New paper. GPF-Tribonacci sequences

Starting with 5, 13, 7, each subsequent term is the greatest prime factor of the sum of the previous three terms. More about this type of sequences - in a new Fibonacci Quarterly article by Greg Back and Mihai Caragiu ("The Greatest Prime Factor and Recurrent Sequences" - Fibonacci Quarterly 48 (2010), no. 4, 358–362) - abstract here. 

In the main result on GPF-Fibonacci sequences (Theorem 3) we prove that all GPF-Fibonacci sequences (that is, prime sequences in which each subsequent term is the greatest prime factor of sum of the previous two terms) that are non-constant eventually enter the same 4-cycle 7,3,5,2.

The "phi-bonacci" sequence - an update

More data on the "phi-bonacci" sequence introduced previously here, after computing the first 500 terms:

• a plot of the sequence of quotients X(n+1)/X(n) for n = 1,2,...,499:

X(n) is a multiple of 4 for n between 13 and 500.
Here is the updated raw data (last previously calculated term marked in red):

GPF stability?...

A visual on the behavior of the same recurrence as before, only with a different initial condition (an 8-digit prime, picked at random)

L[1] = 11631013

L[N] = P(26390*L[N-1] + 1103)

The limit cycle is the same. The choice

L[1] = 7654237825827857857221111238572389123865443346789678979

leads, again to the same limit cycle. This raises an interesting question of "stability" (that is, assuming ultimate periodicity holds, are there finitely many - if not a single one - limit cycles?*) Note that in this particular case, the limit cycle is not unique: for example, the choice

L[1] = 2250957258971258907129712971234237484736596896123596812363

leads to a different limit cycle, of period 18:

A recurrence with primes inspired by a formula of Ramanujan

Inspired by the linear component appearing in the numerators of the terms of the Ramanujan's formula for pi, I looked into the recurrent sequence of primes defined as follows:   

L[1] = 2

L[N] = P(26390*L[N-1] + 1103)

where P is the greatest prime factor function. The prime sequence (L[N]) turns out to be ultimately periodic, with period

(1459, 30011, 15529243, 409816723873, 292299009270529, 701251895877205583, 15696384675317604187, 451826639233, 109391789076697, 151939437564949207, 74396630251, 29303389139179, 26646743, 2111734381, 55728670315693, 70865191, 18516162293, 487831, 12873861193, 1132987, 1921687, 330167, 968123137, 2901559, 14505047, 2091738751, 675347, 5940802811, 57192517, 137210047703, 938359, 24763295113, 6250999, 2795997707, 6922448587, 30040009, 46632696389, 3290305727, 282788861, 24151449977, 212452254964711, 5606615008518724393, 4871183188935589, 733134419023, 19347417318018073, 29451120121, 218661017, 242650193, 36178184149, 16311737023, 6797093683, 179375302295473, 29858734728029, 1862817989297131, 4993054962517, 15723597697, 311287129201, 13424039, 6946282163, 252731, 741063577, 106787, 26293, 10356319, 23227, 612961633, 511873, 4640443, 346915841, 3511741099, 33101671, 3527213, 6883, 20719, 433, 57427, 1529263, 510851287, 4456663, 117611337673, 999421747, 33078701, 2389547, 2335560979, 4285298911, 113089038262393, 30741542832733, 1801335599663, 688945601088517, 4790671, 7790117, 769966999, 239280127, 8821951, 756227, 34513, 4289, 37729271, 914302537, 11594639093, 39465611, 11110373, 478571, 11549, 859, 22670113, 176369, 1551459671, 40453493, 6714262147, 14927447, 43770591937, 70138194257, 2603300909203, 51190093, 942712181, 8292724819231, 24011017, 168928483, 6954793553, 270216913, 16301459, 1158523, 30573423073, 21129377, 5023461803, 1979, 17408971, 70381, 1798021, 88289, 467017, 540149, 39929, 4231837, 223803967, 5906186690233, 155864266755249973, 144041128039, 604427630617, 8896176894581, 1577342687387, 17151245784979, 452621376265596913, 878711663, 19359229, 2829679, 12128509, 8627, 41177, 65371, 1663589, 4878012757, 207798167, 609310403137, 1409652983, 1377805267499, 526960594337677, 86463252140809, 8419797874523803, 15369684199, 10962323405749, 1278583667, 132525119, 129531033019, 1221680363, 7825861, 206524472893, 5450180839647373, 24708859707660913, 652066807685171495173, 1941559636106473627171, 193285868819, 100015962316363, 1111967795377, 29344830120000133, 81694832084891, 718642206240091531, 18964967822676015504193, 4407663528935561, 857965267921, 42742695533, 20327255503, 11413537717559, 250324962137, 1275058048793, 11216260635882791, 857911268779, 13341354380129, 39119815787956157, 209783083, 325657385969, 72031783, 2035949, 163601, 72953, 7247, 21249937, 560785838533, 1345376207171543, 1075893275977485481, 9099479420859497, 98635636088557, 320878911329, 2822664823324471, 110249906411, 14055531546799, 499608015161, 548195730743, 160630728871, 385367721355163, 32609594737, 1641473, 83465267, 122840243, 140682811, 161071, 4250664793, 153443, 449929097, 26904379, 116579, 405499, 7715299, 3579647, 33083, 2881391)

