Promenade around Pascal Triangle { Number Motives · Pascal [Pas1665] wrote his Trait e du Triangle...

Bull. Math. Soc. Sci. Math. RoumanieTome 56(104) No. 1, 2013, 73–98

Promenade around Pascal Triangle – Number Motives

byCristian Cobeli and Alexandru Zaharescu

Dedicated to the memory of Nicolae Popescu (1937-2010)on the occasion of his 75th anniversary

Abstract

We survey classical results and recent developments, old and new problems, conjecturesand ideas selected from the endless theme of iterated application of the fundamental ruleof addition. The multitude of forms created by this spring, like the commandment “Letthere be light” from the first day of creation, is emphasized by the role played by the primenumbers. The subject sounds harmoniously between poetry and astronomy or geometry,finding its origin in the East at Pingala (4-2nd century BC) and in the West at Apolloniusof Perga (3rd century BC). Our work is divided in two parts: the present paper is mostlydedicated to playing with numbers, while the second one, which will follow in a companionpaper, is based on geometrical motives.

Key Words: Pascal triangle, Sierpinski gasket, fractals, Collatz conjecture, Thwaitesconjectures, Ducci game, absolute differences, Gilbreath conjecture, greatest primefactor, continued fractions.2010 Mathematics Subject Classification: Primary 11-02,

Secondary 11B65, 11L05, 11B50, 11B85.

1 From history to content

Repeated application of simple rules often produce very complex objects. This fact is revealed innumerous situations (cf. Wolfram [Wol’02]). Pascal Arithmetic Triangle is generated recursivelyby two basic rules:

(i) boundary conditions; (ii) insertion method. (R)

To obtain a row: (i) copy the row above and attach a 1 at each endpoint; (ii) between anytwo consecutive elements that come from the previous row insert their sum; then delete theolder entries. Small changes in (i) and (ii) may produce very different constructions. Although,

74 Cristian Cobeli and Alexandru Zaharescu

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 11 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 11 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 11 11 55 165 330 462 462 330 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 11 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1

1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 11 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1

1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 11 17 136 680 2380 6188 12376 19448 24310 24310 19448 12376 6188 2380 680 136 17 1

1 18 153 816 3060 8568 18564 31824 43758 48620 43758 31824 18564 8568 3060 816 153 18 11 19 171 969 3876 11628 27132 50388 75582 92378 92378 75582 50388 27132 11628 3876 969 171 19 1

1 20 190 1140 4845 15504 38760 77520 125970 167960 184756 167960 125970 77520 38760 15504 4845 1140 190 20 11 21 210 1330 5985 20349 54264 116280 203490 293930 352716 352716 293930 203490 116280 54264 20349 5985 1330 210 21 1

1 22 231 1540 7315 26334 74613 170544 319770 497420 646646 705432 646646 497420 319770 170544 74613 26334 7315 1540 231 22 11 23 253 1771 8855 33649 100947 245157 490314 817190 1144066 1352078 1352078 1144066 817190 490314 245157 100947 33649 8855 1771 253 23 1

1 24 276 2024 10626 42504 134596 346104 735471 1307504 1961256 2496144 2704156 2496144 1961256 1307504 735471 346104 134596 42504 10626 2024 276 24 11 25 300 2300 12650 53130 177100 480700 1081575 2042975 3268760 4457400 5200300 5200300 4457400 3268760 2042975 1081575 480700 177100 53130 12650 2300 300 25 1

1 26 325 2600 14950 65780 230230 657800 1562275 3124550 5311735 7726160 9657700 10400600 9657700 7726160 5311735 3124550 1562275 657800 230230 65780 14950 2600 325 261

Figure 1: The first 27 rows of Pascal triangle with entries written in base 10.

they usually share remarkable arithmetic, geometric, topologic, and probabilistic properties. Ineach case, a ’kaleidoscope of mirrors’, or the multitude of symmetries involved is perhaps themost appealing. We survey various such constructions, recent results and open questions.

Known for more than a millenium in Asia and Europe (cf. Burton [Bur’07]), the origins ofthe Pascal triangle are lost in the mist of time. In Chandah. sastra, the Hindu scholar Pingala hasclassified meters (chandas) or rhythm of poems that are closely allied to music (Bag [Bag’66]).He enumerated and counted the meters of a given length n that have exactly r syllables of a kind.In doing this, he obtained Meruprastara (the stairway to the mythical mountain Meru). ThenHalayudha (cca. 975) in Mr.tasanjıvanı, a text of commentaries on Pingala’s Chandah. sastra,clearly described Meruprastara as what is today known as the arithmetic triangle of Pascal.Among those who considered the triangle before Pascal, we find: Al-Karaji (953-1029), Persia;Jia Xian (1010-1070), China; Al-Karaji (953-1029); Al-Samawal al-Maghribi (1030-1080) inIraq, Marocco, Iran in his treatise The brilliant in algebra; Omar Khayyam (1048-1131), it iscalled Khayyam triangle in Iran; Yang Hui’s (1238-1298) triangle in China; in 1303 was namedthe Old Method by Chu Chijie in Siyuan yujian (Jade Mirror of the Four Unknown). Andin the West: Ramon Llull, a Majorcan theologian (1232–1316); Niccolo Tartaglia in GeneraleTrattato (1556) (the triangle bears his name in Italy); Michael Stifel in Arithmetica Integra(1544); in 1527, Petrus Apianus puts the triangle on the frontispiece of his book. De Moivrenamed it Triangulum Arithmeticum Pascalianum in 1730.

Pascal [Pas1665] wrote his Traite du Triangle Arithmetique in 1654. It was the result ofa fruitful correspondence with Pierre de Fermat about calculating the odds in some games ofchance. The relations between binomial coefficients proved by Pascal, later led Newton to thediscovery of the binomial theorem for negative and fractional exponents, and Leibniz to thediscovery of infinitesimal calculus1. Pascal triangle is extremely rich in displaying remarkablesequences of integers, triangular numbers, Fibonacci and Padovan numbers, squares, powers,Catalan, Bernoulli and Stirling numbers, etc. Consequently, there are so many relations, that’when someone finds a new identity, there aren’t many people who get excited about it anymore, except the discoverer’ (cf. Donald Knuth, The art of computer science, Vol. I, Chap. 1).

1’On several occasions Leibniz was to declare that he was led to the invention of calculus more by studyingPascal’s writings than anything else.’ (cf. Burton [Bur’07, pages 412–413])

Promenade around Pascal Triangle – Number Motives 75

The following matrices [BC’04], [MP’09] nicely relate the triangle in the form used by Pascalwith one close to what we use today. Let L(∞)i,j =

(ij

), i, j ≥ 0, where,

(ij

)= 0 if i < j.

L(∞) =

11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 1...

......

......

......

. . .

= exp

01 00 2 0

0 3 0. . .

and

P (∞) =

1 1 1 1 1 · · ·1 2 3 4 5 · · ·1 3 6 10 15 · · ·1 4 10 20 35 · · ·1 5 15 35 70 · · ·...

......

......

. . .

Then

P (∞) = L(∞) · L(∞)T ; tr(P (n)

)=

n−1∑k=0

(2k)!

(k!)2and det(P (n)) = 1.

But looking further than pure identities, a more optimistic thought becomes appropriate,that of Jacobi Bernoulli [Ber1713, Part II, Chapter 3, page 88], who used the triangle to get intothe intricacies of sums of consecutive p powers. He admired the exceptional properties of thetriangle, which conceals within itself mysteries of combinations and top secrets of mathematics.

2 The Shape of the game

The mountain shape in Figure 1 captures some magic of the triangle. One can use Stirling’sformula to study the shape of the outer curve. More generally, let b ≥ 2, n and S be fixedpositive integers and denote by db(m) the number of digits of m in base b. Then denote:

mb(n, S) := mina1+···+an=Sa1,...,an∈N\{0}

n∑k=1

db(ak) and Mb(n, S) := maxa1+···+an=Sa1,...,an∈N\{0}

n∑k=1

db(ak) .

Question 1. Estimate mb(n, S) and Mb(n, S).

One may ask about the distribution of digits in Figure 1. The first digit of the binomialcoefficients appears to satisfy Benford’s law. The triangle has a different shape in anothernumber system, such as the Chinese one used by Chu Chijie in 1303 or the Arabic one employedby Al-Samawal al-Maghribi.


Question 2. Find the shape of the triangle in closed form.

Problems similar to Questions 1 and 2 may be addressed for the shape of individual rowsof the triangle (see Figure 2). The thickness of the lens shape figure that corresponds to then-the row of the Pascal triangle is ≈ n. Remark that if n = 2m is even, the number that givesthat thickness is (m+ 1)Cm, where Cm is the m-th Catalan number.

