## Thursday 31 August 2023

### On This Day in Math - August 31

The pursuit of the good and evil are now linked in astronomy as in almost all science. ... The fate of human civilization will depend on whether the rockets of the future carry the astronomer's telescope or a hydrogen bomb.
~Sir Bernard Lovell

The 243rd day of the year; 243 is the largest three digit number that can be expressed as a fifth power (35).

243 is also the sum of five consecutive prime numbers (41 + 43 + 47 + 53 + 59).

Venus' day is 243 Earth days. *Derek Orr

243 is a Harshad number, divisible by the sum of it's digits, and every permutation of its digits is also. Students should try to figure out why.

I have heard it was Feynman who first noted that 1/243 = .00 4 11 5 22 6 33 7 44 8 55 9 67 0 78 1 89 3..  If you think of the zero in 67 0 78 as the zero of a ten, you can understand the 67, 78, etc

243, like all odd numbers, is the difference in consecutive squares, 122^2-121^2 = 243, because it is 3 mod6 it is also the difference of two squares of integers that differ by 3, 42^2 - 39^2. And with multiple 3's you get more differences, 18^2 - 9^2

243 is a palindrome in base 8 (363)

3^3 + 6^3 = 243

On April 14, 2014,  Almost exactly a year after Yitang Zhang announced a proof (see April 17) that there are infinitely many pairs of prime numbers which differ by 70 million or less Terrance Tao's online group attack on the problem reduced the number to 243. Zhang's proof is the first to establish the existence of a finite bound for prime gaps, resolving a weak form of the twin prime conjecture.

EVENTS

1682 Michael Rolle published an elegant solution to a difficult problem publicly posed by Ozanam: Find four integers the difference of any two of which is a perfect square as well as the sum of the first three will be a perfect square. This brought him public recognition. *VFR Ozanam believed that the smallest of the four numbers that would satisfy these properties would have at least 50 digits. Rolle found four numbers, all satisfying the conditions Ozanam posed, containing only seven digits in each of the four numbers. *Michel Rolle and His Method of Cascades, Christopher Washington

He published his most important work Traité d'algèbre  in 1690 on the theory of equations. In this treatise, he invented the notation $\sqrt[�]{�}$ for the $�$th root of  and, as a consequence, it became the standard notation.  He is best known for Rolle's theorem (1691), and  is also the co-inventor in Europe of Gaussian elimination (1690).

