Deza, Elena; Deza, Michel Marie

Hackensack, NJ: World Scientific (ISBN 978-981-4355-48-3/hbk; 978-981-4355-49-0/ebook). xviii, 456 p. (2012).

I start this review with some general remarks before entering into the contents of the book in more detail.
From the outside this book looks very attractive and appealing so I started to read it full of expectation. After reading some Chapters, I became somewhat disappointed. The first three chapters turned out not to be that interesting, they just contained long listings of formulae, generating functions, and references to Sloane’s Handbook of integer sequences. Chapters 4 and 5 turned out to be more interesting. Chapter 6 is a summing up of all kind of recreational facts on some special numbers. For the time being I did not look at the exercises in Chapter 7. The whole book breaths sloppiness both in the use of language and in the formulae. The language may be excused but the formulae should have been checked with more care.
Now I will go into detail with respect to the separate Chapters.
The book starts with 4 pages of Notation. The definition in words not always covers the formal definition, e.g. the definition of the Euler totient function lacks in the wording the limitation that the integers counted should be smaller than \(n\).
The Notation Section is followed by a preface giving historic background and an outline of the book.
Chapter 1 is on plane figurate numbers. It starts with some definitions, formulae and generating functions for polygonal numbers. A shorter proof for the generating function might have been obtained by starting with the recurrence relation on page 5, multiplying both sides by \( x^{n+1} \), summing from \(n=0\) to infinity and solving the in this way obtained equation for the generating function.
Next it is shown that there are several formulae connecting polygonal numbers by linear dependencies. Polygonal numbers can be expressed into other (especially triangular) polygonal numbers. The chapter continues with looking for numbers that are \(m\)-gonal for at least 2 different \(m\), for instance square triangular numbers. The last question leads to solving Pell’s equation \( x^2 - 2y^2 = 1 \), which boils down to finding the convergents in the continued fraction expansion of \( \sqrt{2} \).
Next pentagonal triangular numbers, pentagonal squares and several other bi-polygonal numbers are discussed. They all correspond to solutions of Pell like equations, and the first few values are calculated for several cases. It is here that the sloppiness starts manifesting itself, below find the list of errors I have found. In Section 1.4.7 the formula should be \( (10u-3)^2 - 5(2v+1)^2 = 4 \). The substitution in Section 1.4.11 has to be \( x = 2(3u-1) \). The formula in Section 1.4.14 should read \( 8(3u-1)^2 - 3(4v-1)^2 = 5 \). In Section 1.4.17 \(u\) and \(v\) should be interchanged in the first formula. Also in Section 1.4.18 and 1.4.19 the \(u\) and \(v\) should be interchanged in the first formula. In Section 1.4.20 the initial formula should read \(\frac{1}{2}v(7v-9)=\frac{1}{2}u(5u-3) \). The second equation should read \( 5(14v-5)^2-7(10u-3)^2=62\) , and the Pell like equation becomes \(5x^2-7y^2=62 \). In Section 1.4.21 in the first formula \(u\) and \(v\) should again be interchanged, the second formula is incorrect and should read \( 3(14v-5)^2 - 56(3u-1)^2=19 \). The Pell like equation is correct in this case.
The first few of these bunch of errors can be forgiven, but as the number keeps growing the reader is pushed beyond the point of annoyance.
Section 1.5 handles the question of the number of times a given number is a polygonal number.
Section 1.6 discusses so called centered polygonal numbers. Moreover it looks at centered polygonal numbers that are also polygonal deriving Diophantine equations for some cases. Section 1.7 starts with composite numbers in connection with rectangles. Also in this Section there are some errors.
In Section 1.7.2 : 65 = 5 * 13 and not 5 * 12 as stated. On page 65 the sequence of 9 consecutive integers centered at nine giving sum 18 starts with \(-2\) in stead of \(-3\). The section continues with trapezodial numbers and L-shape numbers (just the odd numbers). The Section finishes with some special types of figurative numbers like aztec numbers, diamond numbers. I did not check out the math in these cases.
Section 1.8 is the final section of Chapter 1, and deals with generalized plane figurate numbers. These are obtained by extending the domain of the formulae to include the negative numbers. It recalculates the results of the previous sections for the negative values.
Chapter 2 is on Space figurate numbers. Here configurations of points in space are studied. First the platonic solids can be used for the generation of polygonal configurations. This leads to so called pyramidal, cubic, octahedral, dodecahedral and icosahedral numbers. Of course also other solids can be used. The connections between pyramidal numbers and polygonal numbers are discussed. In Section 2.1.5 the \(u\) and \(v\) are again interchanged either in the formula or in the summing up of the solutions. In III the formula should read \( u^2 = \frac{1}{6}v(v+1)(v+2) \).
