"mathematics is not yet ready for such problems"
~Paul Erdos in reference to Collatz's problem
[What is the Collatz Conjecture?The rules: Pick any positive whole number. If it is even, divide it by 2. If it is odd, multiply it by 3 and add 1.The process: Repeat these steps with your new number.The conjecture:You will always get to the number one.]
It takes 29 iterations for the number 269 to reach 1. This is the 269th day of the year, (on non-leap years, the 269th day is Sep 26, and the date is written 26/9 in much of Europe. This is the only day of the year which presents itself in this way. (Are there any days that work using month/day?)
269 is a regular prime, an Eisenstein prime with no imaginary part, a long prime, a Chen prime, a Pillai prime, a Pythagorean prime, a twin prime, a sexy prime, a Higgs prime, a strong prime, and a highly cototient number. So many new terms to look up... Well? Look them up.
269 is also the Hypotenuse of a Primitive Pythagorean triple, (69, 260 269)
269 is the smallest natural number that cannot be represented as the determinant of a 10 × 10 (0,1)-matrix
Prime Curios offers this interesting convention of prime numbers, "The longest official game of chess on record (269 moves) took place in Yugoslavia on 2/17/89 and ended in a draw. Note that 2, 17, 89, and 269 are all prime numbers." And I'm guessing that Yugoslavia was a Prime country in its day.
Prime Curios also had this interesting tidbit, "The smallest prime whose square, 72361, is a concatenation of primes in two ways, i.e., (7, 23, 61) and (7, 2, 3, 61).)
269 is the largest prime factor of 9! + 1 = 362881.
269 Like all Pythagorean Primes (of the form 4n+1) is the sum of two squares, conjectured by Fermat, proved by Euler. 269 = 10^2 + 13^2.
269 is also the difference of two squares, as 135^2 - 134^2,
EVENTS
1679 On September 26, 1679, a fierce fire consumed the Stellaburgum — Europe’s finest observatory, built by the pioneering astronomer Johannes Hevelius in the city of Danzig, present-day Poland, decades before the famous Royal Greenwich Observatory and Paris Observatory existed.
And while he rebuilt the observatory, it simply did not compare with the original. He never fully recovered from the loss. His resilience in continuing was in large part fueled by the miraculous salvation of one of his manuscripts — his fixed-star catalog, which contained the results of thousands of calculations of the positions of the stars made over decades of patient observation. The small leather-bound notebook was the sole manuscript to survive the fire, presumably saved by Hevelius’s 13-year-old daughter Katharina Elisabeth, the sole family member in Danzig at the time of the fire, who had a key to her father’s study. Half a millennium later, it was rediscovered. In 1971, it made its way to Utah’s Brigham Young University, becoming the one-millionth acquisition by the institution’s library.
Nearly two centuries before Maria Mitchell, Elisabeth Hevelius essentially became the first Western female astronomer. After his death, Elisabeth, who had assisted him in the catalog all along, took it upon herself to finish Hevelius’s lifelong quest. She completed the book, dedicating it to the generous Polish monarch. The finished catalog included more than 600 new stars that Johannes and Elisabeth had observed, as well as a dozen new constellations, whose names, as given by Hevelius, astronomers still use today.
*History of Astronomy @HistAstro
*Maria Popova at brainpickings.org
One of the constellations named by Hevelius was Leo Minor, nestled between the Big Bear and the Big Lion.
1732, In Dec. of 1729, Goldbach wrote to Euler to ask, "Do you know about Fermat's remark that all numbers of the form \( 2^{2^{x}} +1 \) are prime?" Less than three years later Euler shows that F5, 4294967297, the fifth Fermat "prime" is, in fact, not prime, but divisible by 641. He goes on to show that it is also the sum of two squares, in two different ways.
