~William Whewell
The 144th day of the year; 144 is the only non-trivial square in the Fibonacci Sequence.In fact, there are only four Fibonacci numbers that are perfect powers, 0, 1, 8, and 144. And we haven't known that for so very long. Here is the story from , Professor Stewart's Incredible Numbers.
In 1913 R. D. Carmichael proved his conjecture that for any Fibonacci Number F(n), greater than F(12)=144, has at least one prime factor that is not a factor of any earlier Fibonacci number.
144 is the largest possible SP (sum times product) number. If you take the sum of the digits of a number, and also the product of the digits, and then multiply the two outcomes, there are only three positive numbers for which you will get the original value. One works, trivially. The other two are 135 and 144. 135-> (1+3+5)*(1*3*5) = 9*15=135. 144->(1+4+4)*(1*4*4)= 9 * 16 = 144.
.A Good exercise for students is to take the SP product in a iteration to find out if it goes to zero, or repeats some pattern, or lands eventually on one of these three fixed points(that's four fixed points if you count zero). (Try it with your students). 23->5*6 = 30. 30-> 3*0 = 0.... fixed point.
\(144^5 = 27^5 + 84^5 + 110^5 + 133^5 \) This counter-example disproved Euler's Conjecture that n nth powers are needed to sum to an nth power. It is also part of one of the shortest papers ever published in a math journal(two sentences)
(Squares are important in knowing if a number, n is Fibonacci or not. N is Fibonacci IFF one or both of \(5n^2 \pm 4\) is a perfect square. )
And in a "Talking Numbers" post by John Golden I re-learned that 144 is the smallest perimeter of a primitive Pythagorean triple that is a square number. (see below)
144 is also the smallest square number which is also a square when its digits are reversed 144 = 12 2 while 441= 212
144 is a Sum-Product number, (n = (sum of digits of n)x(product of digits of n) 144=(1+4+4)x(1x4x4) . It is the largest such number. There is one more multi-digit example that is slightly smaller.
The sum of the first 144 decimal digits of pi (don't use the 3.) is 666, "The Number of the Beast." One person wrote that they thought that was gross! ( sorry! :-{ , bad pun)
144 is the only year day that is a square number that is the perimeter of a primitive Pythagorean Triangle. (16, 63, 65) *Benjamin Vitale @BenVitale
EVENTS
1032 The renowned "Arab" scientist Ibn Sina noted, “I saw Venus as a spot on the surface of the sun,” This is the first known record of witnessing the transit of Venus. The first recorded observation of a transit of Venus was made by Jeremiah Horrocks from his home at Carr House in Much Hoole, near Preston in England, on 4 December 1639. Kepler predicted the 1761 transit of Venus, the first such prediction in Western recorded history, and one that inspired several astronomical expeditions. *Sky and Telescope (A.M.S @amoshaye pointed out that he was NOT Arabic, {Persian I believe}. It was once common to lump mid-eastern scholars who wrote in Arabic as Arabian. I write in English, but am not English, a fact that I, and all of England are equally grateful for.)
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| Transit of Venus, 2004 |
1543 An advance copy of his work De revolutionibus orbium coelestium was presented to Copernicus. On the same day he died. *VFR
1547 Ferrari replied to Tartaglia’s letter of 21 April 1547 by sending 31 challenge problems of his own. Tartaglia solved all but the five dealing with cubic equations. *VFR For more on this "math wars" story, see this Renaissance Mathematicus blog.
1683 The Ashmolean, Britain's first Museum, first opened to the public on 24 May 1683. *Ashmolean (in full the Ashmolean Museum of Art and Archaeology) on Beaumont Street, Oxford, England, is the world's first university museum. Its first building was built in 1678–1683 to house the cabinet of curiosities Elias Ashmole gave Oxford University in 1677. *Wik
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1727 Euler arrived at St. Petersburg for the first time on 24 May 1727, only seven days after the Russian empress Katherine I, the wife of Peter I (an adopted daughter of Ernst Glilck, the provost of Marienburg, now Aliiksne) died, and he worked at the St. Petersburg Academy of Sciences until 1741 when he moved to
Berlin, there to stay until 1766. *Historia Mathematica
1844 Samuel F. B. Morse dispatched the first telegraphic message over an experimental line from Washington, D.C., to Baltimore. The message, taken from the Bible, Numbers 23:23 and recorded on a paper tape had been suggested to Morse by Annie Ellsworth, the young daughter of a friend. {Nice to have influential friends, she was the teenage daughter of the Commissioner of Patents. Congress appropriated $30,000 for a telegraph wire to be strung the 80 miles between Washington and Baltimore}..Morse sent the message from the chamber of the Supreme Court, then in the United States Capitol, to his assistant Albert Vail at the Mount Clair depot in Baltimore in 1844.*Library of Congress
A photo of the actual paper tape with raised dots and dashes in the Library of Congress is here. Across the top of this artifact of his historic achievement Morse has given credit to Annie Ellsworth for suggesting the message.