$�$

1831, New London Bridge opened to traffic in London. In 1821, a committee was formed by Parliament to consider the poor condition of the existing centuries-old bridge. The arches had been badly damaged by the Great Freeze, so it was decided to build a new bridge. Building commenced under John Rennie in 1825, and completed in 1831, at the expense of the city. The bridge is composed of five arches, and built of Dartmoor granite. It was opened with great splendor by King William the fourth, accompanied by Queen Adelaide, and many of the members of the royal family, August 1st, 1831. In the 1960's it was auctioned and sold for $2,460,000 to Robert McCulloch who moved it to Havasu City, Arizona. The rebuilt London Bridge was completed and dedicated on 10 Oct 1971. *TIS In 1842, the U.S. Naval Observatory was authorized by an act of Congress, one of the oldest scientific agencies in the U.S. James Melville Gilliss (1811-1865) is considered its founder, who in 1842 he secured the Congressional appropriation for the Depot of Charts and Instruments (est. 1830) to become the Naval Observatory. Its primary task was to care for the Navy's charts, navigational instruments and chronometers, which were calibrated by timing the transit of stars across the meridian. Initially located at Foggy Bottom, the observatory moved in 1893 to its present facility in Washington, DC. Gillis visited Europe to procure instruments, and the books that formed the core of the Naval Observatory Library. Matthew Fontaine Maury was the first director, followed by Gillis (1861-65)*TIS 1846 Le Verrier's announces his prediction of the location of the yet to be discovered planet Neptune. Using only mathematics and astronomical observations of the known planet Uranus and encouraged by physicist Arago, Director of the Paris Observatory, Le Verrier was intensely engaged for months in complex calculations to explain small but systematic discrepancies between Uranus's observed orbit and the one predicted from the laws of gravity of Newton. At the same time, but unknown to Le Verrier, similar calculations were made by John Couch Adams in England. Le Verrier announced his final predicted position for Uranus's unseen perturbing planet publicly to the French Academy on 31 August 1846, two days before Adams's final solution, which turned out to be 12° off the mark, was privately mailed to the Royal Greenwich Observatory. Le Verrier transmitted his own prediction by 18 September letter to Johann Galle of the Berlin Observatory. The letter arrived five days later, and the planet was found with the Berlin Fraunhofer refractor that same evening, 23 September 1846, by Galle and Heinrich d'Arrest within 1° of the predicted location near the boundary between Capricorn and Aquarius. Some of the earliest recorded observations ever made through a telescope, Galileo Galilei's drawings on 28 December 1612 and 27 January 1613 contain plotted points that match up with what is now known to have been the positions of Neptune on those dates. On both occasions, Galileo seems to have mistaken Neptune for a fixed star when it appeared close—in conjunction—to Jupiter in the night sky. *Wik 1869 Mary Ward was an Anglo Irish amateur scientist who was killed when she fell under the wheels of an experimental steam car built by her cousins. As the event occurred in 1869, she is the world's first known fatal motor vehicle accident victim." Ward was a keen amateur astronomer, sharing this interest with her cousin William Parsons, 3rd Earl of Rosse. Parsons built the Leviathan of Parsonstown, a reflecting telescope with a six-foot mirror which remained the world's largest until 1917. *Wik In 1886, the first U.S. earthquake on record with significant human consequence - the loss of some 100 lives - hit Charleston, S.C. and its massive effect spread through many eastern States. The epicenter was 15 miles northwest of Charleston, where 41 people died, 90 percent of the city's 6,956 brick buildings were damaged, and nearly all of its 14,000 chimneys were broken off at the roof. However, geologically the most severe earthquakes in U.S. history had occurred earlier in the century near the present town of New Madrid, Missouri (16 Dec 1811). The epicenter then was in a sparsely populated region and caused no known casualties, so the human consequences were relatively not significant, although the violent movement of the ground changed the course of the Mississippi River and created many new lakes.*TIS 1899 Cantor, in a letter to Dedekind, remarked that his “diagonal process” can be used to show that the power set of a set has more elements than the set itself. *VFR 1946 The New Yorker Magazine devoted its cover, and the entire issue to a ground breaking article by war correspondent John Hersey, Describing the actual after-effects of the Hiroshima bomb, exposing the suffering, and long-lasting effects of the bomb. The US military, US government, and General Groves had pushed the white-wash that there were no significant after-effects, and one spokesman described the bomb as a "nice way to die." 1950 G¨odel addressed the International Congress of Mathematicians, in Cambridge, Massachusetts, on his work in relativity theory. *VFR At the International Congress at Toronto in 1924 it had been decided that at each international mathematical congress two gold medals should be awarded. Professor J C Fields, the Secretary of the 1924 Congress, presented a fund to subsidise these medals. They were first awarded in Oslo in 1936. The Committee to select the winners of the 1950 medals was: Professor Harald Bohr (Chairman), Professors L V Ahlfors, Karol Borsuk, Maurice Fréchet, W V D Hodge, A N Kolmogorov, D Kosambi, and Marston Morse. The medals in 1950 were awarded to Professor Laurent Schwartz of the University of Nancy and to Professor Atle Selberg of the Institute for Advanced Study. Professor Bohr gave an excellent résumé of the work of Schwartz on distributions and of the work of Selberg on the Riemann zeta function and his elementary proof of the celebrated prime number theorem. Professor Bohr gave an excellent résumé of the work of Schwartz on distributions and of the work of Selberg on the Riemann zeta function and his elementary proof of the celebrated prime number theorem. 1994 Aldus Corp. and Adobe Systems Inc. finalized their merger. The two companies hoped to combine forces in creating powerful desktop publishing software, building on the field Aldus founder Paul Brainerd had created in 1985 with his PageMaker software. Pagemaker was one of three components to the desktop publishing revolution. The other two were the invention of Postscript by Adobe and the LaserWriter laser printer from Apple. All three were necessary to create a desktop publishing environment. 1994 David Charles Hahn, later called the "Radioactive Boy Scout" or the "Nuclear Boy Scout", attracted the attention of local police when he was stopped on another matter and they found material in his vehicle that troubled them and he warned that it was radioactive. The gift of a Chemistry set at age twelve sparked his interest, first to make nitroglycerin and then, at age seventeen, to build a homemade breeder nuclear reactor. A Scout in the Boy Scouts of America, Hahn conducted his experiments in secret in a backyard shed at his mother's house in Commerce Township, Michigan. His mother's property was cleaned up by the Environmental Protection Agency ten months later as a Superfund cleanup site. Hahn attained Eagle Scout rank shortly after his lab was dismantled. *R. R. Johnson, Romancing the Atom 2012 A Blue Moon, or the second of two full moons in a single month. August 2012 will have a blue moon on August 31 The last month with two full moons was March of 2010 March 1 and March 30. The next month with a blue moon will be in January of 2018. Once in a Blue moon really isn’t all that often. Their are alternate definitions for blue moon,f or instance the fourth moon in a quarter. For that you have to wait until June of 2019. Tonight in 2023 will be the first Blue Moon since 2018. The nexr blue moon will be on May 31, 2026. 2012 "Japanese mathematician Shinichi Mochizuki posted four papers on the Internet. The titles were inscrutable. The volume was daunting: 512 pages in total. The claim was audacious: he said he had proved the ABC Conjecture, a famed, beguilingly simple number theory problem that had stumped mathematicians for decades." From The Paradox of the Proof by Caroline Chen . BIRTHS 1663 Guillaume Amontons (31 Aug 1663; 11 Oct 1705)French physicist, who developed the air thermometer - which relies on increase in volume of a gas (rather than a liquid) with temperature - and used it (1702) to measure change in temperature in terms of a proportional change in pressure. This observation led to the concept of absolute zero in the19th century. Deaf since childhood, Amontons worked on inventions for the deaf, such as the first telegraph, which relied on a telescope, light, and several stations to transmit information over large distances. Amontons' laws of friction, relied upon by engineers for 300 years, state that the frictional force on a body sliding over a surface is proportional to the load that presses them together and is also independent of the areas of the surfaces. *TIS 1821 Hermann Ludwig Ferdinand von Helmholtz (31 Aug 1821; 8 Sep 1894) was a German scientist who contributed much to physiology, optics, electrodynamics, mathematics, and meteorology, including the law of the conservation of energy (1847). He also developed thermodynamics, in particular introducing concept of free energy. In 1850, he measured the speed of a nerve impulse and, in 1851, invented the ophthalmoscope. He discovered the function of the cochlea in the inner ear and developed Thomas Young's theory of color vision (published 1856). His study of muscle action led him to formulate a much more accurate theory concerning the conservation of energy than earlier proposed by Julius Mayer and James Joule. *TIS 1864 Robert Hardie (31 Aug 1864 in George Street, Edinburgh, Scotland - 9 March 1942 in Edinburgh, Scotland) graduated from Oxford and occupied various posts in the Philosophy department of Edinburgh University. He was a founder member of the EMS. *SAU 1880 Heinrich Franz Friedrich Tietze (August 31, 1880 – February 17, 1964) contributed to the foundations of general topology and developed important work on subdivisions of cell complexes. The bulk of this work was carried out after he took up the chair at Munich in 1925.*SAU He is remimbered for the Tietze extension theorem. He also developed the Tietze transformations for group presentations, and was the first to pose the group isomorphism problem. He was born in Schleinz, Austria and died in Munich, Germany. *Wik 1884 Birthdate of George Alfred L´eon Sarton, (August 31, 1884; Ghent, Belgium - March 22, 1956, Cambridge, Massachusetts) ,historian of science and founder of the journal Isis. *VFR Sarton intended to complete an exhaustive nine volume history of science — which, during the preparation of the second volume, induced him to learn Arabic and travel around the Middle East inspecting original manuscripts of Islamic scientists — but at the time of his death only the first three volumes had been completed. (I. From Homer to Omar Khayyam. — II. From Rabbi Ben Ezra to Roger Bacon, pt. 1-2. — III. Science and learning in the fourteenth -century, pt. 1-2. 1927-48.) The project was inspired by his study of Leonardo da Vinci but the period of Leonardo's life was not reached before the death of Sarton. *Wik 1885 Herbert Westren Turnbull (31 Aug 1885; 4 May 1961)English mathematician who made extensive and notable contributions to the study of algebraic invariants and concomitants of quadratics. Turnbull was also interested in the history of mathematics, writing The Mathematical Discoveries of Newton (1945), and began work on the Correspondence of Isaac Newton.