On This Day in Math - August 16

 

Projective geometry is all geometry.

~Arthur Cayley

The 228th day of the year; 228 is the number of ways, up to rotation and reflection, of dissecting a regular 11-gon into 9 triangles.

If the sum of its digits,12, is subtracted from 228, you get a cube, 228 - 12 = 216 = 6^3.  If the product of its digits, 32, is subtracted, you get a square.  228 - 32 = 196 = 14^2

228 + 1,  822 + 1, and (228 + 822) + 1 are all primes. Is there another such year day?

228 in binary is written 11100100 notice that this is all four possible two digit binary combinations in descending order, 11, 10, 01, 00. (Just figured out that the equivalent (all possible three digit numbers concatenated in order) in base three is a little over 10³⁹)

228 is between twin primes
See More Math Facts for every Year Day, here

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EVENTS

1565  The Grand Duchess Christina of Lorraine was born Aug. 16, 1565. Christina was the grand-daughter of Catherine d’ Medici, and she re-cemented her ties to the family in 1589, when she married Ferdinando I de’ Medici of Florence in a lavish wedding.  Christina hired Galileo Galilei, then at Padua, to tutor her eldest son, Cosimo II, and when Christina’s husband died in 1609, Cosimo succeeded him as Grand Duke of Tuscany, and Christina stayed on at the court.   Galileo gave Cosimo the telescope with which he discovered the four moons of Jupiter in 1610, naming them the “Medicean stars” in his honor.  After Galileo joined the Medici court, he became well acquainted with the Duchess (who was actually a year younger than Galileo), and on several occasions she asked Galileo how the Copernican idea of a moving earth could be compatible with those passages of Scripture that discuss a fixed earth and a moving sun. In response, Galileo wrote, in 1615, what is usually called the Letter to the Grand Duchess Christina, in which he suggested that the language of the Bible was written to “accommodate” the understanding of the ordinary person and was not intended to be taken literally.  He further argued that the Bible was intended to indicate the road to salvation, and was not meant to provide instruction in natural philosophy.

This letter circulated in manuscript and was brought to the attention of Cardinal Bellarmine, the principal theological advisor to the Pope. Bellarmine ruled that accomodationism was acceptable when one could prove that the Bible had to be read some way other than literally, but first you needed proof, and Galileo had no proof that the earth moved. Therefore scriptural passages suggesting a fixed Earth should be read literally.   A committee then pronounced in 1616 that Copernicanism was heretical, and Copernicus’ book On the Revolutions (1543) was, for the first time, placed on the Index of Prohibited Books. Galileo’s trial was still 16 years away, but the stage had now been set, thanks to the Letter to Christina.  *Linda Hall Org

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1811   “Having to conduct my grandson through his course of mathematics, I have resumed the study

with great avidity. It was ever my favorite one. We have no theories there, no uncertainties

remain on the mind; all is demonstration and satisfaction.” So wrote Thomas Jefferson (1743–

1826) to Benjamin Rush. Taken from The Writings of Thomas Jefferson, edited by A. A. Lipscomb, vol. 13 (1903), p. 75, as quoted from Cajori, Mathematics in Liberal Education, p. 109,

which is a collection of interesting quotations on the value of mathematics.

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1825 A royal decree granted Neils Henrik Abel, then 23, sufficient funds for a year’s travel in France and Germany.  *VFR

 In 1815 Niels entered the cathedral school in Oslo, where his mathematical talent was recognized in 1817 with the arrival of a new mathematics teacher, Bernt Michael Holmboe, who introduced him to the classics in mathematical literature and proposed original problems for him to solve. 

Abel’s father died in 1820, leaving the family in straitened circumstances, but Holmboe contributed and raised funds that enabled Abel to enter the University of Christiania (Oslo) in 1821. Abel obtained a preliminary degree from the university in 1822 and continued his studies independently with further subsidies obtained by Holmboe.