Wolfram alpha will let you convert any message to Morse code, or from Morse code by typing [Morse code "input"] to encode, or Morse Code message to decode.
1937 A temporary science exhibition called Le Palais de la Decouverte opened its doors in the west wing of the Grand Palais in time for the 1937 International Exposition of Art and Technology in Modern Life, which was to be held in Paris. It was primarily the inspiration of French physicist Jean Perrin, who won the Nobel prize in 1926 for his work on the atom. An interesting mathematical anecdote relates to the museum.
The museum contains a circular room known as the "pi room". On its wall is inscribed 707 digits of the number π. The digits are large wooden characters attached to the dome-like ceiling. The digits were based on an 1853 calculation by English mathematician William Shanks, which included an error in the 528th digit. The error was detected in 1946 and corrected in 1949
Shank's 707 digits was the record at that time, taken from his teacher William Rutherford, whose record was 404 digits.His record, now adjusted to the 528 digits he had right, was the last record of hand computation. D. F. Ferguson's 1946 calculation of 606 digits was done with a desk top calculating machine.
1961 MIT's Clark Begins Work on LINC Computer :
Wes Clark began his work on LINC, or the Laboratory Instrument Computer, at MIT's Lincoln Laboratory. His plan was to create a computer for biomedical research, that was easy to program and maintain, that could be communicated with while it operated, and that could process biotechnical signals directly. Building on his previous experience in developing the Whirlwind, TX-0, and other early computers, Clark set to work on one of the earliest examples of a "user friendly" machine -- setting the standard for personal computer design in the following decades. *CHM
The first LINC included two oscilloscope displays. Twenty-one were sold by DEC at $43,600 (equivalent to $390,600 in 2021),
"If you have to ask, you can't afford it!"
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| *Wik |
Weise acted as supervisor of PhD students from a wide range of mathematical fields, a dozen of them went on to become professors, among them Wolfgang Gaschütz (finite groups), Wolfgang Haken (knot theory and the solution of the four-colour-problem), Wilhelm Klingenberg (differential geometry) and Jens Mennicke (topology). Let us look in a little more detail at Weise's influence on one of these students, Wolfgang Haken, who studied mathematics, physics and philosophy at the University of Kiel. Haken attended Heinrich Heesch's talk on his contributions to the Four Colour Problem, but he was most enthused by Weise's lectures on topology. In these lectures, Weise described three long-standing unsolved problems - the Poincaré Conjecture, the Four Colour Problem, and a problem on knot theory. Haken decided to attempt to solve all three problems and began this challenge while studying for a doctorate at Kiel with Weise as his thesis advisor. His thesis, submitted in 1953, was Ein topologischer Satz über die Einbettung (d-1)-dimensionaler Mannigfaltigkeiten in d-dimensionale Mannigfaltigkeiten. He had solved the knot theory problem and this led to his appointment at the University of Illinois in the United States. Eventually, assisted by Kenneth Appel, he solved the Four Colour Problem in 1976 with the aid of computer techniques.
Weise was retired on 30 September1977, and in the following year the Christian Albrechts Universität conferred on him the title of 'Ehrensenator' (honorary senator). *SAU
1914 Federico Cafiero (24 May 1914 in Riposto, Catania, Sicily, 7 May 1980 in Naples, Italy) Cafiero played an important role in building a vigorous mathematical school at Naples which included (in alphabetical order) Luigi Albano, Ugo Barbuti, Antonio Chffi, Paolo De Lucia, Nicola Fedele, Renato Fiorenza, Francesco Guglielmino, Giuseppe Pulvirenti, Giuseppe Santagati and Antonio Zitarosa. We have already seen that Cafiero made contributions to the theory of ordinary differential equations and to the theory of measure and integration.