Here is a logarithmic plot of this sequence:

This special case illustrates a general conjecture on the ultimate periodicity of GPF sequences. For this, and related sequences and algebraic structures, see

• Greg Back and Mihai Caragiu, The Greatest Prime Factor and Recurrent Sequences, Fibonacci Quarterly (accepted for publication);
• Mihai Caragiu and Greg Back, The Greatest Prime Factor and Related Magmas, JP J.of Algebra, Number Theory and Appl. 15 (2), 127-136 (December 2009);
• Mihai Caragiu and Lisa Scheckelhoff, The Greatest Prime Factor and Related Sequences, JP J.of Algebra, Number Theory and Appl. 6(2), 403-409 (2006);

A "phi-bonacci" sequence and its consecutive quotients

A most interesting sequence:
"phi-bonacci" ?...

X(0)=0, X(1)=1
X(n)=phi (X(n-1)+X(n-2)+1) 
if n is at least 2, where phi is the Euler's totient function.

This ensures that X(n) is never greater than the 'regular' Fibonacci number F(n)
 
Plotted - the sequence of quotients X(n+1)/X(n) for n = 1,2,...,324

The raw list of the first 325 non-zero terms follows:

1, 1, 2, 2, 4, 6, 10, 16, 18, 24, 42, 66, 108, 120, 228, 348, 576, 720, 1296, 2016, 3312, 5256, 7200, 12456, 17860, 25200, 40256, 37368, 39600, 72900, 112500, 185400, 282204, 364800, 517600, 805392, 1133988, 1939380, 2788176, 4727556, 6819120, 11539840, 18324852, 28220080, 46471680, 70297856, 77663160, 98640672, 173595168, 256221952, 408844800, 613907760, 1020322800, 1598868000, 2614401972, 3650502240, 6204873360, 9219832128, 14163287040, 23375208496, 37533203556, 59869153008, 77921885248, 136242824256, 171331767600, 280988047872, 412648088320, 492483317760, 759235553856, 1248565926960, 1825274073460,

Eadem mutata resurgo

Commemorating Jacob Bernoulli...

Eadem mutata resurgo
"Though changed I shall rise the same"

Inscribed on Jacob Bernoulli's tombstone (he died on August 16, 1705 in Basel), this motto refers to the logarithmic (equiangular) spiral (N.B. the spiral that was actually imprinted on the tombstone is not equiangular). Through this self-similar object Jacob Bernoulli symbolically points to the ‘fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self ’ - as quoted in Mario Livio's book "The Golden Ratio..." (via here).

Jacob Bernoulli at the Mathematics Genealogy Project.

The Whirlpool Galaxy...