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360

1

Figure 2: The 360th row of Pascal triangle. The binomial coefficients(360k

), 0 ≤ k ≤ 360, written

in base 10 are displayed centered, and the figure is rotated by 90 degrees in the counterclockwisedirection.

3 Singmaster Conjecture

Singmaster [Sng’71], [Sng’75] guesses that there are not too many points on any ’parabola’ thatcuts Pascal triangle passing only through entries greater than one of the same size. He provedthat N(t) = O

(log t

)for t ≥ 2, where

N(t) :={m : m binomial coefficient, t = m

},

and made the following conjecture.

Conjecture 1 (Singmaster). The function N(t) is bounded.

This seems very difficult to prove, although Singmaster believed that N(t) ≤ 10 or 12 shouldbe the real margin. The best upper bound so far is:

N(t) = O

((log t)(log log log t)

(log log t)3

),

due to Kane [Kan’07], who improves on previous estimates of himself, Abbott, Erdos andHanson.

As for lower bounds, the results obtained so far are sporadic: N(2) = 1, N(3) = N(4) =N(5) = 2, N(6) = 3; if t < 1028 then N(t) = 6 for t = 120, 210, 1540, 7140, 11628, 24310;N(3003) = 8 since

(30031

)=(782

)=(155

)=(146

), the single known t for which N(t) = 8.

Singmaster proved that N(t) ≥ 6 infinitely often, but there are not known solutions of equationsN(t) = 5 or 7.


4 Odds and Ends in Pascal Triangle – Sierpinski Gasket

Recognition of primes is an extremely difficult problem. The Pascal triangle encodes enoughinformation to offer a genuine image of every positive integer. Looking at the entries in Pascal

Figure 3: The first 107 rows of Pascal triangle modulo 2 on the left and modulo 7 on the rightwith residue classes drown in distinct colors.

triangle modulo n, for n = 2, 3, 4, 5, . . ., using colors to distinguish distinct residue classes, onemay see the face of a prime. It is striking to see the relative order in these images when nis prime (two examples are shown in Figure 3), as opposed to the ’chaos’ when n is highlycomposite. A systematic study to discover some sort of 2-dimensional distribution is yet to bedone.

Pascal triangle modulo prime powers brought the attention of many mathematicians. Amongthem are Legendre, Cauchy, Gauss, Kummer, Hermite, Hensel, Lucas, Granville [Gra’95].Singmaster[Sng’80] studied the distribution of entries that are equal to zero in the Pascaltriangle modulo n. Actually, the most attention by far was given to the Pascal triangle mod-ulo 2, the Sierpinski triangle. Glaisher [Gla1899] proved that the odd numbers in any rowof the Pascal triangle is always a power of 2. For higher powers of 2 and similar problemsfor other primes the reader is referred to Granville [Gra’92], [Gra’97], Huard et al. [HSW’97],[HSW’98], Mihet[Mih’10], Rowland [Row’11a], [Row’11b] and Shevelev [She’11]. Another in-triguing property that relates the Pascal triangle modulo 2 with Fermat numbers was discoveredby Hewgill [Hew’77]. Let c(n, j) :=

(nj

)(mod 2) and let Fj := 22

j

be the Fermat numbers forj = 0, 1, . . . Then

n∑j=0

c(n, j)2j = F d0

0 F d1

1 · · ·F drr , (4.1)

where d0, . . . , dr ∈ {0, 1} are the digits of n in base 2, that is n = d0 + 2d1 + · · · + dr2r and

dr 6= 0. Reading the rows of the Pascal triangle modulo 2 as numbers written in base 2, we


obtain the expressions from (4.1). These numbers are:

1, 3, 5, 15, 17, 51, 85, 255, 257, 771, . . . (4.2)

The 32-nd is 232 − 1 = 4294967295, which is the product of 3, 5, 17, 257 and 65537, theonly known Fermat primes (cf. Cosgrave [Cos’99], Dubner and Gallot [DG’01]). Gauss provedthat the regular N -gons constructible with the ruler and the compass are those for which Nis a square free product of Fermat primes multiplied by a power of 2. Gardner [Gar’77] andWatkins [CG’96], [KLS’01] observed that the known constructible N -gons with N odd arethose for which N is one of the first 32 numbers of the sequence (4.2). Close resemblance toGlaisher’s result are relations between binomial coefficients and Bernoulli numbers, such as

n∑j=0

Bj

(n

j

)= Bn .

In a work of Lehmer [Leh’35], one finds:

n∑j=0

B6j

(6n+ 3

6j

)= 2n+ 1

andn∑j=0

B6j+2

(6n+ 5

6j + 2

)=

1

3(6n+ 5) .

Kummer and later Vandiver [Van’39] obtained similar relations modulo powers of primes, whichhave various applications (cf. [Car’68]). Their congruences employ sums of binomial coefficientswith weights Euler numbers or Bernoulli numbers. Later, Carlitz (see [Car’68] and the refer-ences therein) obtained more general results of this kind. Sierpinski triangle, first outlined inmathematical form by Sierpinski [Sie’15] appeared beforehand in the XIII-th century Cosmatimosaics in cathedrals of Rome region (see Wolfram [Wol’02, page 43]). There are several waysto define the Sierpinski triangle (one of them being nondeterministic2). The most commonone is to start with a given triangle, delete from its interior the triangle determined by themidpoints of the edges, then do the same thing with the three remaining triangles, and repeatthe process forever. The set of remaining points is the Sierpinski Gasket (Figure 3, left). Itis a basic type of fractal, and its dimension δS is easy to find: doubling its size, it replicatesthree times the original triangle, so 3 = 2δS , and δS = log 3/ log 2 ≈ 1.58496. Fraenkel andKontorovich [FK’07] recover Sierpinski triangle and its connection to binomial coefficients inthe context of a p-variety XY with a Nim-product and p-sieve with p prime. Sierpinski gasketis a fertile ground of investigation, where one uses tools from analysis, pde, harmonic functiontheory, number theory, see Ben-Bassat et al. [BST’99], Needleman et al. [NSTY’04], Ben-Galet al. [BSSY’06], Hinoa and Kumagai [HK’06], Teplyaev [Tep’07], DeGrado et al. [DRS’09],image recognition and processing, theoretical or applied physics Huzler [Huz’08], Daerden andVanderzande [DV’98], Pradhan et al. [PCRM’03], Belrose [Bel’04].

2Take a triangle and label its vertices by the numbers 1, 2, 3. Pick a point P in its interior. Cast a 3-facedie. Draw the midpoint of the segment determined by P and the chosen vertex, and let P be this drawn point.Repeat the process endlessly. The collection of drowned points is the Sierpinski triangle.


5 Cousins of Pascal Triangle

Replacing the 1-s on the edges of the triangle with different sequences produces interestingoutcomes. Hosoya [Hos’76] proposes a Fibonacci triangle where each entry is the sum of theprevious two numbers in either the row or the column:

HF (∞) =

1 1 2 3 5 8 · · ·1 1 2 3 5 8 · · ·2 2 4 6 10 16 · · ·3 3 6 9 15 24 · · ·5 5 10 15 25 40 · · ·8 8 16 24 40 64 · · ·...

......

......

.... . .

Actually, this is the multiplication table of Fibonacci numbers. Here the sum of entries ajkwith j + k = n, are the first convolved Fibonacci numbers (see Koshy [Kos’01] and Klavzarand Peterin [KP’07]). Falcon and Plaza [FP’07] study a Pascal 2-triangle using the k-Fibonaccisequence, while Fahr and Ringel [FR’12] change the rule of insertion to get Fibonacci partitiontriangles.

The sequences on the boundary can be quite arbitrary. Given u = (u0, . . . , un−1) andv = (v0, . . . , vn−1), with u0 = v0, put P (j, 0) = uj , P (j, j) = vj for 0 ≤ j ≤ n and

P (j, k) = P (j − 1, k − 1) + P (j − 1, k), for 2 ≤ j ≤ n, 1 ≤ k ≤ n− 1.