*TIS 1913 Sir Alfred Charles Bernard Lovell (31 Aug 1913, ) is an English radio astronomer who established and directed (1951-81) Jodrell Bank Experimental Station, Cheshire, England, with (then) the world's largest steerable radiotelescope, now named after him Prior to WW II, he worked at Manchester University on cosmic ray research. During the war, he helped develop aircraft onboard radar systems. After the war, to escape interference to radar equipment from city trams, he moved his research to the University's more remote Jodrell Bank property. In 1946, he showed that radar echoes could detect optically invisible daytime meteor showers. He gained funding to build the 250-ft-diam. telescope. When completed in 1957, it was able to track the first artificial satellite, Sputnik I. *TIS  The Lovell Telescope at Jodrell Bank, *wik 1916 Robert Hanbury Brown (31 Aug 1916; 16 Jan 2002) British astronomer who was a pioneer in radar and observational astronomy. During and after WW II he worked with R.A. Watson-Watt and then E.G. Bowen to develop radar for uses in aerial combat. In the 1950s he applied this experience to radio astronomy, developing radio-telescope technology at Jodrell Bank Observatory and mapping stellar radio sources. He designed a radio interferometer capable of resolving radio stars while eliminating atmospheric distortion from the image (1952). With R.Q. Twiss, Brown applied this method to measuring the angular size of bright visible stars, thus developing the technique of intensity interferometry. They set up an intensity interferometer at Narrabri in New South Wales, Australia, for measurements of hot stars.*TIS DEATHS 1721 John Keill (1 Dec 1671, 31 Aug 1721) Scottish mathematician and natural philosopher, who was a major proponent of Newton’s theories. He began his university education at Edinburgh under David Gregory, whom he followed to Oxford, where Keill lectured on Newton's work, and eventually became professor of astronomy. In his book, An Examination of Dr. Burnett's Theory of the Earth (1698), Keill applied Newtonian principles challenging Burnett's unsupportable speculations on Earth's formation. In 1701, Keill published Introductio ad Veram Physicam, which was the first series of experimental lectures and provided a clear and influential introduction to Isaac Newton’s Principia. He supported Newton against priority claims by Leibnitz for the invention of calculus.*TIS 1811 Louis-Antoine de Bougainville (12 November 1729 – 31 August 1811) was a French soldier and explorer who wrote a calculus book, but is better known for his other exploits.*SAU A contemporary of James Cook, he took part in the French and Indian War and the unsuccessful French attempt to defend Canada from Britain. He later gained fame for his expeditions to settle the Falkland Islands and his voyages into the Pacific Ocean.*Wik 1918 André-Louis Cholesky (October 15, 1875, August 31, 1918, ) was a French military officer and mathematician. He worked in geodesy and map-making, was involved in surveying in Crete and North Africa before World War I. But he is primarily remembered for the development of a matrix decomposition known as the Cholesky decomposition which he used in his surveying work. He served the French military as engineer officer and was killed in battle a few months before the end of World War I; his discovery was published posthumously by his fellow officer in the "Bulletin Géodésique". *Wik 1945 Stefan Banach died. (30 Mar 1892, 31 Aug 1945) Polish mathematician who founded modern functional analysis and helped develop the theory of topological vector spaces. In addition, he contributed to measure theory, integration, the theory of sets, and orthogonal series. In his dissertation, written in 1920, he defined axiomatically what today is called a Banach space. The idea was introduced by others at about the same time (for example Wiener introduced the notion but did not develop the theory). The name 'Banach space' was coined by Fréchet. Banach algebras were also named after him. The importance of Banach's contribution is that he developed a systematic theory of functional analysis, where before there had only been isolated results which were later seen to fit into the new theory. *TIS 1950 Subbayya Sivasankaranarayana Pillai (April 5, 1901 Nagercoil, Tamil Nadu - 31 August 1950, Cairo, Egypt) was an Nagercoil native Indian mathematician specializing in number theory. His contribution to Waring's problem was described in 1950 by K. S. Chandrasekharan as "almost certainly his best piece of work and one of the very best achievements in Indian Mathematics since Ramanujan". In number theory, a Pillai prime, named for him, is a prime number p for which there is an integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. To put it algebraically, $n! \equiv -1 \mod p$ but $p \not\equiv 1 \mod n$. The first few Pillai primes are 23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, ... (sequence A063980 in OEIS). *Wik 2005 Sir Joseph Rotblat (4 Nov 1908, 31 Aug 2005)Polish-born British physicist who is a leading critic of nuclear weaponry. Rotblat and the Pugwash Conferences, "for their efforts to diminish the part played by nuclear arms in international politics and in the longer run to eliminate such arms," received the Nobel Peace Prize in 1995. Forty years earlier, he and other scientists, with philosopher Bertrand Russell and Albert Einstein, published a manifesto calling on researchers to take responsibility for their work, particularly those working on the atomic bomb. This led to the Pugwash Conferences on Science and World Affairs, first convened in 1957 in Pugwash, Nova Scotia, Canada. He was secretary-general (1957-73), and president (from 1988) of this London-based worldwide organization. *TIS 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 ## Wednesday 30 August 2023 ### This Day in Math - August 30 Frustra fit per plura, quod fieri potest per pauciora. It is vain to do with more what can be done with less. ~William of Ockham The 242nd day of the year; 242 has six divisors...but 243, 244, and 245 also each has six divisors. 242 is the smallest integer to begin a run of four consecutive integers all of which have the same number of divisors. (What is the smallest integer that begins a run of three consecutive integers with an equal number of divisors?) 242 is not only a palindrome in base ten, it is also a palindrome in base 3, ${22222}_{3}$ and base 7, ${464}_{7}$. (What palindrome in base ten is also a palindrome in the most other bases 2-9?) 242 = 2 x 11^2, it is divisible by 2, 11, 22, 121 all of which are also palindromes. 242 is the nth prime + n for n = 45. 197 + 45 = 242 See More Math Facts for every Year Day, here EVENTS In 1831, Charles Darwin replied to the letter from Revd. Henslow telling him of the offer to sail on the H.M.S. Beagle. Darwin's had learned natural history from Henslow, who had recommended him for the unpaid position as a naturalist. Darwin told Henslow that his father would not permit him to leave on such a voyage. Meanwhile, his father had written to his brother-in-law, Josiah Wedgwood II, about his concerns regarding the proposed two-year jaunt. This afternoon Darwin prepared to join the Wedgwoods for the next day's beginning of the shooting season by riding to Maer Hall, the Wedgwood home. The Darwin family was related to the Wedgwood family through the marriage of Darwin's father to the daughter of the first Josiah Wedgwood, the famous potter. *TIS 1791 Thomas Jefferson sends a letter to Benjamin Banneker after receiving his almanac and a letter announcing that, "Sir, I freely and cheerfully acknowledge, that I am of the African race, and in that color which is natural to them of the deepest dye". Jefferson responds that, " I thank you sincerely for your letter of the 19th. instant and for the Almanac it contained. no body wishes more than I do to see such proofs as you exhibit, that nature has given to our black brethren, talents equal to those of the other colours of men, that the appearance of a want of them is owing merely to the degraded condition of their existence both in Africa and America. I can add with truth that no body wishes more ardently to see a good system commenced for raising the condition both of their body & mind to what it ought to be, as fast as the imbecillity of their present existence, and other circumstance which cannot be neglected, will admit. I have taken the liberty of sending your almanac to Monsieur de Condorcet, Secretary of the Academy of sciences at Paris, and member of the Philanthropic society because I considered it as a document to which your whole colour had a right for their justification against the doubts which have been entertained of them. I am with great esteem, Sir, Your most obedt. humble servt. Th. Jefferson" *Mathematicians of the African Diaspora, SUNY at Buffalo In 1831, Michael Faraday demonstrated the first electrical transformer.*TIS 1908 A committee appointed by the Swiss society of naturalists reported its willingness, provided sufficient ﬁnancial assistance could be secured, to publish the complete works of Euler in about 40 volumes. Today 80 volumes of Euler’s Opera Omnia have been published, and the end is hardly in sight.*VFR 1950 Harald Bohr presented a Fields Medal to Lauren Schwartz, French mathematician who is best known for his work in the theory of distributions, at the International Congress of Mathematicians in Harvard for his work on the theory of distributions. Harald Bohr described Schwartz's 1948 paper as one:- ... which certainly will stand as one of the classical mathematical papers of our times. ... I think every reader of his cited paper, like myself, will have left a considerable amount of pleasant excitement, on seeing the wonderful harmony of the whole structure of the calculus to which the theory leads and on understanding how essential an advance its application may mean to many parts of higher analysis, such as spectral theory, potential theory, and indeed the whole theory of linear partial differential equations ... Harald August Bohr (22 April 1887 – 22 January 1951) was a Danish mathematician and footballer. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the Nobel Prize-winning physicist Niels Bohr. He was a member of the Danish national football team for the 1908 Summer Olympics, where he won a silver medal. In 1963, the "Hot Line" communications link between the White House, Washington D.C. and the Kremlin, Moscow, went into operation to provide a direct two-way communications channel between the American and Soviet governments in the event of an international crisis. This was one year after the Cuban Missile Crisis. It consisted of one full-time duplex wire telegraph circuit, routed Washington- London- Copenhagen- Stockholm- Helsinki- Moscow, used for the transmission of messages and one full-time duplex radiotelegraph circuit, routed Washington- Tangier- Moscow used for service communications and for coordination of operations between the two terminal points. Note, this was not a telephone voice link.*TIS In 1979, Comet Howard-Koomen-Michels (SOLWIND I) collided with the Sun, the first recorded comet to collide with Sun and the first discovered by a spacecraft. The coronographs taken on 30 and 31 Aug 1979 from the satellite P78-1 used to monitor solar corona activity were not inspected until Sep 1981, by Russ Howard. The recording instruments were designed and operated by Martin Koomen and Don Michels. The remarkable series of images showed the comet heading around the Sun. Its perihelion distance was too small, and the head did not reappear from behind the Sun, presumably disintegrated by the heat of the sun. The decapitated comet's tail continued, becoming fan-like, brightening the corona, until dissipated and blown away from the Sun*TIS 2019 National Slinky Day As its jingle once cheered: “A spring, a spring, a marvelous thing! Everyone knows it’s Slinky.” The coiled toy certainly is a marvelous, if simplistic, thing. In 1943, mechanical engineer Richard James was designing a device that the Navy could use to secure equipment and shipments on ships while they rocked at sea. As the story goes, he dropped the coiled wires he was tinkering with on the ground and watched them tumble end-over-end across the floor. After dropping the coil, he could have gotten up, frustrated, and chased after it without a second thought. But he—as inventors often do—had a second thought: perhaps this would make a good toy. Richard James went home and told his wife, Betty James, about his idea. In 1944, she scoured the dictionary for a fitting name, landing on “slinky,” which means “sleek and sinuous in movement or outline.” Together, with a$500 loan, they co-founded James Industries in 1945, the year the Slinky hit store shelves.