Abel’s first papers, published in 1823, were on functional equations and integrals; he was the first person to formulate and solve an integral equation. His friends urged the Norwegian government to grant him a fellowship for study in Germany and France. In 1824, while waiting for a royal decree to be issued, he published at his own expense his proof of the impossibility of solving algebraically the general equation of the fifth degree, which he hoped would bring him recognition. He sent the pamphlet to Gauss, who dismissed it, failing to recognize that the famous problem had indeed been settled.

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In 1858, Queen Victoria sent the first official telegraph message across the Atlantic Ocean from London to the US. (Test messages had been exchanged for 10 days). Her message to President Buchanan, in Washington DC, began transmission at 10:50am and was completed at 4:30am the next day, taking nearly 18-hrs to reach Newfoundland. With 99 words, consisting of 509 letters, it averaged about 2-min per letter. The message was forwarded across Newfoundland by an overhead wire supported on poles; across Cabot Strait by submarine cable to Aspy Bay (Dingwall), Cape Breton; and by an overhead wire across eastern Canada and Maine, via Boston to New York. This earliest Transatlantic cable went dead within a month.

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1878 Hermite writes to Sylvester at Johns Hopkins concerned about his accepting a Math Chair in America to questioned the ability of the American people to contribute to research-level mathematics. Only three years later he would be reading the paper of Fabian Franklin, a young assistant mathematics instructor at Johns Hopkins, before the French Academy. The paper was on a short, purely graphic, proof of Euler's theorem on pentagonal numbers. *Karen Hunger Parshall, David E. Rowe; The Emergence of the American Mathematical Research Community, 1876-1900 Franklin would later wed Christine Ladd-Franklin (1882 ir 1884?).

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1890  The US Census Bureau announces the U.S. population of 62,622,250, determined for the first time by using an automated method, the Hollerith Census Machine. The Hollerith machine sorted returns by completing an electrical circuit wherever a hole existed in a punch card and could process almost 10 times the number of census data than a human clerk.

Census workers used the Punch Pantograph to enter data. Hollerith formed the Tabulating Machine Company in 1896. This company merged with two others in 1924 to become the International Business Machines company or IBM.  Hollerith died November 17, 1929.

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In 1898, the loop-de-loop Roller Coaster was patented by Edwin Prescott.*TIS The vertical loop is not a recent roller coaster innovation. Its origins can be traced back to the 1850s when centrifugal railways were built in France and Great Britain. In 1901 Prescott built the Loop-the-Loop at Coney Island. This ride used the modern teardrop-shaped loop and a steel structure, however more people wanted to watch the attraction, rather than ride. No more looping roller coasters were built until 1976 when Revolution opened at Six Flags Magic Mountain.*Wik

*Smithsonian

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1941 When Herbert Robbins saw the proof sheet of the title page of What is Mathematics? with only the name Richard Courant on it, his first reaction was “My god, the man’s a crook.” Realizing that a quiet meeting on their co-authorship of the book would be impossible, Robbins wrote Courant on this date that, while the custom might be different in Europe, in this country the junior author did receive credit. Courant backed down, and so today we know this lovely book as one by Courant and Robbins.

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1966 Stephen Smale, University of California, Berkeley, received the Fields Medal at the International Congress of Mathematicians in Moscow for his work on dynamical systems. Ten days later on the steps of Moscow University he will make a speech condemning American
military activity in Vietnam and Soviet military involvement in Hungary. *VFR

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1983 Poland issued a stamp celebrating the 50th anniversary of the Enigma Decoding Machine. VFR

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BIRTHS

1744 Pierre (-François-André) Méchain (16 Aug 1744; 20 Sep 1804). a French astronomer and hydrographer at the naval map archives in Paris recruited by Jean Delambre. He was a mathematical progidy. In 1790, they were chosen by the National Assembly to establish a decimal system of measurement based on the meter. Since this was defined to be one ten-millionth of the distance between the Earth's pole and the equator, Mechain led a survey of the meridian arc from Dunkirk, France, to Barcelona, Spain. Through his astronomical observations, Mechain discovered 11 comets and provided 26 additions to Messier's catalog. He calculated the orbits of the two comets he found in 1781. Mechain died of yellow fever while making further surveys for the meridian measurement. *TIS

 Méchain discovered a number of deep-sky nebulous objects, including M101 (the Pinwheel Galaxy; shown), M87 (the Owl Nebula), and M104 (the Sombrero Galaxy). 