Section 2.2 is on cubic numbers giving some interesting facts. Section 2.3 and 2.4 are devoted to octahedral numbers and other polyhedral numbers, respectively. Recurrence formulae for polyhedral numbers are derived involving the parameters of the platonic solids. In Section 2.4.4 the final result for \(D(n)-D(n-1)\) is correct, the formula before the final one should read \(19+27(n-2)+9(\frac{3n^2-11n+10}{2})\).
Next in Section 2.5 truncated cubic, truncated octahedral, truncated dodecahedral and truncated icosahedral numbers are discussed, and some formulae are given. Again I found some errors. The formula for \(SO(n)\) on page 119 should read \(SO(n)=S^3_4(n) + S^3_4(n-1) + 8S^3_3(n-1) \).
Section 2.6 is on centered space figurate numbers. On page 121, the third level is the cube formed by 26 points (and not 28 as stated wrongly two times). Formulae, generating functions and recursive relations are given consecutively for centered pyramidal, centered tetrahedron, centered square pyramidal, centered \(m\)-gonal pyramidal, centered octahedral, centered icosahedral, centered dodecahedral, truncated tetrahedron, truncated cube and truncated octahedral numbers.
Section 2.7 is on other space figurate numbers like prism numbers. It starts with hex pyramidal numbers, then generalizes to \(m\)-gonal pyramidal numbers and finally specialises again to \(m=2,3,...,12\). The hex pyramidal numbers are thus handled twice. Next prism numbers and rhombic dodecahedral numbers are handled and a connection is revealed.
Section 2.8 is on generalizations of the afore mentioned numbers by allowing negative values in the derived formulae. Of course, we can find the usual sloppiness in this section. In the formula for generalized tetrahedral numbers in the middle of page 146 a minus sign is missing twice. The formula for centered tetrahedral numbers listed at the top of page 155 should read \( (4n-1)(n^2-n-3)/3 \). In Section 2.8.8. the formula for \(\bar S^3_4\) is listed wrongly see for instance page 129 for a correct version. Moreover the listed formula for \( \bar S^3_3 \) is not what we were looking for in this section, actually it should read \( \bar S^3_4 = - \frac{(2n+1)(n^2+n+2)}{2} \). Section 2.8.9 states that we obtain the ordinary centered, square pyramidal numbers where it should be the ordinary rhombic dodecahedral number. In Section 2.8.11 the generating function \(f(x)\) does not match the one previously given. It should read \( f_1(x)= \frac{1+10x+7x^2}{(1-x)^4} \). Further on the formula should read \( x^2f_1(x^2)+x^3f_2(x^2)\) so in the original formula in the book a minus sign has to be altered.
Chapter 3 is on multidimensional figurate numbers. It starts with pentatope numbers and multidimensional analogues. The formula for \( S^4_3(n) \) is wrong , the second \( S^3_3(n)\) should be an \( S^3_3(2) \).
In Section 3.1 formulae and generating functions for the number of points in \(k\)-dimensional hypertetrahedra are given.
Section 3.2 is on biquadratic numbers and higher dimensional analogues. On page 170 its stated that \(1+4n+6n^2+4n^3=(n+1)^3 \) which is obviously wrong. Moreover, in Section 3.2.2 a very clumsy way is used to obtain the generating function for \( \sum n^4x^n \). Starting from the identity \(\frac{1}{1-x} = \sum x^n \) and applying four times the operator \(x\frac{d}{dx}\) to both sides will do the trick much faster. After a massive calculation a generating function for the \(k\)-hypercube numbers is derived.
Section 3.3 is on other polytope numbers. It starts with the 4-dimensional case giving formulae, generating functions and recurrence relations for hypertetrahedron numbers (section 3.3.2), hypercube numbers (Section 3.3.4), hypericosahedron numbers (Section 3.3.5), hyperdodecahedral numbers (Section 3.3.6) and polyoctahedral numbers (Section 3.3.7). Next the \(k\)-dimensional analogues are discussed more specific \(k\)-dimensional hypertetrahedral numbers (Section 3.3.8), \(k\)-dimensional hypercube numbers (Section 3.3.9) and \(k\)-dimensional hyperoctahedraon numbers (Section 3.3.10). In Section 3.3.9 on the fly Eulerian numbers and some of their properties are introduced and derived. Sections 3.3 ends with Section 3.3.11 showing that regular polytope numbers can be written as linear combinations with nonnegative integer coefficients of \(k\)-dimensional hypertetrahedron numbers, and for some cases the coefficients are given.
Section 3.4 is on Nexus numbers, these are differences of two consecutive higher dimensional hypercubes. Again generating functions and formulae are given.
Section 3.5 is on second order pyramidal numbers and multidimensional analogues. This section again contains some mistakes. In the formula of Section 3.5.1 I, the left hand side should read \( S^4_m(n) \) and the last formula on page 213 should start with \( \frac{1}{4} \) and end with \( S^3_3(n) \). In the formula above Section 3.5.4. a 3 is missing, the right hand side should start with \( S^3_3(k) \).