On August 30, 2019  National Slinky Day, the Pennsylvania Historical and Museum Commission installed a historical marker to commemorate the invention of the toy in Clifton Heights, the Philadelphia suburb where it was first manufactured. *Smithsonian Mag

BIRTHS

1804 Ernst Wilhelm Grebe (30 August, 1804 - 14 January, 1874) is remembered only for a thoughtful paper that appeared in 1847 concerning some interesting properties of the triangle: If on each side of a given (arbitrary) triangle ABC one describes a square ( exterior to ABC ), then the extended outside sides of the squares, thus obtained, form a similar triangle A'B'C'. The center of similarity of both triangles is the meeting point of the straight lines AA', BB', CC'. In German this point was first called _Grebe'schen Punkt_ [Grebe's point], a TERM which seems to have been first referred to by E. Hain as early as 1875, in his paper "Ueber den Grebe'schen Punkt" [ _Archiv der Mathematik und Physik_ (= Grunert's _Archive_) volume LVIII (1876), pp. 84-89 ] *Julio Gonzalez Cabillon , Posting to the Historia Matematica discussion group
He also named the Vecten point.  (The Greeb Point is also called the Lemoine Point, and the Symmedian {medians reflected at the associated angle bisectors}
point)   The French mathematician Émile Lemoine proved the existence of the symmedian point in 1873,
The symmedian point of a triangle with side lengths a, b and c has homogeneous trilinear coordinates [a : b : c].
Ross Honsberger called its existence "one of the crown jewels of modern geometry"

### Martin Lukarevski informed me that, " Grebe was 26 years ahead of Lemoine, but in fact the symmedian point had already been noted in 1809 by the Swiss geometer Simon L'Huilier. "  My Thanks!