Two notable awards the Cafiero received for his mathematical contributions were the Tenore prize of the Accademia Pontaniana (awarded in 1953 for his monograph Funzioni additive d'insieme e integrazione negli spazi astratti) and the Golden medal 'Benemeriti della Scuola, della Cultura, dell'Arte' which he received from the President of the Italian Republic in 1976. *SAU
1734 Georg Ernst Stahl (22 October 1659 – 24 May 1734 [NS}) was a German chemist, physician and philosopher. He was a supporter of vitalism, and until the late 18th century his works on phlogiston were accepted as an explanation for chemical processes
Stahl used the works of Johann Joachim Becher to help him come up with explanations of chemical phenomena. The main theory that Stahl got from J. J. Becher was the theory of phlogiston. This theory did not have any experimental basis before Stahl. He was able to make the theory applicable to chemistry.[4] Becher's theories attempted in explaining chemistry as comprehensively as seemingly possible through classifying different earths according to specific reactions. Terra pinguis was a substance that escaped during combustion reactions, according to Becher.[10] Stahl, influenced by Becher's work, developed his theory of phlogiston.People who dismiss Phlogiston theory as early ignorance should read The Renaissance Mathematicus blog, The Phlogiston Theory – Wonderfully wrong but fantastically fruitful.
There exists a manner of viewing geometry that could be called géométrie analytique, and which would consist in deducing the properties of extension from the least possible number of principles, and by truly analytic methods.
... should be treated separately, as far apart as they can be, and that the results in each should serve for mutual clarification, corresponding, so to speak, to the text of a book and its translation.
1896 Luigi Menabrea (4 Sept 1809 in Chambéry, Savoy, France - 24 May 1896 in St Cassin (near Chambéry), France) was a French-born soldier and engineer who made contributions to elasticity theory and became prime-minister of Italy. *SAU
1904 Cecil John Alvin Evelyn (25 August 1904 in London, England, 24 May 1976 in Deptford, Kent, England) He graduated with a B.A. in 1927. At Oxford he had become friendly with Hubert Linfoot who was one year older than Evelyn. Linfoot graduated in 1926 but had remained at Oxford undertaking research advised by G H Hardy. Both Evelyn and Linfoot were interested in number theory at this time and they worked together.
Between 1929 and 1933, Evelyn and Linfoot produced six joint papers, all with the title On a problem in the additive theory of numbers.
A book was to be Evelyn's final mathematical publication. He published the book (with G B Money-Coutts and J A Tyrrell) The seven circles theorem and other new theorems (1974) which was translated into French and published as (with G B Money-Coutts and J A Tyrrell) Le théorème des sept cercles (1975). R D Nelson, Ampleforth College, York, writes :-
This elegant book will please all geometers, amateur and professional, and deserves a place in every library. Using a variety of essentially elementary methods, the authors present and prove a number of new or little known theorems in plane geometry. To emphasise the aesthetic appeal of these results and to assist the argument in places, over forty of its pages carry diagrams of high quality. The book has three independent sections but the style of writing is uniform. The authors invite and sometimes require the co-operation of the reader as he works through the book and, in this way, they prepare him for the intricacies of the final and most difficult section. The first part re-introduces an algebra of vectors, due to Silberstein, in which the laws of addition and equivalence are such that few of the usual properties are obvious. For example, associativity of addition requires two applications of Desargues' theorem for its proof. No use is made of this algebra. The second section opens with a delightful new theorem concerning seven Pascal lines derived from a heptagon inscribed in a conic. This is followed by a number of extensions and generalisations of the theorems of Pascal and Brianchon. ... Finally there are four new theorems about closed chains of six circles ... In the first theorem each circle touches a seventh, in the second the circles alternately touch a pair of parallel lines, in the third each circle touches two of the sides of a triangle and in the fourth each circle touches two out of three fixed circles making a configuration of nine circles in all. The first theorem, beautifully proved by inversion, gives the book its title.
1968 Frances Sarnat Hugle (August 13, 1927 – May 24, 1968) was an American scientist, engineer, and inventor who contributed to the understanding of semiconductors, integrated circuitry, and the unique electrical principles of microscopic materials. She also invented techniques, processes, and equipment for practical (high volume) fabrication of microscopic circuitry, integrated circuits, and microprocessors which are still in use today.
In 1962, Hugle co-founded Siliconix, one of Silicon Valley's first semiconductor houses. She is the only woman included in the "Semiconductor Family Tree *Wik
*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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