Then Pu,v = (P (j, k))j,k is the generalized Pascal triangle with u and v the generating edges.One can also consider triangles with no boundary conditions. Steinhaus’ definition is as fol-

lows. Let x = (x0, . . . , xn−1) and ∂x := (x0 +x1, . . . , xn−2 +xn−1). Then, recursively ∂0x = x,∂1x = ∂x, and ∂ix = ∂∂i−1x. This generates the shrinking triangle S(x) = (x, ∂x, . . . , ∂n−1x),whose lines are shorter and shorter, the last one, ∂n−1x, containing just one number. One pro-blem is to characterize the x-s based on the shape of the produced triangles. Molluzzo [Mol’76]and J. Chappelon [Cha’08] considered Steinhaus triangles with entries in Zm. Most questions onSteinhaus or generalized Pascal triangles refer to triangles with components in F2, also namedBoolean triangles. Steinhaus [Ste’58], Harborth and Hurlbert [Har’72], [HH’05] estimate thenumber of triangles that are balanced on the number of entries. Kutyreva and Malyshev [KM’06]estimate the number of Boolean Pascal triangles of size n containing a positive proportion ofones. Eliahou et al. [EH’04], [EH’05], [EMR’07] investigate binary sequences that generatetriangles with symmetries, while Brunat and Maureso [BM’11] give explicit formulae for thenumber of these binary triangles having rotational and dihedral symmetries.

6 Euclid-Mullin sequences

The sequence P of prime numbers, like the Pascal arithmetic triangle, is also generated by asimple rule (Eratostene’s sieve). Gallagher [Gal’76], [Gal’81] showed that primes appear like ina Poissonian process. Goldston and Ledoan [GL’12] show Poisson distribution for individualspacings between neighboring primes. The result is implicitly contained in an earlier work of


Odlyzko, Rubinstein and Wolf [ORW’99] on jumping champions. All these results are provedassuming the prime k-tuple conjecture of Hardy and Littlewood.

Subsequences of primes constructed recursively are natural places to look for symmetries.If p1, p2, . . ., pr is a list of known primes, then the integer p1 · p2 · · · pr + 1 is not divisible byany of p1, p2, . . ., pr, so it is either prime or divisible by a different prime. In any case, a newprime pr+1 may be added to the list. Choosing various selection rules of primes from the setof divisors of nr = p1 · p2 · · · pr + 1, we get a large class of sequences of primes (cf. Caldwelland Gallot [CG’01]). For example, one may work with spf(nr), the smallest prime factor ofnr, or gpf(nr), the greatest prime factor of nr. These choices produce the first Euclid-Mullinsequence:

2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, . . . (6.1)

and the second Euclid-Mullin sequence:

2, 3, 7, 43, 139, 50207, 340999, 2365347734339, 4680225641471129, . . . (6.2)

These sequences are not monotonic, but their members do not repeat. Also, they are infinite,but since factoring large numbers is a very difficult task, only few terms of each of them areknown: the first 47 members of the first one and 13 of the second one. Mullin [Mul’63] askedif the sequence (6.1) contains all primes, while Cox and Van der Poorten [CV’68] found a fewprimes (5, 11, 13, 17, and a some others) that are absent from (6.2) and conjectured thatthere are infinitely many of them. This conjecture was proved by Booker [Boo’12]. Sloane’sEncyclopedia of Sequences, started in 1964, today in the form of OEIS foundation [OEIS], andthe references therein collect more information about the Euclid-Mullin sequences and someothers that are related to them.

7 Baby-Fractal sequences of numbers

Kimberling [Kim’95] defines a fractal sequence of numbers as one that contains itself as a propersubsequence. For example, the sequence

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, . . .

is fractal because we get the same sequence after we delete from it the first appearance ofall positive integers. This sequence appears in a card sorting algorithm of Kimberling andShultz [KS’97], [Kim’97]. Simple examples of fractal sequences of numbers are constant or cyclicsequences, or sequences that repeat periodically except for finitely many terms. We call thembaby-fractal sequences. The ‘3n+ 1’ type problems, iterated applications of absolute differences(Thwaites [Thw’96a]) and the gpf-sequences produce examples galore of such sequences.

7.1 The ’3n+ 1’ conjecture

For any positive integer n, let C(n) = n/2, for n even and C(n) = 3n + 1, for n odd. Forexample, starting with n = 7, repeated application of C(·) gives:

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8,4,2,1, 4, 2, 1, . . .


The ’3n + 1’ conjecture asserts that for any n ≥ 1, the sequence{C(k)(n)

}k≥1 eventually

enters into the cycle 1, 4, 2. The number of iterations necessary to reach the cycle may be largeeven when n is small (for example, starting with n = 27, it takes 111 steps to reach 1). Collatz[Cox’71] mentions the ’3n + 1’ problem for the first time in a mathematics journal, but theproblem entered the mathematical folklore in the early 1950’s. It was first posed by Collatzin 1937 and independently by Thwaites (cf. Wirsching [Wir’98]). Thwaites [Thw’85] indicateseven the day and the hour of his discovery, while Collatz [Col’86] says that because he couldn’tsolve it, he did not publish anything, but the problem was publicized in seminars in differentcountries after he told it to Helmut Hasse. In spite of numerous tries to solve it, the conjectureis wide open. Wirsching [Wir’98], Chamberland [Cha’03b] and Lagarias [Lag’11], [Lag’12] wrotesurveys on the problem. Functions analogue to C(·) and similar problems gather into a largecategory, but a general statement on periodicity of the generated sequences looks similar toone about a Turing machine, which falls beyond the undecidability line. This was proved byConway [Con’72], while Kurtz and Simon [KS’07] showed that a certain generalization of the’(3n+ 1)’-problem is undecidable.

7.2 The greatest prime factor, a resourceful tool and a startling phenomenon

Let spf(n) be the smallest prime factor and gpf(n) the greatest prime factor of any integer n ≥ 2.These functions are simple to define but very deep in nature. In recent studies, mostly, thegpf(·) function was employed and many generated sequences are like baby fractals, terminatingin surprising cycles. Back and Caragiu [BC’10] combined the Fibonacci growing rule with thegpf(·) function in the role of a molifier. The new terms are obtained by a fixed combinationof neighbor terms. In this sense, the Fibonacci growing rule resembles exactly the Pascal-rule,since Fn = Fn−1 +Fn−2 for n ≥ 2 and F0 = 0, F1 = 1. The general idea is to consider recursivesequences of primes produced by the linear formula:

pj = gpf(a0 + a1pj−1 + a2pj−2 + · · ·+ arpj−r) . (7.1)

(Here, the coefficients a0, . . . , ar are non negative integers.) The case r = 1 and a0 > 0 whereinvestigated by Caragiu and Scheckelhoff [CS’06] and by Caragiu and Back [BC’09]. In thiscase, they showed that when a0 divides a1 the generated sequences are always ultimately peri-odic. Furthermore, based also on computer investigations, they conjecture that this propertycharacterizes all these sequences that do not necessarily satisfy the divisibility constrain on thecoefficients.

The gpf-Fibonacci sequences are those generated by pj = gpf(pj−1 + pj−2), for j ≥ 2. Forinstance, if p0 = 509, p1 = 673, the first elements are:

509, 673, 197, 29, 113, 71, 23, 47,7,3,5,2, 7, 3, 5, 2, . . .

Back and Caragiu [BC’10] proved that if the first two terms of a gpf-Fibonacci sequence aredistinct, then it eventually enters into the 4-cycle 7, 3, 5, 2. Similarly, the gpf-Tribonacci se-quences are given by pj = gpf(pj−1 + pj−2 + pj−3), for j ≥ 3. Two examples of gpf-Tribonaccisequences are:

9431, 563, 523, 809, 379,59,43,37,139,73,83, 59, 43, 37, . . . . . . (7.2)


and

31,13,7,17,37,61,23,11,19,53,83,31,167,281,479,103,863,

17,983,23,31,61,23,23,107,17,7,131, 31, 13, 7, . . .(7.3)

Again, both sequences (7.2) and (7.3) enter into cycles. The length of the first one is 6, whilethe length of the second one is 28. This phenomenon is widely spread, but these two cyclesare quite rare. In fact, most astonishingly, in the large class of gpf-Tribonacci sequences, only4 distinct cycles where discovered so far. The other two have length 100 and 212. Back andCaragiu observed that almost 99% of gpf-Tribonacci sequences with p0, p1, p2 ≤ 1000 enter intoone of the longer cycles and the one of length 100 is encountered about three times more oftenthan the one of length 212. They state the following conjecture.

Conjecture 2 (Back, Caragiu [BC’10]). All recurrent sequences of primes defined by relation(7.1) are ultimately periodic.

A multidimensional version of the problem with vector analogues of gpf-sequences is formu-lated by Caragiu, Sutherland and Zaki [CSZ’11]. Using bounds on the spacings between consecu-tive primes, Caragiu, Zaki and one of the authors [CZZ’12] found the rate of growth of the relatedinfinite order recurrent sequences of primes defined by qj = gpf(q1+q2+ · · ·+qj−1), for j ≥ 2.They showed that qj = j/2 +O

(j0.525

).