Image:  A triangle with medians (black), angle bisectors (dotted) and symmedians (red). The symmedians intersect in the symmedian point L, the angle bisectors in the incenter I and the medians in the centroid G.

1819 Joseph Alfred Serret (30 Aug 1819 in Paris, France - 2 March 1885 in Versailles, France) He was a French mathematician best remembered for the Serret-Frenet formulas for a space-curve. In 1860 Serret succeeded Poinsot in the Académie des Sciences. In 1871 he retired to Versailles as his health began to deteriorate.
Serret also worked in number theory, calculus and mechanics. He edited the works of Lagrange which were published in 14 volumes between 1867 and 1892. He also edited the 5th edition of Monge in 1850.*SAU

The Frenet–Serret formulas are: where d/ds is the derivative with respect to arclength, κ is the curvature, and τ is the torsion of the curve. The two scalars κ and τ effectively define the curvature and torsion of a space curve.

 *Wik

1856 Carl David Tolmé Runge worked on a procedure for the numerical solution of algebraic equations and later studied the wavelengths of the spectral lines of elements. *SAU In numerical analysis, the Runge–Kutta methods that are named for him are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900  Runge and M.W. Kutta.*Wik When  your regular walking partners include Felix Klein, David Hilbert, and Hermann Minkowski, you can't count on easily impressing them with your mental math skills, but it seems that Runge did so frequently.  Once on their regular walks Klein brought up some departmental event that required them to know what date Easter would occur the next year.  The group immediately turned to the idea of where they might acquire a calendar for the following year along the walk; all that is, except Runge who fell silent for a few yards, and then announced the date.

1871 Sir Ernest Rutherford (30 Aug 1871; 19 Oct 1937). (baron) New Zealand-born British physicist who laid the groundwork for the development of nuclear physics. He worked under Sir J. J. Thomson at Cambridge University (1895-98). Then he collaborated with Frederick Soddy in studying radioactivity. In 1899 he discovered alpha particles and beta particles, followed by the discovery of gamma radiation the following year. In 1905, with Soddy, he announced that radioactive decay involves a series of transformations. In 1907, with Hans Geiger and Ernest Marsden, he devised the alpha-particle scattering experiment that led in 1911 to the discovery of the atomic nucleus. In 1919 he achieved the artificial splitting of light atoms. In 1908 he was awarded the Nobel Prize for Chemistry. *TIS A story is told by A.. V. Hill that Rutherford had told him once, "I've just been reading some of my early papers, and when I had read them for a bit I said to myself, "Ernest my boy, you used to be a darn clever fellow.'" *Walter Gratzer, Eurekas and Euphorias, pg 27
[I love this quote from a few years before his death. "The energy produced by the breaking down of the atom is a very poor kind of thing. Anyone who expects a source of power from the transformation of these atoms is talking moonshine."]

1906 Olga Taussky-Todd She received her Ph.D. in 1930 under Philip Furtwangler at Vienna in number theory. Her ﬁrst job involved editing Hilbert’s papers on number theory.*VFR

1907 John Mauchly (30 Aug 1907; 8 Jan 1980) American physicist and engineer, who with John P. Eckert invented (1946) the Electronic Numerical Integrator and Computer (ENIAC), the first general-purpose electronic computer. Mauchly initially conceived of the computer's architecture, and Eckert possessed the engineering skills to bring the idea to life. ENIAC was developed (1946) for the US Army Ordnance Department as what was probably the first general-purpose electronic computer. It was a vast machine, consuming 100 kW of electric power and containing 18,000 electronic valves. Their successful UNIVAC computer (1951) was the first commercial computer, and introduced magnetic tape for programming.*TIS

1907 Gordon Brown founded the Servomechanisms Laboratory at MIT, which pioneered the development of feedback-control theory, computer technology, and automatic control of machine tools and had many famous graduate students who went on to become major contributors in the fields of Electrical Engineering and Computer Science. *CHM

1912 E.M. Purcell (30 Aug 1912; 7 Mar 1997) American physicist who shared, with Felix Bloch of the United States, the Nobel Prize for Physics in 1952 for his independent discovery (1946) of nuclear magnetic resonance in liquids and in solids. Nuclear magnetic resonance (NMR) has become widely used to study the molecular structure of pure materials and the composition of mixtures. The method detects and measures the magnetic fields of atomic nuclei. *TIS

DEATHS

1621 Baha' ad-Din al-Amili (27 Feb 1547 in Baalbek, now in Lebanon - 30 Aug 1621 in Isfahan, Iran) was a Lebanese-born mathematician who wrote influential works on arithmetic, astronomy and grammar. Perhaps his most famous mathematical work was Quintessence of Calculation which was a treatise in ten sections, strongly influenced by The Key to Arithmetic (1427) by Jamshid al-Kashi. *SAU

1844 Francis Baily   (28 Apr 1774, 30 Aug 1844)English astronomer who detected the phenomenon called "Baily's beads" during an annular eclipse of the Sun on 15 May 1836. His vivid description aroused new interest in the study of eclipses. After retiring in 1825 from a successful business career, Baily turned to science. Baily revised several star catalogs, repeated Henry Cavendish's experiments to determine the density of the Earth, and measured its elliptical shape. His protests regarding the British Nautical Almanac, then notorious for its errors, were instrumental in bringing about its reform.*TIS  A really nice discussion of the many contributions of Baily is in this post at The Renaissance Mathematicus, To whet your appetite, " Baily’s Flamsteed memoir had a major influence on the history and the  historiography of science; he had succeeded in pricking Newton’s  hagiographic bubble. St Isaac had been taken down a peg or two. Baily’s  work marks a turning point in our understanding of Newton moving him  along the road from plastic saint to real, if somewhat unpleasant, human  being. Sometimes editing star catalogues can lead to unexpected results  for the history of science."