Beauty in Mathematics

Michael Atiyah - Beauty in Mathematics

Structure and Randomness in the Prime Numbers (Terence Tao)

Terence Tao: Structure and Randomness in the Prime Numbers, UCLA
Slides: pdf, powerpoint

Lecture for a general audience: Terence Tao is UCLA's Collins Professor of Mathematics, and the first UCLA professor to win the prestigious Fields Medal.

Johann Bernoulli (1667 - 1748) anniversary

Johann Bernoulli (1667 - 1748) was born on July 27, 7^3 years ago, in Basel, Switzerland. He was Euler's "mathematical parent".

And here is the... "sophomore's dream" - neat identities due to Johann Bernoulli (1697):

A traffic flow simulation

This traffic flow educational project with Matlab features a gradually increasing car density starting from an initial value of 0.2. There are 250 cells. The update rule (describing the transition from time t to time t + 1): apply rule 184, after which randomly select a position - if occupied, nothing happens, while if empty, introduce a car at the selected place with probability 0.7. The image follows the first 500 time units. Notice the transition to a congested regime happening at some point (emerging shock waves). In the picture, free cells are blue, while cells occupied by "cars" are red.

Lothar Collatz anniversary

Lothar Collatz, who proposed (in 1937) the celebrated "3n+1 problem", was born 100 years ago in Arnsberg, Germany.

Mac Tutor Biography

Lothar Collatz at MGP

A Thermodynamic Classification of Real Numbers

I just found a very interesting paper presentation (JNT on YT - link) by Thomas Garrity (Williams College) - "A Thermodynamic Classification of Real Numbers":


On arXiv - A Thermodynamic Classification of Real Numbers.

Élie Cartan (1869-1951)

Élie Cartan had significant contributions in areas such as Lie theory, differential geometry, exterior differential forms, the theory of spinors (introduced by him in 1913), etc. Cartan died on May 6, 1951.

Dieudonné places Cartan right after Poincaré and Hilbert when it comes to the lasting influence in shaping modern mathematics. He was a speaker at the 1924, 1932 and 1936 International Mathematical Congresses. He lectured in Romania in 1931. The letters that he exchanged with Albert Einstein, Gheorghe Ţiţeica, Alexandru Pantazi and Gheorghe Vrânceanu, have been published (as mentioned in M. A. Akivis and B Rosenfeld - Élie Cartan (1869-1951), Providence R.I., 1993).

Paul Montel (1876-1975)

Paul Antoine Aristide Montel, Honorary Member of the Romanian Academy, advisor of Henri Cartan, Jean Dieudonné, Miron Nicolescu, Tiberiu Popoviciu and others (see Montel's entry at MGP), was born on April 29, 1876...

Max Planck on consciousness

"I regard consciousness as fundamental. I regard matter as derivative from consciousness. We cannot get behind consciousness. Everything that we talk about, everything that we regard as existing, postulates consciousness." (Max Planck - born on April 23, 1858)

link to quotation source
link to top picture source/credits (grave of Max Planck in Göttingen)
link to  bottom picture source/credits (NGC7090)
Mac Tutor Biography
Mathematics Genealogy Project - Max Planck
Planck units

Marius Dabija (13 ianuarie 1969-22 iunie 2003)

De pe blogul lui Florin:

Un prieten căruia îi păstrez o vie amintire este Marius Dabija. Minte strălucită, scormonitoare, imprevizibil in acţiuni, căutând mereu soluţia surpriză. În clasele a VII-a şi a VIII-a am lucrat împreună la matematică, pregătindu-ne pentru Olimpiade. Ca să variem, jucam şah până nu mai ştiam de noi. Era talentat şi la fotbal, tenis de masă etc.

În liceu ne-am văzut mai rar. Eu eram la liceul I. L. Caragiale, el, la Mihai Viteazul. L-a avut profesor pe Eugen Onofraş. În clasa a XI-a, Marius a luat locul I la Olimpiada de matematică, faza Naţională şi la Olimpiada Internaţională de Matematică. După ce a absolvit Facultatea de matematică, a plecat la doctorat în America, unde munca de cercetare i-a fost încununată de reuşită. A luat doctoratul şi a scos nişte articole remarcabile.