7.3 Ultimately periodic sequences in Ducci games

Let d ≥ 2 be a positive integer, let Nd be the set of sequences with nonnegative integer en-tries that are periodic of period d, and define the application φd : Nd → Nd, φd(a0, a1, . . .) :=(a′0, a

′1, . . .), where a′j = |aj−aj−1| for j ≥ 0. Let a ∈ Nd and denote by ||a|| the largest element

of a. Since in Nd there are finitely many sequences whose components are ≤ ||a||, it followsthat starting with any a ∈ Nd and applying φd repeatedly, produces a sequence that eventuallyenters into a cycle. The first interesting fact is that the length of each cycle is 1 if and only ifd is a power of 2. Thus,

φ(n)

2k(a) = (0, 0, . . .), for any k ∈ N, a ∈ N2k and sufficiently large n. (7.4)

Pairing neighbor numbers placed around a circular table and taking absolute differencesrecursively is just a different setting of the same problem. The original Ducci game beginswith only four numbers, and the evolution may be viewed in a Diffy box. Start by placing thenumbers on the corners of a square. Then, for the next generation, the absolute differences ofneighbor numbers are recorded on the middle of the edges. This produces a new square, onwhich the operation is repeated. The iterated process eventually ends with 4 zeros. Duringthe action a graph is generated and one may consider a more developed game, by taking theabsolute differences of neighbor numbers on the other edges. In the same way, a similar gamemay be played on different graphs. This leads tangentially to Golomb rulers problems, codetheory and even worldly applications (Malkevitch [Mal’12]).

The origin of the problem is not completely clear (cf. Thwaites [Thw’96b] and Behn et al.[BKP’05]). The article of Ciamberlini and Marengoni [CM’37] seems to be the first published


source on the subject. Ciamberlini and Marengoni begin their work by stating that Prof. Duccihas communicated the problem to them long before. This might had happened in the XIXthcentury, as both Enrico Ducci (1864–1940) and Corrado Ciamberlini (1861–1944) lived at thetime. Anyhow, Thomas et al. [CT’04] and [CST’05] place the first reference to Ducci gamesat the end of the XIXth century. But for this, their cited support is Honsberger [Hon’98],whose single reference from page 73 attributes the cyclic quadruples game to an observationmade by professor E. Ducci in the 1930’s. The problem was discovered repeatedly, and severalindependent proofs where given (see Ciamberlini and Marengoni [CM’37], Meyers [Mey’82],Pompilli [Pom’96], Thwaites [Thw’96b], Andriychenko and Chamberland [AC’00], Crasmaruand the authors[CCZ’00], Chamberland [Cha’03a] and the references therein).

Following a note of Campbell [Cam’96] that advertized Thwaites conjectures [Thw’96a] andbeing unaware of the other previous results, Crasmaru and the authors [CCZ’00] gave threedistinct proofs of problem (7.4), the second £100–conjecture of Thwaites. Regarding the cycles,it turns out that the essence of the evolution function φ is captured by its restriction to Ud, thesubset of all the elements of Nd with components in {0, 1}. The advantage is that the evolution

function restricted in this way is additive. Then the j-th component of φ(n)|Ud (a0, a1, . . .) is equal

ton∑k=0

(n

j

)aj+k (mod 2).

Moreover, one only needs to understand the transforms of the unitary sequence e0, whosecomponents are all equal to zero, except those of rank divisible by d, which are equal to one.In this case we have

φ(n)(e0) =(Sd(n, 0), Sd(n,−1), Sd(n,−2), . . .

)(mod 2), (7.5)

where

Sd(n, r) =∑

1≤k≤nk≡r (mod d)

(n

k

).

Expression (7.5) is used in [CCZ’00] to investigate the general problem of finding the length ofthe cycles for arbitrary d. It is shown that the periods (multiples of the length of cycles) dependon the order of 2 modulo the largest odd factor of d. Short periods occur when d = 2p − 1 isa Mersenne prime (47 such primes are known so far: 3, 7, 31, 127, 8191, . . . , 243 112 609 − 1).In this case d is the length of the cycle. Long cycles may happen when d is prime and 2 is aprimitive root modulo d. (For example 3, 5, 11, 13, 19, 29, 37, . . . and Artin’s conjecture states

that there are infinitely many such primes.) In these cases d(2d−1

2 − 1)

is a period. Anyhow,very long cycles do occur, because known partial results on Artin’s conjecture allow to deduce

that there are infinitely many primes d for which the length of the cycle is larger than 2d1/4

.A more general evolution function is defined as follows. Let S = {0, 1}d. Denote by ρ(x) the

circular rotation to the right of the vector x ∈ S (e.g., for d = 5, ρ(1, 0, 1, 0, 0) = (0, 1, 0, 1, 0) )and · : S × S → S the componentwise addition modulo 2. Let s and α1, . . . , αs be positiveintegers and define θ : S → S by

θ(x) = ρ(α1)(x) · · · ρ(αs)(x).


Since θ(x) has an accentuated chaotic character, Crasmaru [Cra’01] initiated the constructionof a cryptosystem based on the fact that there is an effective way to calculate θ(x). For s = 2,α1 = 0 and α2 = 1 the function θ(x) replicates Ducci’s evolution function. There is an efficientalgorithm to calculate θ(n)(x) in Os(log n) steps (cf. [CCZ’00]). In the case θ(x) = xρ(x), theprocedure is based on the formula

θ(n)(x) = x∏

R⊂P(Rn)

ρ

(∑r∈R

r)(x), x ∈ S, (7.6)

where n = 2l0 + 2l1 + · · ·+ 2lµ with l0 < · · · < lµ is the representation of n in base 2, and

Rn ={r : r ≡ 2li (mod d), 0 ≤ r ≤ d− 1, for some 0 ≤ i ≤ µ

}.

For any m ∈ {1, . . . , d} denote

νn,d(m) = #{R ⊂ Rn :∑r∈R

r ≡ m (mod d)}.

The representation (7.6) and the fact that (0, . . . , 0) and (1, . . . , 1) ∈ S are the only fixedpoints of ρ(x) yield a criterion of periodicity, which states that a positive integer n is a periodfor θ(x) = xρ(x) if and only if the numbers νk,d(m), 1 ≤ m ≤ d, have the same parity (cf.[CCZ’00, Corollary 3]). Furthermore, computer experiments show that when n is the length ofthe shortest period, the numbers νn,d(m) are most of the time equal.

Conjecture 3 ( [CCZ’00]). Suppose d is prime, s is the order of 2 mod d, s is even andn = d(2s/2 − 1). Then

νn,d(1) = νn,d(2) = · · · = νn,d(d− 1) = νn,d(d) + 2.

As consequence of Conjecture 3, we get precise values for the length of the periods of Ducci’sevolution function θ(x) = xρ(x). The stretch of the initial iterations before the games enterinto the cycle and various results on the length of cycles in the n-Ducci game for particularvalues and for n satisfying divisibility constrains were obtained by Webb [Web’82], Ludington[Lud’88], Ehrlich [Ehr’90], Ludington-Young [LY’90] and [LY’99], Creely [Cre’88], Calkin et al.[CST’05], Lidman and Thomas [LT’07], Brown and Merzel [BM’07]. Some generalizations ofthe Ducci game with weights, with p-adic integers or with algebraic numbers are considered inChamberland [Cha’03a], Breuer [Bre’07] and [Bre’10], Baxter and Caragiu [BC’07], Caragiu,Zaki and the second author [CZZ’11], while the close relation between the length of the cyclesand Pascal triangle modulo 2 is studied by Glaser and Schoffl [GS’95], and Breuer [Bre’98].

8 p-adic functions and convergents to e

The convergents of continued fractions of linear fractional transformations involving the Eulernumber e and the special exponentials e2/h reveal augmented symmetries. Farther, runningacross the sequence of denominators of convergents of the continued fraction of e, one finds


that their divisibility properties exhibit attractive ’supercongruences’ modulo powers of primesfrom a distinguished set B (cf. Berndt et al. [BKZ’12]), which are encoded by six remarkablep-adic functions. These functions satisfy certain functional equations and are represented bybinomial coefficient series, whose coefficients also carry the signature of the entries from thePascal triangle.