1901 Biquard Pierre, (30 August 1901 in Paris - 28 April 1992 in Paris)In 1932, He discovered light diffraction by ultrasonic waves: Pierre Biquard, born 30 Aug 1901, friend of Frédéric Joliot Curie *Arjen Dijksman ‏@materion

1928 Wilhelm Wien  (13 Jan 1864, 30 Aug 1928)German physicist who received the Nobel Prize for Physics in 1911 for his displacement law concerning the radiation emitted by the perfectly efficient blackbody (a surface that absorbs all radiant energy falling on it). While studying streams of ionized gas Wien, in 1898, identified a positive particle equal in mass to the hydrogen atom. Wien, with this work, laid the foundation of mass spectroscopy. J J Thomson refined Wien's apparatus and conducted further experiments in 1913 then, after work by E Rutherford in 1919, Wien's particle was accepted and named the proton. Wien also made important contributions to the study of cathode rays, X-rays and canal rays.*TIS [I find it curiously interesting that all three of  the  great physicists mentioned here either were born or died this day]

1940 Sir J(oseph) J(ohn) Thomson (born 18 Dec 1856, 30 Aug 1940 )was an English physicist who helped revolutionize the knowledge of atomic structure by his discovery of the electron (1897). He received the Nobel Prize for Physics in 1906 and was knighted in 1908. Thomson experimented with currents of  electricity inside empty glass tubes, investigating a long-standing puzzle known as "cathode rays." His experiments prompted him to make a bold proposal: these mysterious rays are streams of particles much smaller than atoms. He called these particles "corpuscles," and suggested that they might make up all of the matter in atoms. It was startling to imagine a particles inside the atom at a time when most people thought that the atom was indivisible, the most fundamental unit of matter.*TIS

1995 Fischer Sheffey Black (January 11, 1938, August 30, 1995) was an American economist, best known as one of the authors of the famous Black–Scholes equation.In 1973, Black, along with Myron Scholes, published the paper 'The Pricing of Options and Corporate Liabilities' in 'The Journal of Political Economy'. This was his most famous work and included the Black–Scholes equation. The Nobel Prize is not given posthumously, so it was not awarded to Black in 1997 when his co-author Myron Scholes received the honor for their landmark work on option pricing along with Robert C. Merton, another pioneer in the development of valuation of stock options. In the announcement of the award that year, the Nobel committee prominently mentioned Black's key role.*Wik

2004 Fred Lawrence Whipple  (5 Nov 1906, 30 Aug 2004) was an American astronomer who proposed the "dirty snowball" model for comet nuclei. In the 1930s, using a new, two-station method of photography, he determined meteor trajectories and found that nearly all visible meteors are made up of fragile material from comets, and that none come from beyond the solar system. Whipple suggested (1950) that comets have icy cores inside thin insulating layers of dirt, and that jets of material ejected as a result of solar heating were the cause of orbital changes. This model was confirmed in 1986 when spacecraft flew past comet Halley. Whipple’s work on tracking artificial satellites led to improved knowledge of the shape of the earth and greatly improved positions on earth. *TIS

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

## Tuesday 29 August 2023

### On This Day in Math - August 29

 Jeannie at VLA

In most sciences one generation tears down what another has built,
and what one has established, another undoes.
In mathematics alone each generation adds a new story to the old structure.
~Hermann Hankel

The 241st day of the year; 241 is the larger of a pair of twin primes. The larger of a pair of twin primes is always one more than a multiple of six; the smaller is always one less than a multiple of six.

2+4+1 is prime. 241 is the 53rd prime. (53 is also prime) *Derek Orr

241 is also The smallest prime p such that p plus the reversal of p equals a palindromic prime.  241 + 142 = 383; which is a prime palindrome.

And it is the largest known prime p such that the reversal of (p! + p) is prime.  (241! + 241 ends with a string of fifty-five zeros, and then 241 :
980360372638941007038951797078339359751464353463061342202811
188548638347461066010066193275864531994024640834549254693776
854464608509281547718518965382728677985343589672835884994580
815417004715718468026937051493675623385569404900262441027874
255428340399091926993707625233667755768320823071062785275404
107485450075779940944580451919726756974354635829128751944137
27644867102380111026020691554782580923999494640500736
000000000000000000000000000000000000000000000000000000

P. Honaker at Prime Curios points out that the sequence of primes formed by n!+239 begins, 241, 263, 359... Maybe I mis-searched but I did not find this sequence in OEIS. Seems like a good computer programming project for students, pick a prime and find primes of the form n! + p
John Cook posted that "if k is relatively prime to b, there is a multiple of k whose base b representation contains all ones. If I understand that then since 241 is prime, it is relatively prime to ten. Can you find the base ten multiple of k that has all ones for its digits? Be the first to share the answer with me and get immortality by being listed here. I think it must be a very large multiple of 241.

241 = 15^2 + 4^2 which means 241 is a Pythagorean Prime.

241 is a palindrome in duodecimal, base 12 (181) and a repdigit in base 15(111)

EVENTS

1609 Galileo writes to his brother in Florence to tell him about his telescope presentation to the Doge on the 24th of August.

1654 Fermat to Pascal Saturday, August 29, 1654

Monsieur,
Our interchange of blows still continues, and I am well pleased that our thoughts are in such complete adjustment as it seems since they have taken the same direction and followed the same road. Your recent Trait´e du triangle arithmetique and its applications are an authentic proof and if my computations do me no wrong, your eleventh consequence went by post from Paris to Toulouse while my theorem, on figurate numbers, which is virtually the same, was going from Toulouse to Paris. I have not been on watch for failure while I have been at work on the problem and I am persuaded that the true way to escape failure is by concurring with you. But if I should say more, it would he of the nature of a Compliment and we have banished that enemy of sweet and easy conversation. It is now my turn to give you some of my numerical discoveries, but the end of the parliament augments my duties and I hope that out of your goodness you will allow me due and almost necessary respite.