\(2^{32}+1)^2 = 65536^2+1= 62264^2+204496^2\)
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| Euler |
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| Fermat |
1775 John Adams writes to his wife, Abigail (Smith) Adams to entreat her to teach his children geometry and... "I have seen the Utility of Geometry, Geography, and the Art of drawing so much of late, that I must intreat you, my dear, to teach the Elements of those Sciences to my little Girl and Boys. It is as pretty an Amusement, as Dancing or Skaiting, or Fencing, after they have once acquired a Taste for them. No doubt you are well qualified for a school Mistress in these Studies, for Stephen Collins tells me the English Gentleman, in Company with him, when he visited Braintree, pronounced you the most accomplished Lady, he had seen since he left England.—You see a Quaker can flatter, but don't you be proud. *Natl. Archives
Adams had studied some mathematics at Harvard (where geometry, trigonometry, and surveying were part of the curriculum), but his real talents lay in rhetoric, law, and politics. He was never known for facility with numbers.
Abigail, meanwhile, had almost no access to formal schooling—very typical for girls in colonial New England. What she knew of arithmetic and mathematics she learned at home, largely self-taught. We don’t have evidence that she was a “better mathematician” in the technical sense, but she was certainly better at education in practice. She managed the family farm accounts during John’s long absences, dealt with business matters, and taught their children. John Quincy Adams later praised her influence as formative in his education. *PB notes
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| Abigail Adams |
1874 James Clerk Maxwell in a letter to Professor Lewis Campbell describes Galton, "Francis Galton, whose mission it seems to be to ride other men's hobbies to death, has invented the felicitous expression 'structureless germs'. " *Lewis Campbell and William Garnett (eds.), The Life of James Clerk Maxwell (1884), 299.
1924 Jean Hoerni, a pioneer of the transistor, is born in Switzerland. A physicist, Hoerni in 1959 invented the planar process, which, combined with Robert Noyce's technique for placing a layer of silicon dioxide on a transistor, led to the creation of the modern integrated circuit. Hoerni's planar process allowed the placement of complex electronic circuits on a single chip. *CHM
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| early integrated circuit |
1960 On this day in 1960, U.S. presidential candidates Richard Nixon and John F. Kennedy debated each other. It was a landmark event: Never before had the major-party nominees for president faced each other in a nationally televised debate. One candidate looked tired and ill, gray, and sweaty, while the other was (comparatively) energetic, neat, and clean—with a fresh tan too. The latter was Kennedy, who went on to win by one of the narrowest majorities in the history of U.S. presidential elections. *Britannica
The 1960 United States presidential debates were a series of debates held during the 1960 presidential election. Four presidential debates were held between Republican nominee Richard Nixon and Democratic nominee John F. Kennedy. All four presidential debates were the first series of debates conducted for any US presidential election. The next presidential debate did not occur until 1976, after which debates would become a regular feature of all presidential campaigns.*Wik
1991 The first two year closed mission of Biosphere 2 began just outside Tucson, Arizona. Four men and four women entered the Biosphere 2 on this day in 1991. For two years, the eight participants lived in this huge glass and steel structure in the Arizona desert completely closed off from the rest of the world. It also contained 4,000 species of plants, animals and microbes. *On This Day in Chemistry
Biosphere II , or Biosphere 2 , is an experimental facility built to replicate a closed, artificial ecological system located in Oracle , Arizona , in the desert at the foot of the Santa Catalina Mountains . Biosphere 2 was constructed between 1987 and 1991 by Space Biosphere Ventures, a company founded by John Polk Allen and Margret Augustine. The structure aimed to recreate a viable ecosystem within a massive, enclosed dome. One of its objectives was to assess the feasibility of similar biospheres for future space colonization . The experiment was named Biosphere 2 , considering Earth to be " Biosphere 1. " The necessary funding, estimated at $200 million, was provided by Edward Bass between 1985 and 2007. . *Wik