Since 1873, when Hermite [Coh’06] proved that e is not an algebraic number, continuedfractions became an important instrument not only in diophantine approximation, but also infinding the barrier between transcendence and algebraicity. There is a regularity in the con-tinued fraction of e, more precisely e = [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, . . .]. Hurwitz (cf. Perron[Per’54, §33]) proved a general statement, which roughly asserts that if the terminal compo-nents of the continued fraction for a number r consists of a few arithmetic progressions, then(ar + b)/(cr + d), were a, b, c, d ∈ Z and ad − bc 6= 0 has the same property. For example, wehave:

1

2e− 5= [2, 3, 2, 3, 1, 2, 1, 3, 4, 3, 1, 4, 1, 3, 6, 3, 1, 6, 1, 3, 8, 3, 1, 8, 1, 3, . . .] (8.1)

and

−36e+ 98

78e− 212= [5, 2, 5, 1, 2, 5, 1, 2, 1, 5, 4, 5, 1, 4, 5, 1, 4, 1, 5, 6, 5, 1, 6, 5, 1, 6, 1, 5, . . .] . (8.2)

A similar result occurs for generalized continued fractions. Inspired by Angel’s [Ang’10] in-vestigation of a family of generalized continued fractions that converge to rational numbers,Gottfried Helms found, by computer experiments, a few examples that converge to rationalexpressions involving e. In particular, the expressions that are analogous to relations (8.1) and(8.2) are:

1

2e− 5= 2 +

1

3+

2

4+

3

5+

4

6+· · ·

and−36e+ 98

78e− 212= 5 +

3

6+

4

7+

5

8+

6

9+· · ·

The result of Hurwitz applies as well to some exponentials r = ea/h, with a, h positive integers.The following expressions were already known by Euler, Stieltjes and Hurwitz:

e1

q = [1, (2j − 1)q − 1, 1]∞j=1 = [1, q − 1, 1, 1, 3q − 1, 1, 1, 5q − 1, 1, 1, 7q − 1, . . .] (8.3)

for q ≥ 1, and

e2

2q+1 = [1, (1 + 6j)q + 3j, (12 + 24j)q + 6 + 12j, (5 + 6j)q + 2 + 3j, 1]∞j=0, (8.4)

for q ≥ 0. Thakur [Tha’96] further extended Hurvitz’s result. He found the patterns dis-played by the simple continued fractions for the analogues of the fractional transform (ae2/h +b)/(ce2/h + d) in function fields Fq[t]. We remark that relations (8.4) and (8.3) are special, inthe sense that no such relations are known for e3 or for other powers of e. Also, the number ofarithmetic progressions needed to express the tail of (8.3) and (8.4) is small (three, respectivelyfive), but their fractional transforms may need many. For example, 94 arithmetic progressionsare needed to describe the tail of the continued fraction of 3e1/3 − 1/3.


Ramanujan, and later Davis, found best possible diophantine approximations of e2/a (seeBerndt et al. [BKZ’12] for a detailed description). Sondow and Schlam’s investigations [Son’06],[SS’08], [SS’10], compare convergents of the simple continued fraction of e to the partial sumsof the Taylor series of e. Berndt, Kim and the second author [BKZ’12] proved that at mostOa(

log n)

of the first convergents of e2/a may coincide with partial fractions of the series

e2/a =

∞∑j=0

2j

aj j!. (8.5)

Furthermore, as a consequence of a finer analysis in the case a = 2, they proved that equalityoccurs only twice, which settles a conjecture of Sondow [Son’06]. This required the study ofthe symmetries of the sequence {qj}j∈Z of the denominators of convergents. (Here, for j ≤ 0the definition of qj is made naturally, by the same recursive defining relation of the convergentsfor j ≥ 0.) An important role is played by their images modulo powers of primes selected froma distinguished set of primes: B :=

{p : qj+6p ≡ qj (mod p), for any j ∈ Z

}. There are other

equivalent ways to define the set B, whose first elements are:

B :={

3, 7, 11, 17, 47, 53, 61, 67, 73, 79, 89, 101, 139, . . .}.

Based in part on numerical data, in [BKZ’12] the following conjecture is made:

Conjecture 4 (Berndt, Kim, Zaharescu [BKZ’12]).

limx→∞

#{p ∈ B : p ≤ x}π(x)

=1

e.3

The divisibility and the congruence properties of the sequence {qj}j∈Z are captured by a fewfunctions fr : Z → Z defined by fr(l) := q6l−r, for l ∈ Z. These functions are 1-Lipschitzianwith respect to the p-adic absolute value | · |p on the field of p-adic numbers Qp, normalized by|p|p = 1/p. Consequently, by a result of Mahler [Mah’58], they can be represented in a Pascalseries

fr(x) = ar,0 + ar,1x+ ar,2x(x− 1)

2+ · · ·+ ar,j

(x

j

)+ · · · ,

where

ar,j =

j∑k=0

(−1)k(j

k

)f(j − k) ,

that is, ar,0 = fr(0) = q−r, ar,1 = fr(1)− fr(0) = q−r+6 − q−r, ar,2 = fr(2)− 2fr(1) + fr(0) =q−r+12 − 2q−r+6 + q−r, . . . This is reminiscent of continued fractions of Hurwitz type, whosedefinition involves binomial coefficients and the iterated difference operator (cf. Perron [Per’54,§32]). Of these functions, only six of them fr(x) for r = 1, . . . , 6 are essential. This sextupleof functions is intrinsically linked by a few functional relations. Moreover, each fr(x) has aunique extension by continuity to a function defined on Zp, the ring of integers in Qp, and the

3 Unrelated to B, but based on similar heuristic arguments, the proportion of missed residue classes modulop by the set of factorials {1!, 2!, . . . , p!} is conjectured by Richard K. Guy to have the same limit [CVZ’00].


extensions satisfy the same functional equations. Among the properties of any function fr(x)that are reckoned as interesting to investigate, Berndt et al. [BKZ’12] highlight the naturalquestions:

Question 3 (BKZ2012). (a) Is fr(x) differentiable? (b) Does fr(x) has any zeros exceptcertain trivial ones?

We remark that Question 3 a. is equivalent to the limit ar,j/j → 0 as j →∞.

9 A whopping class of fractals

The authors [CZ’12] consider triangles generated by a multiplicative rule. These are similarto Pascal arithmetic triangle, except that the length of the new lines appears as decaying. Inreality, triangles may have any size, since we view them as chunks cut off from an infinitetriangle, if the first generating row is so. Let us consider the following function

Z(a, b) :=ab

gcd(a, b)2.

Starting with a given sequence of integers, the subsequent generations are obtained as follows:under two consecutive terms, say a and b, in the next generation put Z(a, b). For example,Figure 4 shows the triangle of order n = 10 that starts on the first line with the sequence ofnatural numbers. More precisely, this is triangle

TN(n) :={aj,k : 1 ≤ j ≤ k ≤ n

}, (9.1)

where a1,k = k for k ≥ 1 and aj+1,k = Z(aj,k−1, aj,k), for j ≥ 1, k ≥ 2.

1 2 3 4 5 6 7 8 9 102 6 12 20 30 42 56 72 90

3 2 15 6 35 12 63 206 30 10 210 420 84 1260

5 3 21 2 5 1515 7 42 10 3

105 6 105 3070 70 14

1 55

Figure 4: Triangle TN(10), that is, the cut-off triangle of order 10 generated by the repeteadapplication of the Z(·, ·) rule on the sequence of positive integers.

This is reminiscent of the Gilbreath’s Conjecture from 1958 (for which an incorrect proofwas given by Proth in 1878). In that case, the first line of the triangle from Figure 5 lists theprimes and the following generations are obtained by taking the absolute difference of neighbornumbers and iterating this operation. The conjecture states that the left edge of that trianglecontains only ones.


2 3 5 7 11 13 17 19 23 291 2 2 4 2 4 2 4 6

1 0 2 2 2 2 2 21 2 0 0 0 0 0

1 2 0 0 0 01 2 0 0 0

1 2 0 01 2 0

1 21

Figure 5: Triangle generated by iterated absolute differences staring with the sequence of thefirst 10 primes.

Odlyzko [Odl’93] confirmed Gilbreath’s earlier checks and verified the conjecture for trianglesof order < π(1013). He had also offered heuristics to support it for analogue triangles producedby starting with many other sequences whose spacings are sufficiently random and not too large.The analogous property of the triangle from Figure 4 is that the maximal powers of all primesthat divide the numbers situated on the left edge are ones.

Conjecture 5. The left edge of the triangle TN(n), for n ≥ 1, contains only square free numbers.