In the same letter he states that, "Meditate however, if you find it convenient, on this theorem: The squared powers of 2 augmented by unity [I.e. 22n+1] are always prime numbers. [That is,] The square of 2 augmented by unity makes 5 which is a prime number;The square of the square makes 16 which, when unity is added makes 17, a prime number; The square of 16 makes 256 which, when unity is added, makes 257, a prime number; The square of 256 makes 65536 which, when unity is added, makes 65537, a prime number; and so to infinity. This is a property whose truth I will answer to you. The proof of it is very difficult (impossible, since the statement, as Euler would show later, is not true) and I assure you that I have not yet been able to find it fully." * York University Maths Dept

1692 For his services to the field of astronomy, Johann Philipp von Wurzelbauer was ennobled in 1692 by Leopold I, Holy Roman Emperor and added the von to his name. *Wik

1740 In a letter to Euler dated August 29th, 1740, Philippe Naudé (the Younger) asked Euler in how many ways a number n can be written as a sum of positive integers. In his answer written on September 12th (23rd), Euler explained that if we denote
this “partition number” by p(n), then

*Correspondence of Leonhard Euler with Christian Goldbach, Springer

1831 Michael Faraday discovered electrical induction. *VFR In 1831, Michael Faraday wound a thick iron ring on one side with insulated wire that was connected to a battery. He then wound the opposite side with wire connected to a galvanometer. He found that upon closing the battery circuit, there was a deflection of the galvanometer in the second circuit. Then he was astonished to see the galvanometer needle jump in the opposite direction when the battery circuit was opened. He had discovered that a current was induced in the secondary when a current in the primary was connected and an induced current in the opposite direction when the primary current was disconnected.*TIS

1859 Amateur English astronomers Richard Carrington and Richard Hodgson, independently observed a "white light flare" emanating from the surface of the sun. Less than a day later, Earth's magnetic field was knocked awry. Across America and Europe, telegraph wires sparked and failed.
Fewer than 18 hours elapsed between the flare and the geomagnetic storm on Earth. That meant whatever had exploded off the sun must have traveled at more than 5 million miles per hour. *NY Times

1899 Dedekind sends a letter to Georg Cantor that includes a proof of the Schroder-Bernstein Theorem (Let A and B be sets. If there is a 1-1 correspondence from A to B and a 1-1 corespondence from B to A, then the sets have the same cardinality.) *Cantorian Set Theory and Limitation of Size By Michael Hallett

In 1940, Sir Henry Tizard led a mission of leading British and Canadian scientists to the USA to brief official American representatives on devices under active development for war use and to enlist the support of American scientists. Thus began a close cooperation of Anglo-American scientists in such fields as aeronautics and rocketry. His influence probably made the difference between defeat or victory at the Battle of Britain in 1940. *TIS

1949 the USSR tested their first atomic device, "First Lightning." It was an an implosive type plutonium bomb, detonated at the Semipalatinsk test range, giving up to a 20 kiloton yield. In the U.S. it was called Joe No. 1 ("Joe" was nickname for Y. Stalin.) This event came five years earlier than anyone in the West had predicted, largely due to one man, the spy Klaus Fuchs. As a Los Alamos physicist, Fuchs had passed detailed blue prints of the original American Trinity bomb design to the Russians. With the emergence of the USSR as a nuclear rival, America's monopoly of atomic weaponry was ended giving the U.S. strong motivation for intensifying its program of nuclear testing. Thus the Cold War was launched.*TIS

1970 Oscar Morgenstern writes in his diary that Gödel would NOT publish his ontological proof for the existence of God. The first version of the ontological proof in Gödel's papers is dated "around 1941". Gödel is not known to have told anyone about his work on the proof until 1970, when he thought he was dying. In February, he allowed Dana Scott to copy out a version of the proof, which circulated privately. In August 1970, Gödel told Oskar Morgenstern that he was "satisfied" with the proof, but Morgenstern recorded in his diary entry for 29 August 1970, that Gödel would not publish because he was afraid that others might think "that he actually believes in God, whereas he is only engaged in a logical investigation (that is, in showing that such a proof with classical assumptions (completeness, etc.) correspondingly axiomatized, is possible) *Wik

1990 The British Computer Misuse Act goes into effect One of the earliest laws anywhere designed to address computer fraud, the Act resulted from a long debate in the 1980s over failed prosecutions of hackers -- in one well-publicized case, two men hacked into a British Telecom computer leaving messages in the Duke of Edinburgh's private mailbox. *CHM

BIRTHS

1756 Jan Śniadecki (August 29, 1756– November 9, 1830) was a Polish mathematician, philosopher and astronomer at the turn of the 18th and 19th centuries.
Born in Żnin, Śniadecki studied at Kraków University and in Paris. He was rector of the Imperial University of Vilnius, a member of the Commission of National Education, and director of astronomical observatories at Kraków and Vilnius. He died at Jašiūnai Manor near Vilnius.
Śniadecki published many works, including his observations on recently discovered planetoids. His O rachunku losów (On the Calculation of Chance, 1817) was a pioneering work in probability. *Wik He is considered as the best Polish mathematician born in the 18th century.

1876 Charles F. Kettering (29 Aug 1876; 25 Nov 1958) was an American engineer whose 140 patents included the electric starter, car lighting and ignition systems. In his early career, with the National Cash Register Co., Dayton (1904-09), he created the first electric cash register with an electric motor that opened the drawer. When he co-founded the Dayton Engineering Laboratories Company (DELCO, with Edward A. Deeds) he invented the key-operated self-starting motor for the Cadillac (1912) and it spread to nearly all new cars by the 1920's. As vice president and director of research for General Motors Corp. (1920-47) he developed engines, quick-drying lacquer finishes, anti-knock fuels, and variable-speed transmissions.*TIS

1881 Ferdinand Springer born, The founder of an important publishing house,. Today Springer-Verlag is one of the most important publishers of advanced work on mathematics. *VFR

1904 Leonard Roth (29 August 1904 Edmonton, London, England – 28 November 1968 Pittsburgh, Pennsylvania) British Mathematician who worked primarily in Algebraic Geometry. *SAU