In Sections 3.5.4, 3.5.5 and 3.5.6 the authors go up to higher dimensions generalizing the previous 4-dimensional results.
Section 3.6 is on centered multidimensional numbers. It gives formulae and generating functions for centered biquadratic numbers (Section 3.6.1), centered \(k\)-dimensional hypercube numbers (Section 3.6.2), centered polytope numbers (Section3.6.3), \(k\)-dimensional centered hypertetrahedron numbers (Section 3.6.4), centered hyperoctahedral numbers also orthoplex numbers (Section 3.5.6) and \(k\)-dimensional centered hyperoctahedron numbers (Section 3.6.6). The formula just above Section 3.6.3 should not contain a \(z\), and it is the formula for 5-dimensional centered hypercube numbers (which is mentioned nowhere).
In Section 3.7 the generalization of the previous numbers is discussed, allowing negative numbers to be entered in the variety of formulae. Apparently the authors had enough of it since they do not generalize all the numbers mentioned before, and I tend to agree with them.
Chapter 4 is on the relation between number and figurate numbers. It starts with pointing out connections between addition and multiplication tables on one hand and certain figurate numbers. Next Pascal’s triangle is studied in relation to figurate numbers. Then connections with Diophantine equations are revisited. In Section 4.3.3. the pair \((5,55)\) should read \((5,11)\). Furthermore, in this Chapter some interesting results are quoted like Fermat’s Last Theorem, and the Catalan conjecture. Section 4.3.6 deals with the Bachet-Meziriac problem of finding five numbers that if multiplied in turn give a triangular, square, cube, pentagonal and biquadratic number, respectively. Section 4.3.7 elaborates on a problem by Saint Croix. Section 4.3.8 touches the Ramanujan-Nagell equation. Section 4.4 is on perfect numbers. It is stated that all even perfect numbers are triangular. Connections with figurate numbers are given for several types of special numbers, more specific Mersenne and Fermat numbers (Section 4.5), Fibonacci and Lucas numbers (Section 4.6), Palindromic numbers (Section 4.7), Catalan numbers, Stirling numbers, Bell numbers, Bernoulli numbers and some other rare types of numbers (Section 4.8), prime numbers (Section 4.9), Magic numbers (Section 4.10). Section 4.11 is on certain types of partitions and their connection with generalized pentagonal numbers. Section 4.12.1 is on Waring’s problem which involves \(m\)-dimensional hypercube numbers. Section 4.12.2 deals with Lagrange’s Four Square Theorem.
Chapter 5 is on Fermat’s Polygonal number Theorem. This Theorem states that every number can be written as sum of at most \(m\) \(m\)-gonal numbers. It starts with a historical overview, which is followed by a prove of the Four Square Theorem. Next the Three Triangular Number Theorem is considered and proved. Then sums of squares in relation to Minkowski’s convex body Theorem are studied. Cauchy’s proof of Fermat’s polygonal number theorem is discussed. Pépin’s proof of this Theorem is shown. Section 5.4.1 contains again some errors. On page 324 the second formula’s left hand side should read \( ax^2+2bxy+cy^2=.....+c(\gamma t + \delta u)^2 \) the rest of the formula is correct. On page 327 in the middle of the page it should state: \( c = \frac{b^2+D}{a} = \frac{0+1}{1}=1 \) in stead of \(\frac{1+1}{1} =1 \). In Section 5.7.2 on page 363, \(R\) should be of the form \(2m+3\) in stead of \(2x+3\). In Section 5.7.3 on page 367 again an error in the middle of the page: \(k = 2B -2l +s \). The formula \( s > \frac{1}{2} + \sqrt{6B-3} \) is wrong, and hence the rest of the proof goes wrong, too. It can be fixed, checking the value of the function \( \frac{2}{3} + \sqrt{8B-8} - \frac{1}{2} - \sqrt{6B-\frac{15}{16}} \) for \(B = 110\) it turns out to be \( 4.024... > 4 \). Since the listed function is monotonic increasing for \(B\) greater than or equal to 110 the function values are always bigger than 4 for \(B\) bigger than 110. So after all the rest of the proof goes through (sigh!). The example on page 370 contains the same error. Next some other results related to the polygonal number theorem are discussed referring to an alternative proof by Dickinson. There are a lot of tables playing a role and also some results of Nathanson are mentioned.
Chapter 6 gives a long list of figurate numbers stating why a particular number in the list is special. Browsing through the list it makes one wonder which is the smallest nonspecial figurate number (just kidding!).
Chapter 7 consist of a lot of Exercises comprising a long list of more or less interesting facts on figurate numbers. There is a solution Section giving hints for solving the exercises.

Reviewer: Herman J. Tiersma (Den Haag)

MSC:

11-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11Axx	Elementary number theory
05-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics
05B99	Designs and configurations

Keywords:

Figurate numbers; Configurations of points; Polygonal numbers; Polytope numbers; Four Square Theorem.

Software:

OEIS