1999 The Kobe meteorite fell on September 26 (local time 20:23), 1999, in Kita-ku in the north of Kobe city, Japan. The meteorite fall was widely observed in Kobe and the surrounding area, and was photographed by an amateur photographer in Imabari city, 200 km southwest of Kobe. The meteorite struck a house with an explosive sound but otherwise caused only minor property damage. The approximately 20 fragments of the meteorite had a total mass of 136 g. *terrapub.co.jp
2011 Astronauts had this view of the aurora on September 26, 2011. Credit: NASA
We’ve had some great views of the aurora submitted by readers this week, but this one taken from the International Space Station especially highlights the red color seen by many Earth-bound skywatchers, too. Karen Fox from the Goddard Space Flight Center says the colors of the aurora depend on which atoms are being excited by the solar storm. In most cases, the light comes when a charged particle sweeps in from the solar wind and collides with an oxygen atom in Earth’s atmosphere. This produces a green photon, so most aurora appear green. However, lower-energy oxygen collisions as well as collisions with nitrogen atoms can produce red photons — so sometimes aurora also show a red band as seen here. *Universe Today
BIRTHS
1688 Willem 's Gravesande (26 September 1688 – 28 February 1742)was a Dutch mathematician who expounded Newton's philosophy in Europe. In 1717 he became professor in physics and astronomy in Leiden, and introduced the works of his friend Newton in the Netherlands.
His main work is Physices elementa mathematica, experimentis confirmata, sive introductio ad philosophiam Newtonianam or Mathematical Elements of Natural Philosophy, Confirm'd by Experiments (Leiden 1720), in which he laid the foundations for teaching physics. Voltaire and Albrecht von Haller were in his audience, Frederic the Great invited him in 1737 to come to Berlin.
His chief contribution to physics involved an experiment in which brass balls were dropped with varying velocity onto a soft clay surface. His results were that a ball with twice the velocity of another would leave an indentation four times as deep, that three times the velocity yielded nine times the depth, and so on. He shared these results with Émilie du Châtelet, who subsequently corrected Newton's formula E = mv to E = mv2. (Note that though we now add a factor of 1/2 to this formula to make it work with coherent systems of units, the formula as expressed is correct if you choose units to fit it.) *Wik
's Gravesande is also remembered for his invention of a simple experiment you may have tried in middle school science. It is a simple experiment demonstrating thermal expansion, which has been used in physics education since. This is known today as "'s Gravesande's experiment" or "'s Gravesande's ring". The apparatus consists of a small metal ball on a chain or handle, and a metal ring on a stand. The ring is just big enough so that when the ring and ball are at the same temperature, the ball fits through the ring. However, if the ball is heated by dipping it into boiling water or playing the flame of a spirit lamp over it, the metal will expand, and the ball will no longer fit through the ring. When the ball has cooled down, it will fit through the ring again.
1754 Joseph-Louis Proust (26 Sep 1754; 5 Jul 1826) French chemist who proved (1808) that the relative quantities of any given pure chemical compound's constituent elements remain invariant, regardless of the compound's source, and thus provided crucial evidence in support of John Dalton's “law of definite proportions,” which holds that elements in any compound are present in fixed proportion to each other. *TIS
1731 Giovanni Francesco Giuseppe Malfatti, also known as Gian Francesco or Gianfrancesco (26 September 1731 – 9 October 1807) was an Italian mathematician. He was born in Ala, Trentino, Italy and died in Ferrara.
Malfatti studied at the College of San Francesco Saverio in Bologna where his mentors included Vincenzo Riccati, F. M. Zanotti and Gabriele Manfredi. He moved to Ferrara in 1754, and became a professor at the University of Ferrara when it was re-established in 1771. In 1782 he was one of the founders of the Societa Italiana delle Scienze, later to become the Accademia nazionale delle scienze detta dei XL.