Spectacular properties of the Z(·, ·)-generated triangles are revealed by the scans of theirp-localized versions. Each of these renderings may be considered as a face of the correspondingprime, an analogue facet of the Pascal arithmetic triangle modulo p. The collection of thesep-generated images build on the unique ’p-print face’ and inherent character. For instance,for the prime p = 3, Figure 6 is just a bit from this collection. There, darker colors indicatehigher powers. Notice that each number m that is part of the triangle carries a ’potentialenergy’ proportional to the maximum power of p in m. This energy radiates in the subsequentgenerations along the ’force lines’ on the left and on the right, and downward. Also, whenforces coming from opposite directions meet, their strength cancels like in a physical dynamicclash. These carry flavors from the world of billiards, in which techniques involving Fareyseries play a significant role in solving problems that involve successive insertions (see Bocaet al. [BGZ’03a], [BGZ’03b], [BZ’07], [BG’09], [Boca’10] and Alkan et al. [ALZ’06]). Thepropagation of the potential energy continues endlessly as long as the sequence from the firstline does not terminate. This does not happen when the Z(·, ·) rule is applied repeatedly onthe n-th row of the Pascal triangle. In that case, triangles are radically distinctive, changingabruptly with the change of n. Looking at their localizations, even modulo the same p, onehardly notices any similarity in figures for successive values of n. This is a result of mixing theadditive insertion rule used in the construction of the Pascal triangle with the multiplicativefibers distilled by the Z(·, ·) rule.

Let TB(n) be the Z(·, ·)-generated triangles, whose first line are the binomial coefficients:(n0

),(n1

),(n2

), . . . ,

(nn

). The 5-local renderings of TB(n) with n = 24, 74, 124, 374, 624 show that


they are entirely flat. This is a consequence of the fact that for these values of n, no binomialcoefficient

(nk

), for 0 ≤ k ≤ n, is divisible by 5.

Question 4. For fixed p, are there infinitely many n > 1 for which the p-localized triangleTB(n) is completely flat?

We conclude our promenade by tempting the reader to explore patterns such as those thatstart to reveal in Figure 6., where the constraints imposed by the small size of the triangle leadto the appearance of a central darker figure which looks like a dwarf.

Figure 6: The 3-adic scan of order 64 of the Fibonacci–Z(·, ·)-generated triangle.


Acknowledgements

The first author is partially supported by the CNCS grant PN-II-ID-PCE-2012-4-0376.The authors are grateful to Emilio Ambrisi, Alberto Burato, Mauro Caselli, Andrea Cen-

tomo, Salvatore Rao, Sergio Savarino, Massimo Squillante, Pasqualina Ventrone, and Alessan-dro Zampa for their assistance in clarifying some blurry issues regarding the first written noteson Ducci’s game.

References

[ALZ’06] E. Alkan, A. H. Ledoan, A. Zaharescu, A parity problem on the free path lengthof a billiard in the unit square with pockets, Funct. Approx. Comment. Math. Volume35, no. 1 (2006), 19–36. 88

[AC’00] O. Andriychenko, M. Chamberland, Iterated Strings and Cellular Automata, Ma-thematical Intelligencer 22 (4) (2000), 33–36. 83

[Ang’10] D. Angell, A family of continued fractions, Journal of Number Theory, 130 (2010),904–911. 85

[BC’04] R. Bacher, R. Chapman, Symmetric Pascal matrices modulo p, European J. Com-binatorics 25 (2004), 459–473. 75

[BC’09] G. Back, M. Caragiu, The greatest prime factor and related magmas, JP J. ofAlgebra, Number Theory and Appl., 15 (2009), 127–136. 81

[BC’10] G. Back, M. Caragiu, The greatest prime factor and recurrent sequences, FibonacciQuart. 48 (4) (2010), 358–362. 81, 82

[Bag’66] A. K. Bag, Binomial theorem in ancient India, Indian J. History Sci., 1 no. 4 (1966),68–74. 74

[BC’07] N. Baxter, M. Caragiu, A note on p-adic Ducci games, JP J. Algebra NumberTheory Appl. 8, no. 1 (2007), 115–120. 84

[BST’99] O. Ben-Bassat, R. Strichartz, A. Teplyaev, What is not in the domain of theLaplacian on Sierpinski gasket type fractals, J. of Functional Analysis 166 (1999),197–217. 78

[BKP’05] A. Behn, C. Kribs-Zaleta, V. Ponomarenko, The Convergence of DifferenceBoxes, The American Mathematical Monthly 112, no. 5 (2005), 426–439. 82

[Bel’04] J. S. Belrose, A wire model for the Sierpinski gasket type monopole antenna and atruly broadband bent-wire antenna, Antennas and Propagation Society InternationalSymposium, IEEE Vol. 4 (2004), 3429–3432. 78

[BSSY’06] N. Ben-Gal, A. Shaw-Krauss, R. S. Strichartz, C. Young, Calculus on theSierpinski gasket II: Point singularities, eigenfunctions, and normal derivatives ofthe heat kernel, Trans. Amer. Math. Soc. 358 (2006), 3883–3936. 78


[BKZ’12] B. C. Berndt, S. Kim, A. Zaharescu, Diophantine approximation of the exponen-tial function and Sondows conjecture, submitted for publication, 2012, pp. 31. 85, 86,87

[Ber1713] J. Bernoulli, Ars Conjectandi, Opus Posthumum, Basel, 1713 (translated in 2006 byE. D. Sylla as The Art of Conjecturing, Johns Hopkins University Press, Baltimore).75

[Boca’10] F. P. Boca, The distribution of the linear flow length in a honeycomb in the small-scatterer limit, New York Journal of Mathematics 16 (2010), 651–735. 88

[BG’09] F. P. Boca, R. N. Gologan, On the distribution of the free path length of the linearflow in a honeycomb, Annales de l’Institut Fourier 59 (2009), 1043–1075. 88

[BGZ’03a] F. P. Boca, R. N. Gologan, A. Zaharescu, The average length of a trajectoryin a certain billiard in a flat two-torus, New York Journal of Mathematics 9 (2003),303–330. 88

[BGZ’03b] F. P. Boca, R. N. Gologan, A. Zaharescu, The statistics of the trajectory of acertain billiard in a flat two-torus, Commun. Math. Phys. 240 (2003), 53–73. 88

[BZ’07] F.P. Boca, A. Zaharescu, The distribution of the free path lengths in the periodictwo-dimensional Lorentz gas in the small-scatterer limit, Commun. Math. Phys. 269(2007), 425–471. 88

[Boo’12] A. R. Booker, On Mullin’s second sequence of primes, Integers 12A #A4, The JohnSelfridge Memorial Volume (2012), pp. 10. 80

[Bre’98] F. Breuer, A Note on a Paper by Glaser and Schoffl, Fibonacci Quarterly 36 (1998),463–466. 84

[Bre’07] F. Breuer, Ducci sequences in higher dimensions, Integers 7 #A24 (2007), pp. 13.84

[Bre’10] F. Breuer, Ducci sequences and cyclotomic fields, J. Difference Equ. Appl. 16, no.7 (2010), 847–862. 84

[BM’07] R. Brown, L. J. Merzel, The length of Duccis four-number game, Rocky Mt. J.Math. 37, no. 1 (2007), 45–65. 84

[BM’11] J. Brunat, J. Maureso, Symmetries in Steinhaus Triangles and in generalized Pas-cal Triangles, Integers 11 #A1 (2011), pp. 23. 79

[Bur’07] D. Burton, History of Mathematics, McGraw-Hill, 6th ed., 2007, pp. 790. 74

[CG’01] C. K. Caldwell, Y. Gallot, On the primality of n!± 1 and 2× 3× 5× · · ·× p± 1,Mathematics of Computation, 71 no. 237 (2001), 441–448. 80

[CST’05] N. J. Calkin, J. Stevens, D. Thomas, A characterization for the length of cyclesof the n-number Ducci game, Fibonacci Quart. 43 (2005), 53–59. 83, 84

[Cam’96] P. J. Campbell, Reviews, Mathematics Magazine, Vol. 69, No. 4 (1996), pp. 311–313. 83


[CS’06] M. Caragiu, L. Scheckelhoff, The greatest prime factor and related sequences,JP J. of Algebra, Number Theory and Appl. 6 (2006), 403–409. 81

[CSZ’11] M. Caragiu, L. Sutherland, M. Zaki, Multidimensional greatest prime factorsequences, JP Journal of Algebra, Number Theory and Applications, 23 (2), (2011),187–195. 82

[CZZ’11] M. Caragiu, A. Zaharescu, M. Zaki, On Ducci sequences with algebraic numbers,Fibonacci Quart. 49 (2011), 34–40. 84

[CZZ’12] M. Caragiu, A. Zaharescu, M. Zaki, On a class of solvable recurrences withprimes, JP Journal of Algebra, Number Theory and Applications, 26 (2), (2012),197–208. 82

[Car’68] L. Carlitz, Bernoulli Numbers, Fib. Quart. 6, no. 3 (1968), 71–85. 78

[Cha’03a] M. Chamberland, Unbounded Ducci sequences, J. Difference Equ. Appl. 9, no. 10(2003), 887–895. 83, 84