DEATHS

1873 Hermann Hankel (14 February 1839 - 29 August 1873) He studied and worked with, among others, Möbius, Riemann, Weierstrass and Kronecker. His 1867 exposition on complex numbers and quaternions is particularly memorable. For example, Fischbein notes that he solved the problem of products of negative numbers by proving the following theorem: "The only multiplication in R which may be considered as an extension of the usual multiplication in R+ by respecting the law of distributivity to the left and the right is that which conforms to the rule of signs." *Wik

1930 James Bolam (1839 in Newcastle, England - 29 Aug 1930 in St Helen's, Drumchapel, Dumbartonshire, Scotland) was educated at Newcastle. He became head of the Government Navigation School (later the Leith Nautical College). He was a founder member of the EMS and became an honorary member in 1923. *SAU

1937 Otto Ludwig Hölder (December 22, 1859 – August 29, 1937) worked on the convergence of Fourier series and in 1884 he discovered the inequality now named after him. He became interested in group theory through Kronecker and Klein and proved the uniqueness of the factor groups in a composition series. *SAU

1967 Charles Brace Darrow (10 Aug 1889, 29 Aug 1967) was an American inventor who designed the board game Monopoly. He had invented the game on 7 Mar 1933, though it was preceded by other real-estate board games. On 31 Dec 1935, a patent was issued for the game of Monopoly assigned to Parker Brothers, Inc., by Charles Darrow of Pennsylvania (No. 2,026,082). The patent titled it a "Board Game Apparatus" and described it as "intended primarily to provide a game of barter, thus involving trading and bargaining" in which "much of the interest in the game lies in trading and in striking shrewd bargains." Illustrations included with the patent showed not only the playing board and pieces, cards, and the scrip money. *TIS

The history of Monopoly can be traced back to 1903, when American anti-monopolist Lizzie Magie created a game that she hoped would explain the single-tax theory of Henry George. It was intended as an educational tool, to illustrate the negative aspects of concentrating land in private monopolies. She took out a patent in 1904. Her game, The Landlord's Game, was self-published, beginning in 1906.

According to an advertisement placed in The Christian Science Monitor, Charles Todd of Philadelphia recalled the day in 1932 when his childhood friend Esther Jones and her husband, Charles Darrow, came to their house for dinner. After the meal, the Todds introduced Darrow to The Landlord's Game, which they then played several times. The game was entirely new to Darrow, and he asked the Todds for a written set of the rules. After that night, Darrow went on to utilize this, and distribute the game himself as Monopoly.

The Parker Brothers bought the game's copyrights from Darrow. When the company learned Darrow was not the sole inventor of the game, it bought the rights to Magie's patent for \$500.

Parker Brothers began marketing the game on November 5, 1935 *Wik

 The Landlord Game *Wik

1975 Éamon de Valera (14 October 1882, 29 August 1975) was one of the dominant political figures in twentieth century Ireland, serving as head of government of the Irish Free State and head of government and head of state of Ireland. He also introduced the Constitution of Ireland.
De Valera was a leader of Ireland's struggle for independence from Britain in the Irish War of Independence and of the anti-Treaty forces in the ensuing Irish Civil War (1922–23). In 1926, he founded Fianna Fáil and was head of government from 1932–48, 1951–54 and 1957–59 and President of Ireland from 1959–73.
In his youth he had trained as a mathematician and taught mathematics prior to the Easter Rising. Throughout his life he maintained an interest in mathematics and returned to it with a passion in his later life. *Wik

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

## Monday 28 August 2023

### Not The End for the Happy Ending Couple

 *Wik

On 28 August 2005, Esther Klein and her husband George passed away within an hour of each other. An unusual event made even more interesting by a beautiful mathematical problem, that linked them together, and spawned the mathematical areas called Ramsey Theory, and Combinatorial Geometry.

In 1933 a group of mostly male students met regularly in Budapest to discuss mathematics. At one such meeting, one of (perhaps the only??) women present, Esther Klein, asked a simple geometric question: Is it possible to place five points on a plane so that no four of them form a convex quadrilateral? None of the student's present could answer her challenge; a fact made more impressive in that one of the students was Paul Erdos, one of the most prolific problem solvers in mathematics history.  Another student present was George Szekeres, another prolific mathematician working in combinatorial mathematics and a prominent player in the problem Esther submitted.

Esther then went on to illustrate her proof. Today the problem and it's generalization is regarded as one of the foundational works in the field of combinatorial geometry.

Within four years, Esther and George were married, and Paul Erdos dubbed the problem the Happy Ending Problem as he felt it was the start of their relationship.

The common proof used today for the problem is to divide it into simple cases. It is assumed that the points are in general position, that is, no three are collinear. Pick three of the points to form a triangle. If any point(s) is outside the triangle, then a convex quadrilateral can be drawn using four points. For the case when the two other points are inside the triangle, the segment containing these two points and one side of the triangle can be united into a convex quadrilateral. Also see the nice illustration at Theorem of the Day.

Erdos and George Szekeres generalized the problem to the theorem: For any positive integer N, any sufficiently large finite set of points in the plane in general position has a subset of N points that form the vertices of a convex polygon.
But HOW BIG was a "sufficiently large finite set of points" for a given convex n-gon? For a triangle, three points was all that was necessary. For a quadrilateral, Esther had shown that five points would suffice. Erdos and George predicted that for a pentagon, it would require nine points, but the complete proof was not published until 1970. Shortly after the death of George and Esther Szekeres, the solution for a hexagon was published in the ANZIAM Journal. The paper, Computer solution to the 17-point Erdös-Szekeres problem by George Szekeres(deceased)and
Lindsay Peters, showed that for a convex hull of six sides, the required number would be 17 points. (A challenging problem for students would be to create 16 points in general position so that no six formed a convex hexagon.)
Beyond that.... we just don't know. It must be a finite number, and we know from another Erdos-Sekeres proof that for an n-gon, the number of points is greater than or equal to 1 + 2(n-2).  That would mean that to guarantee a convex polygon, there must be at least nine points in the plain with no three co-linear.

Paul Erdos used to talk about "God's Book." A list of all the best solutions to every mathematical problem. Maybe they got a peak after they left this plane. And Happily, the generalized Happy Ending problem has not ended for us still here. Care to try for a convex heptagon.