In 1803, Malfatti posed the problem of carving three circular columns out of a triangular block of marble, using as much of the marble as possible, and conjectured that three mutually-tangent circles inscribed within the triangle would provide the optimal solution. These tangent circles are now known as Malfatti circles after his work, despite the earlier work of Japanese mathematician Ajima Naonobu and of Malfatti's countryman Gilio di Cecco da Montepulciano on the same problem and despite the fact that the conjecture was later proven false. Several triangle centers derived from these circles are also named after both Ajima and Malfatti
For most triangles a larger area can be achieved by a greedy algorithm that inscribes a single circle of maximal radius within the triangle, inscribes a second circle within one of the three remaining corners of the triangle, the one with the smallest angle, and inscribes a third circle within the largest of the five remaining pieces. The difference in area for an equilateral triangle is small, just over 1%, but as Howard Eves (1946) pointed out, for an isosceles triangle with a very sharp apex, the optimal circles (stacked one atop each other above the base of the triangle) have nearly twice the area of the Malfatti circles.
Eventually it was shown that the Malfatti circles are never optimal. *Wik
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| Malfatti's Circles, *Wik |
1784 Christopher Hansteen (26 Sep 1784; 15 Apr 1873) Norwegian astronomer and physicist noted for his research in geomagnetism. In 1701 Halley had already published a map of magnetic declinations, and the subject was studied by Humboldt, de Borda, and Gay-Lussac, among others. Hansteen collected available data and also mounted an expedition to Siberia, where he took many measurements for an atlas of magnetic strength and declination.*TIS
1854 Percy Alexander MacMahon (26 Sept 1854 , 25 Dec 1929) His study of symmetric functions led MacMahon to study partitions and Latin squares, and for many years he was considered the leading worker in this area. His published values of the number of unrestricted partitions of the first 200 integers which proved extremely useful to Hardy and Littlewood in their own work on partitions. He gave a Presidential Address to the London Mathematical Society on combinatorial analysis in 1894. MacMahon wrote a two volume treatise Combinatory analysis (volume one in 1915 and the second volume in the following year) which has become a classic. He wrote An introduction to combinatory analysis in 1920. In 1921 he wrote New Mathematical Pastimes, a book on mathematical recreations. *SAU
1887 Sir Barnes (Neville) Wallis (26 Sep 1887; 30 Oct 1979) was an English aeronautical designer and military engineer whose famous 9000-lb bouncing "dambuster" bombs of WW II destroyed the German Möhne and Eder dams on 16 May 1943. He designed the R100 airship, and the Vickers Wellesley and Wellington bombers. The specially-formed RAF 617 Squadron precisely delivered his innovative cylindrical bombs which were released from low altitude, rotating backwards at high speed that caused them to skip along the surface of the water, right up to the base of the dam. He later designed the 5-ton Tallboy and 10-ton Grand Slam earthquake bombs (which used on many enemy targets in the later years of the war). Postwar, he developed ideas for swing-wing aircraft. *TIS (His courtship with his wife has been written by his daughter, Mary Stopes-Roe from the actual courtship in the entertaining, but perhaps overpriced book, Mathematics With Love: The Courtship Correspondence of Barnes Wallis, Inventor of the Bouncing Bomb.)