[Cha’03b] M. Chamberland, Una actualizachio del problema 3x + 1, Butletti de la SocietatCatalana 22 (2003), 19–45. 81

[CT’04] M. Chamberland, D. Thomas, The N-number Ducci game, Section of Open Pro-blems and Conjectures from J. Difference Equ. Appl. 10, no. 3 (2004), 339–342. 83

[Cha’08] J. Chappelon, On a problem of Molluzo concerning Steinhaus triangles infinite cyclicgroups, Integers 8, Issue 1 #A37 (2008), pp. 29. 79

[CM’37] C. Ciamberlini, A. Marengoni, Su una interessante curiosita numerica, Periodicodi Matematiche 17 (1937), 25–30. 82, 83

[CCZ’00] C. Cobeli, M. Crasmaru, A. Zaharescu, A cellular automaton on a torus, Por-tugaliae Mathematica, 57, fasc. 3 (2000), 311-323. 83, 84

[CVZ’00] C. Cobeli, M. Vajaitu, A. Zaharescu, The sequence n! (mod p), J. RamanujanMath. Soc. 15 (2000), 135–154. 86

[CZ’12] C. Cobeli, A. Zaharescu, A game with divisors and absolute differences of expo-nents, in preparation. 87

[Coh’06] H. Cohn, A short proof of the simple continued fraction expansion of e, Amer. Math.Monthly 113, No. 1 (2006), 57–62. 85

[Col’86] L. Collatz, On the motivation and origin of the (3n+1)-problem, J. of Qufu NormalUniversity, Natural Science Edition 12, no. 3 (1986), 9–11. 81

[Con’72] J. H. Conway, Unpredictable Iterations, Proceedings of the 1972 Number TheoryConference: University of Colorado, Boulder, Colorado, August 14-18, 1972, pp. 49–52. 81

[CG’96] J. H. Conway, R. K. Guy, The Book of Numbers, Springer-Verlag, New York, pp.310, 1996. 78


[Cos’99] J. Cosgrave, Could there exist a sixth Fermat prime? I believe it is not impossible,Unpublished manuscript, available at http://staff.spd.dcu.ie/johnbcos/fermat6.htm1999, pp. 22. 78

[CV’68] C. D. Cox, A. J. Van der Poorten, On a sequence of prime numbers, J. Austral.Math. Soc. 8 (1968), 571–574. 80

[Cox’71] H. S. M. Coxeter, Cyclic sequences and frieze patterns, The Fourth Felix BehrendMemorial Lecture, Vinculum 8 (1971), 4–7. 81

[Cra’01] M. Crasmaru, A characterization of some linear cellular automata, IEICE Trans.Inf. Syst. E84D, no. 1 (2001), 15–20. 84

[Cre’88] J. Creely, The length of a three-number game, Fibonacci Quarterly 26 (1988), 141–143. 84

[DV’98] F. Daerden, C. Vanderzande, Sandpiles on a Sierpinski gasket, Physica Vol. 256,issue 3-4 (1998), 533–546. 78

[DRS’09] J. L. DeGrado, L. G. Rogers, R. S. Strichartz, Gradients of Laplacian eigen-functions on the Sierpinski gasket, Proc. Amer. Math. Soc. 137 (2009), 531–540.78

[DG’01] H. Dubner, Y. Gallot, Distribution of generalized Fermat prime numbers, Math-ematics of Computation 71, no. 238 (2001), 825–832. 78

[Ehr’90] A. Ehrlich, Periods in Duccis n-number game of differences, Fibonacci Quarterly28 (1990), 302–305. 84

[EH’04] S. Eliahou, D. Hachez, On a problem of Steinhaus concerning binary sequences,Experiment. Math. 13, (2) (2004), 215–229. 79

[EH’05] S. Eliahou, D. Hachez, On symmetric and antisymmetric balanced binary se-quences, Integers 5, Issue 1 #A6 (2005), pp. 18. 79

[EMR’07] S. Eliahou, J. M. Marın, M. P. Revuelta, Zero-sum balanced binary sequences,Integers 7, Issue 2 #A11 (2007), pp. 14. 79

[FR’12] P. Fahra, C. M. Ringel, The Fibonacci partition triangles, Advances in Mathemat-ics 230, Issues 4-6 (2012), 2513–2535. 79

[FP’07] S. Falcon, A. Plaza, The k-Fibonacci sequence and the Pascal 2-triangle, Chaos,Solitons and Fractals 33 (2007), 38–49. 79

[FK’07] A. Fraenkel, A. Kontorovich, The Sierpinski sieve of Nim-varieties and binomialcoefficients, Integers 7, Issue 2 (2007), pp. 19. 78

[Gal’76] P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23(1976), 4–9. 79

[Gal’81] P. X. Gallagher, Corrigendum: On the distribution of primes in short intervals,Mathematika 28 (1981), no. 1, 86. 79

http://staff.spd.dcu.ie/johnbcos/fermat6.htm


[Gar’77] M. Gardner, Mathematical Carnival, Mathematical Association of America, Wash-ington D. C., updated and revised (first ed. 1975) pp. 297, 1989. 78

[Gla1899] W. L. Glaisher, On the residue of a binomial-theorem coefficient with respect to aprime modulus, J. Quart. J. Math., 30 (1899), 150–156. 77

[GS’95] H. Glaser, G. Schoffl, Ducci-Sequences and Pascals Triangle, Fibonacci Quarterly33 (1995), 313–324. 84

[GL’12] D. A. Goldston, A. H. Ledoan, On the differences between consecutive prime num-bers, I, Integers 12B, #A3, Proceedings of the Integers Conference 2011 (2012/13),pp. 8. 79

[Gra’92] A. Granville, Zaphod Beeblebrox’s brain and the fifty-ninth row of Pascal’s Triangle,The American Mathematical Monthly 99 (1992), 318–331. 77

[Gra’95] A. Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficientsmodulo prime powers, Organic mathematics (Burnaby, BC, 1995), 253–276, CMSConf. Proc. 20, Amer. Math. Soc., Providence, RI, 1997. 77

[Gra’97] A. Granville, Correction to: Zaphod Beeblebrox’s brain and the fifty-ninth row ofPascal’s Triangle, The American Mathematical Monthly 104 (1997), 848–851. 77

[Har’72] H. Harborth, Solution of Steinhaus’s problem with plus and minus signs, J. Com-binatorial Theory Ser. A 12 (1972), 253–259. 79

[HH’05] H. Harborth, G. Hurlbert, On the number of ones in general binary Pascal tri-angles, J. Combin. Math. Combin. Comput. 54 (2005), 99–110. 79

[Hew’77] D. Hewgill, A relationship between Pascal’s triangle and Fermat’s numbers, Fi-bonacci Quart. 15 (1977), 183–184. 77

[HK’06] M. Hinoa, T. Kumagai, A trace theorem for Dirichlet forms on fractals, Journal ofFunctional Analysis, Volume 238, Issue 2 (2006), 578–611. 78

[Hon’98] R. Honsberger, Ingenuity in Mathematics, Mathematical Association of America,Series: New Mathematical Library (23) 6th printing, pp. 206, 1998. 83

[Hos’76] H. Hosoya, Fibonacci Triangle, The Fibonacci Quarterly 14, 2 (1976), 173–178. 79

[HSW’97] J. G. Huard, B. K. Spearman, K. S. Willimas, Pascal’s Triangle (mod 9), ActaArithmetica LXXVIII.4 (1997), 331–349. 77

[HSW’98] J. G. Huard, B. K. Spearman, K. S. Willimas, Pascal’s Triangle (mod 8), Europ.J. Combinatorics 19 (1998), 45–62. 77

[Huz’08] S. Huzler, Relation between grain shape and fractal properties in random Apollonianpacking with grain rotation, Phys. Rev. Lett. 101, 120602 (2008), 4 pages. 78

[Kan’07] D. M. Kane, Improved bounds on the number of ways of expressing t as a binomialcoefficient, Integers 7, #A53, (2007) pp. 1–7. 76

[Kim’95] C. Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73(1995), 103–117. 80


[Kim’97] C. Kimberling, Fractal Sequences and Interspersions, Ars Combinatoria 45 (1997),157–168. 80

[KS’97] C. Kimberling, Harris S. Shultz, Card sorting by dispersions and fractal se-quences, Ars Combinatoria 53 (1999), 209–218. 80

[KP’07] S. Klavzar, I. Peterin, Edge-counting vectors, Fibonacci cubes and Fibonacci tri-angle, Publ. Math. Debrecen 71, 3-4 (2007), 267–278. 79