Ca elev şi student, era în stare să-şi conducă profesorii de la agonie la extaz şi invers. Născocea pe loc soluţii din cele mai diverse la câte o problemă, după care, lăudat fiind de profesor, care nu reuşea totuşi să urmărească deplin şirul argumentărilor, revenea şi arătând că greşise într-un loc ştergea totul, scoţând ca din joben o altă demonstraţie fulger. Asta se putea întâmpla de câteva ori la rând...

Avea de regulă o deosebită poftă de viaţă, umor, voioşie, neastâmpăr, o doză sensibilă de nonconformism, atras de situaţii-limită, uneori fiind, e drept chinuit de gânduri şi întrebări, incertitudini existenţiale, întorcând lucrurile pe toate părţile în căutarea unei soluţii, construind şi deconstruind la nesfârşit.

Mie îmi părea câteodată, în unele momente ale sale de graţie, că regăsesc profilul unui Mozart al matematicii. Am aflat cu durere în inimă vestea că în America a trecut pe neaşteptate la cele veşnice, în plină activitate creatoare, la numai 34 de ani. Dumnezeu să-l ierte şi să-l odihnească!

Articole Publicate:

Dabija, M. "Algebraic and Geometric Dynamics in Several Complex Variables". PhD thesis, University of Michigan, 2000. ps.gz
Bonifant, A. and Dabija, M. "Contractive Curves". International Journal of Mathematics and Mathematical Sciences, 30(4), 2002. ps.gz
Bonifant, A. and Dabija,M. "Self-maps of P2 with invariant elliptic curves". Contemporary Mathematics, 311, 2002. ps.gz
Coman,D.and Dabija, M. "On the Dynamics of Some Diffeomorphisms of C2 near parabolic fixed points". Houston Journal of Mathematics, 24(1), 1998. pdf

Articole Nepublicate:

Dabija, M. "Bötcher divisors", 2000. ps
Dabija,M. "Self-maps of projective bundles on projective spaces",2000. ps
Dabija, M."Self-maps of ruled surfaces", 2000. ps
Dabija,M.and Jonsson, M. "Self-maps of P2 with an invariant curve of curves", 2002.

J.S. Bach - Crab Canon on a Möbius Strip

(YT link)

Ph.D. mathematician and NFL champion

In some sense, the stunning 2010 Super Bowl XLIV victory of New Orleans Saints led by Drew Brees against the Peyton Manning's Indianapolis Colts (my favorite team) may be analogue to a similar event that happened in 1964. Then the NFL quarterback Frank Ryan led the Cleveland Browns to the 1964 NFL Championship title in a 27-0 victory against Johnny Unitas' Baltimore Colts. To this one might add the impressive 1966 season in which the Cleveland Browns' legend Ryan threw for 2974 yards and scored 29 touchdowns.

What is especially relevant for this particular blog is that Frank Ryan is also the recipient of a Ph.D. in Mathematics awarded by Rice University in 1965, with a most interesting thesis, "A Characterization of the Set of Asymptotic Values of a Function Holomorphic in the Unit Disc", and that among the references cited in the thesis are Luzin's "Leçons sur les ensembles analytiques et leurs applications", Sierpinski's "General Topology" (University of Toronto Press, 1952), and Stoilow's "Les propriétés topologiques des fonctions analytiques d'une variable", Ann. Inst. H. Poincaré, 2 (1932), 233–266. In 1966 Frank Ryan also published two fundamental papers on the set of asymptotic values of a function holomorphic in the unit disc in Duke Mathematical Journal (he also published in Pacific Journal of Mathematics, Mathematische Zeitschrift, Michigan Mathematical Journal, etc).

I will conclude with mentioning a recent mathematical event - the amazing, super-entertaining after-dinner talk "Resolved, that a Football is a Mathematical Object" delivered by Frank Ryan at the 2007 Ohio MAA Meeting held at Wittenberg (a talk which I will never forget).