1891 Hans Reichenbach (September 26, 1891, April 9, 1953) was a leading philosopher of science, educator and proponent of logical empiricism. Reichenbach is best known for founding the Berlin Circle, and as the author of The Rise of Scientific Philosophy.*Wik
1924 Jean Hoerni, (September 26, 1924 – January 12, 1997)a pioneer of the transistor, is born in Switzerland. A physicist, Hoerni in 1959 invented the planar process, which, combined with Robert Noyce's technique for placing a layer of silicon dioxide on a transistor, led to the creation of the modern integrated circuit. Hoerni's planar process allowed the placement of complex electronic circuits on a single chip. *CHM
1926 Colin Brian Haselgrove (26 September 1926 , 27 May 1964) was an English mathematician who is best known for his disproof of the Pólya conjecture in 1958. the Pólya conjecture stated that 'most' (i.e. more than 50%) of the natural numbers less than any given number have an odd number of prime factors. The conjecture was posited by the Hungarian mathematician George Pólya in 1919.. The size of the smallest counter-example is often used to show how a conjecture can be true for many numbers, and still be false. *Wik
1927 Brian Griffiths (26 Sept 1927 , 4 June 2008) He was deeply involved in the 'School Mathematics Project', he served as chairman of the 'Joint Mathematical Council', and chaired the steering group for the 'Low Attainers Mathematics Project' from 1983 to 1986. This project became the 'Raising Achievement in Mathematics Project' in 1986 and he chaired this from its foundation to 1989. *SAU
There is contemporary evidence that LAMP → RAMP produced positive effects, but the evidence is mostly program reports and case-study/teacher-research data (not large randomized trials), so the results are encouraging rather than definitive. *PB notes
DEATHS
1766 Giulio Carlo Fagnano dei Toschi died. He is important for the identity
and for his rectification of the lemmiscate. *VFR An Italian mathematician who worked in both complex numbers and on the geometry of triangles.*SAU
The lemniscate is of particular interest because, even if it has little relevance today, it
was
the catalyst for immeasurably important mathematical development in the 18th and 19th centuries. The figure 8-shaped curve first entered the minds of mathematicians in 1680, when Giovanni Cassini presented his work on curves of the form, appropriately known as the ovals of Cassini. Only 14 years later, while deriving the arc length of the lemniscate, Jacob Bernoulli became the first mathematician in history to define arc length in terms of polar coordinates.
The first major result of work on the lemniscate came in 1753, when, after reading Giulio Carlo di Fagnano’s papers on dividing the lemniscate using straightedge and compass, Leonhard Euler proved that:
Jacobi called December 23,1751 "the birthday of elliptic functions", as this was the day that Euler began reviewing the papers of Fagnanao who was being considered for membership in the Berlin Academy. *Raymond Ayoub, The lemniscate and Fagnano's contributions to elliptic integrals
1802 Jurij Vega (23 Mar 1754, 26 Sept 1802) wrote about artillery but he is best remembered for his tables of logarithms and trigonometric functions. Vega calculated π to 140 places, a record which stood for over 50 years. This appears in a paper which he published in 1789.
In September 1802 Jurij Vega was reported missing. A search was unsuccessful until his body was found in the Danube near Vienna. The official cause of death was an accident but many suspect that he was murdered. *SAU
1867 James Ferguson (31 Aug 1797, 26 Sep 1867) Scottish-American astronomer who discovered the first previously unknown asteroid to be detected from North America. He recorded it on 1 Sep 1854 at the U.S. Naval Observatory, where he worked 1848-67. This was the thirty-first of the series and is now known as 31 Euphrosyne, named after one of the Charites in Greek mythology. It is one of the largest of the main belt asteroids, between Mars and Jupiter. He was involved in some of the earliest work in micrometry was done at the old U.S. Naval Observatory at Foggy Bottom in the midst of the Civil War using a 9.6 inch refractor. He also contributed to double star astronomy. Earlier in his life he was a civil engineer, member of the Northwest Boundary Survey, and an assistant in the U.S. Coast Survey *TIS
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1868 August Ferdinand Mobius (17 November 1790 – 26 September 1868) 1790 August Möbius (17 Nov 1790; 26 Sep 1868)August Ferdinand Möbius was a German astronomer, mathematician and author. He is best known for his work in analytic geometry and in topology, especially remembered as one of the discoverers of the Möbius strip, which he had discovered in 1858. A Möbius strip is a two-dimensional surface with only one side. It can be constructed in three dimensions as follows. Take a rectangular strip of paper and join the two ends of the strip together so that it has a 180 degree twist. It is now possible to start at a point A on the surface and trace out a path that passes through the point which is apparently on the other side of the surface from A. Although his most famous work is in mathematics, Möbius did publish important work on astronomy.*TIS
He discovered his famous strip in September 1858. Johann Benedict Listing discovered the same surface two months earlier.*VFR (It is somewhat amazing that we call it after Mobius when Listing discovered it first and published, and it seems, Mobius did not. However Mobius did seem to have thought on the four color theorem before Guthrie, or anyone else to my knowledge.)