[Kos’01] T. Koshy, Fibonacci and Lucas Numbers and Applications, Hosoya’s Triangle, Chap.15, John Wiley & Sons, New York, 2001, pp. 654. 79

[KLS’01] M. Krızek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From NumberTheory to Geometry, Springer-Verlag, New York, pp. 257, 2001. 78

[KS’07] S. A. Kurtz, J. Simon, The undecidability of the generalized Collatz problem, J-Y.Cai et al., Ed., Theory and Applications of Models of Computation, 4-th Interna-tional Conference, TAMC 2007, Shanghai, China, May 22-25, 2007. Lecture Notes inComputer Science No. 4484, Springer-Verlag: New York (2007), pp. 542–553. 81

[KM’06] E. V. Kutyreva, F. M. Malyshev, On the distribution of the number of ones in aBoolean Pascal’s triangle, Discrete Math. Appl. 16, 3 (2006), 271–279. 79

[Lag’11] J. C. Lagarias, The 3x + 1 problem: An annotated Bibliography (1963-1999),arXiv:0309224v13, 2011, pp. 74. 81

[Lag’12] J. C. Lagarias, The 3x + 1 problem: An annotated Bibliography, II (2000-2009),arXiv:0608208v6, 2012, pp. 42. 81

[Leh’35] D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler,Ann. Math. 36, No. 3 (1935), 637–649. 78

[LT’07] L. Lidman, D. M. Thomas, Algebraic dynamics of a one-parameter class of mapsover Z2, Atlantic electronic J. of Math. 2, no. 1 (2007), 55–63. 84

[LY’90] A. Ludington-Young, Length of the n-number game, Fibonacci Quart. 28, no. 3(1990), 259–265. 84

[Lud’88] A. Ludington, Length of the 7-number game, Fibonacci Quarterly, 26 (1988), 195–204. 84

[LY’99] A. Ludington-Young, Even Ducci-sequences, Fibonacci Quarterly, 37 (1999), 145–153. 84

[Mah’58] K. Mahler, An interpolation series for continuous functions of a p-adic variable, J.Reine Angew. Math. 199 (1958), 23–34. 86

[Mal’12] J. Malkevitch, Weird Rulers, AMS, Monthly Feature Column, Jan. 2012. 82

[Mey’82] L. F. Meyers, Duccis four-number problem, A short bibliography, Crux Mathemati-corum 8 (1982), 262–266. 83

[Mih’10] D. Mihet, Legendre’s and Kummer’s theorems again, Resonance, Dec. (2010), 1111–1121. 77


[MP’09] A. R. Moghaddamfar, S. M. H. Pooya, Generalized Pascal triangles and Toeplitzmatrices, Electronic Journal of Linear Algebra 18 (2009), 564–588. 75

[Mol’76] J. C. Molluzzo, Steinhaus graphs, Theory ans Applications of Graphs (Proc. In-ternat. Conf. Western Mich. Univ. Kalamazoo, Mich, 1976) 642, Lecture Notes inMath., Springer, 1978, 394–402. 79

[Mul’63] A. A. Mullin, Recursive function theory (A modern look at a Euclidean idea), Bull.Amer. Math. Soc. 69 (1963), 737. 80

[NSTY’04] J. Needleman, R. Strichartz, A. Teplyaev, P.-L. Yang, Calculus on the Sier-pinski gasket I: polynomials, exponentials and power series, J. Functional Anal. 215(2004), 290–340. 78

[Odl’93] A. M. Odlyzko, Iterated absolute values of differences of consecutive primes, Math-ematics of Computation 61 (1993), 373–380. 88

[ORW’99] A. Odlyzko, M. Rubinstein, M. Wolf, Jumping champions, Experiment. Math.8, no. 2 (1999), 107–118. 80

[Pas1665] B. Pascal, Traite du Triangle Arithmetique, avec quelques autres petitez sur lamesme matiere, Paris, 1665. 74

[Per’54] O. Perron, Die lehre von den kettenbrochen, Dritte, verbesserte und erweiterte Au-flage, Band I, B. G. Teubner Verlagsgesellschaft. Stuttgart, 1954, pp. 194+VI. 85,86

[Pom’96] F. Pompili, Evolution of nite sequences of integers..., Mathematical Gazette 80(1996), 322–332. 83

[PCRM’03] S. Pradhan, B. K. Chakrabarti, P. Ray, M. Kanti, Magnitude Distribution ofEarthquakes: Two Fractal Contact Area Distribution, Physica Scripta T 106 (2003),77. 78

[Row’11a] E. Rowland, The number of nonzero binomial coefficients modulo pα, Journal ofCombinatorics and Number Theory 3 (2011), 15–25. 77

[Row’11b] E. Rowland, Two binomial coefficients conjectures, arXiv:1102.1464, (2011), pp. 10.77

[She’11] V. Shevelev, Binomial Coefficient Predictors, Journal of Integer Sequences 14(2011), pp. 8. 77

[Sie’15] W. Sierpinski, Sur une courbe dont tout point est un point de ramification, C. R.Acad. Sci. Paris 160 (1915), 302–305. 78

[Sng’71] D. Singmaster, How often does an integer occur as a binomial coefficient?, AmericanMathematical Monthly 78, (4) (1971), 385–386. 76

[Sng’75] D. Singmaster, Repeated binomial coefficients and Fibonacci numbers, FibonacciQuarterly 13, (4) (1975), 295–298. 76


[Sng’80] D. Singmaster, Divisibility of binomial and multinomial coefficients by primes andprime powers, A collection of manuscripts related to the Fibonacci sequence, Fib.Assoc. Santa Clara, 1980, 98–113. 77

[OEIS] N. J. A. Sloane and OEIS foundation, On-Line Encyclopedia of Inte-ger Sequences, published electronically, sequences: http://oeis.org/A000945 andhttp://oeis.org/A000946. 80

[Son’06] J. Sondow, A geometric proof that e is irrational and a new measure of its irra-tionality, Amer. Math. Monthly 113 (2006), 637–641. 86

[SS’08] J. Sondow, K. Schalm, Which partial sums of the Taylor series for e are convergentsto e?, (and a link to the primes 2, 5, 13, 37, 463), in Tapas in Experimental Mathe-matics, T. Amdeberhan and V. H. Moll, eds., Contemp. Math., vol. 457, AmericanMathematical Society, Providence, RI, 2008, pp. 273–284. 86

[SS’10] J. Sondow, K. Schalm, Which partial sums of the Taylor series for e are convergentsto e? (and a link to the primes 2, 513, 37, 463). Part II, in Gems in ExperimentalMathematics, T. Amdeberhan, L. A. Medina, V. H. Moll, eds., Contemp. Math., vol.517, American Mathematical Society, Providence, RI, 2010, pp. 349–363. 86

[Ste’58] H. Steinhaus, One Hundred Problems in Elementary Mathematics, Pergamon, Elins-ford, 1963 (original in Polish, STO ZADAN, Warshau, 1958). 79

[Tep’07] A. Teplyaev, Spectral zeta functions of fractals and the complex dynamics of poly-nomials, Trans. Amer. Math. Soc. 359 (2007), 4339–4358. 78

[Tha’96] D. S. Thakur, Exponential and continued fractions, Journal of number theory 59(1996), 248–261. 85

[Thw’85] B. Thwaites, My conjecture, Bull. Inst. Math. Appl. 21 (1985), 35–41. 81

[Thw’96a] B. Thwaites, Two Conjectures, or how to win £ 1000, Math. Gazette 80 (1996),35–36. 80, 83

[Thw’96b] B. Thwaites, Correspondence, The Mathematical Gazette 80, no. 488 (1996), 420.82, 83

[Van’39] H. S. Vandiver, Certain congruences involving the Bernoulli numbers, Duke Math.J., 5 (1939), 548–551. 78

[Web’82] W. A. Webb, The length of the four-number game, Fibonacci Quart. 20, no. 1 (1982),33–35. 84

[Wir’98] G. J. Wirsching, The dynamical system generated by the 3n + 1 function, LectureNotes in Math. no. 1681, Springer-Verlag: Berlin 1998, pp. 158. 81

http://oeis.org/A000945

http://oeis.org/A000946


[Wol’02] S. Wolfram, A New Kind of Science, Wolfram Media, Inc., Champaign, IL, 2002,pp. 1197. 73, 78

Received: 20.11.2012,Accepted: 23.12.2012.

Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest, 70700, Romania.

Email: [email protected]

Department of Mathematics, University of Illinois at Urbana - Champaign, 1409 West Green Street,Urbana, IL 61801, USA, and

Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest, 70700, Romania,

Email: [email protected]

Promenade around Pascal Triangle { Number Motives · Pascal [Pas1665] wrote his Trait e du Triangle...

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