1877 Hermann Günther Grassmann (15 Apr 1809, 26 Sep 1877) German mathematician chiefly remembered for his development of a general calculus of vectors in Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (1844; "The Theory of Linear Extension, a New Branch of Mathematics"). *TIS
1910 Thorvald Nicolai Thiele (24 Dec 1838, 26 Sept 1910) He is remembered for having an interpolation formula named after him, the formula being used to obtain a rational function which agrees with a given function at any number of given points. He published this in 1909 in his book which made a major contribution to numerical analysis. He introduced cumulants (under the name of "half-invariants") in 1889, 1897, 1899, about 30 years before their rediscovery and exploitation by R A Fisher. *SAU
1976 Paul (Pál) Turán (18 August 1910, 26 September 1976) was a Hungarian mathematician who worked primarily in number theory. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting 46 years and resulting in 28 joint papers. *SAU
In 1940, because of his Jewish origins, he was arrested by the Nazis and sent to a labour camp in Transylvania, later being transferred several times to other camps. While imprisoned, Turán came up with some of his best theories, which he was able to publish after the war. *Wik
1978 Karl Manne Georg Siegbahn (3 Dec 1886, 26 Sep 1978) Swedish physicist who was awarded the Nobel Prize for Physics in 1924 for his discoveries and investigations in X-ray spectroscopy. In 1914 he began his studies in the new science of x-ray spectroscopy which had already established from x-ray spectra that there were two distinct 'shells' of electrons within atoms, each giving rise to groups of spectral lines, labeled 'K' and 'L'. In 1916, Siegbahn discovered a third, or 'M', series. (More were to be found later in heavier elements.) Refining his x-ray equipment and technique, he was able to significantly increase the accuracy of his determinations of spectral lines. This allowed him to make corrections to Bragg's equation for x-ray diffraction to allow for the finer details of crystal diffraction. *TIS
1990 Lothar Collatz (July 6, 1910, , September 26, 1990) was a German mathematician. In 1937 he posed the famous Collatz conjecture, which remains unsolved. The Collatz-Wielandt formula for positive matrices important in the Perron–Frobenius theorem is named after him. *Wik The Collatz conjeture is an iteration problem that deals with the following algorithm..
If a number n is odd, then f(n)= 3n+1
if n is even, then f(n) = 1/2 (n)
Each answer then becomes the new value to input into the function. The problem, or should I say problems, resolve around what happens to the sequence of outcomes when we keep putting the answer back into the function. For example if we begin with 15 we get the following sequence, also called the orbit of the number:
15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1...
One of the unproven conjectures is that for any number n, the sequence will always end in the number 1. This has been shown to be true for all numbers up to just beyond 10^16. ( A recent note from Wendy Appleby at the Math Connections Group on LinkedIn updated this boundary to ~ 2.95*10^20). A second interesting question is how long it takes for a number to return to the value of 1. For the example above, the number 15 took 17 steps to get back to the unit value. Questions such as which three (or other n) digit number has the longest orbit. There are many variations of the problem, but if you are interested in a good introduction, check this link from Simon Fraser University"
Collatz's Problem is often also called the Syracuse Algorithm, Hasse's problem, Thwaite's problem, and Ulam's problem after people who have worked and written on the problem. It is unclear where the problem originated, as it seems to have had a long history of being passed by word of mouth before it was ever written down. It is often attributed to Lothar Collatz from the University of Hamburg who wrote about the problem as early as 1932. The name "Syracuse Problem" was applied by after H. Hasse, an associate of Collatz, visited and discussed the problem at Syracuse University in the 1950's. During the 1960's Stan Ulam circulated the problem at Los Alamos laboratory. One famous quote about the problem is from Paul Erdos who stated, "mathematics is not yet ready for such problems". *Personal notes
Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell
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