*Wik

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1821 Arthur Cayley, (16 August 1821 – 26 January 1895) English mathematician who played a leading role in founding the modern British school of pure mathematics. He trained first as a lawyer, and from 1849, spent 14 years at the bar, during which time he maintained an interest in mathematics and published about 250 mathematical papers. In 1863, Cayley followed his passion and commenced a new career as professor of Pure Mathematics at Cambridge and during his tenure published 900 papers and notes covering nearly every aspect of modern mathematics. The legacy of his work in n-dimensional geometry was later applied in physics to the study of the space-time continuum. His work on matrices served as a foundation for quantum mechanics developed by Werner Heisenberg in 1925. *TIS

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1836 Marc-Antoine Parseval des Chênes (April 27, 1755 – August 16, 1836) was a French mathematician, most famous for what is now known as Parseval's theorem, which presaged the unitarity of the Fourier transform.
He was nominated to the French Academy of Sciences five times, from 1796 to 1828, but was never elected. His only mathematical publications were, apparently, five papers, published in 1806 as Mémoires présentés à l'Institut des Sciences, Lettres et Arts, par divers savants, et lus dans ses assemblées. Sciences mathématiques et physiques. (Savants étrangers.) This combined the following earlier monographs:

1. "Mémoire sur la résolution des équations aux différences partielles linéaires du second ordre," (May 5, 1798).
2. "Mémoire sur les séries et sur l'intégration complète d'une équation aux différences partielles linéaires du second ordre, à coefficients constants," (April 5, 1799).
3. "Intégration générale et complète des équations de la propagation du son, l'air étant considéré avec ses trois dimensions," (July 5, 1801).
4. "Intégration générale et complète de deux équations importantes dans la mécanique des fluides," (August 16, 1803).
5. "Méthode générale pour sommer, par le moyen des intégrales définies, la suite donnée par le théorème de M. Lagrange, au moyen de laquelle il trouve une valeur qui satisfait à une équation algébrique ou transcendante," (May 7, 1804).

It was in the second, 1799, memoir in which he stated, but did not prove (claiming it to be self-evident), the theorem that now bears his name. He further expanded upon it in his 1801 memoir, and used it to solve various differential equations. The theorem was first printed in 1800 as a part (p. 377) of Traité des différences et des séries by Lacroix.
*Wik

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1837 Joseph-Marie de Tilly,(16 Aug 1837 in Ypres, Belgium - 4 Aug 1906 in Munich, Germany) Belgian mathematician, born. In 1899 he was dismissed from his teaching post at the Ecole Militaire for unduly emphasizing the scientific education of future officers and using the notions of the infinitely small and the differential. *VFR

 In 1858, he became a teacher in mathematics at the regimental school. He began with studying geometry, particularly Euclid's fifth postulate and non-Euclidean geometry. He found similar results as Lobachevsky in 1860, but the Russian mathematician was already dead at that time. Tilly is more known for his work on non-Euclidean mechanics, as he was the one who invented it. He worked thus alone on this topic until a French mathematician, Jules Hoüel, showed interest in that field. Tilly also wrote on military science and history of mathematics. He died in München, Germany.

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1845 Gabriel Lippman (16 Aug 1845; 13 Jul 1921).French physicist, born Hollerich, Luxembourg, who received the Nobel Prize for Physics in 1908 for producing the first colour photographic plate. Lippmann was a giant of his day in classical physics research, especially in optics and electricity. He worked in Berlin with the famed Hermann von Helmholtz before settling in Paris to head (in 1886) the Sorbonne's Laboratories of Physical Research until his death. His inventions include an instrument for precisely measuring minute differences in electrical power and the "coleostat" for steady, long-exposure sky photography.*TIS\

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1884 Hugo Gernsback (August 16, 1884 – August 19, 1967), born Hugo Gernsbacher, was a Luxembourgish-American inventor, writer, editor, and magazine publisher, best known for publications including the first science fiction magazine. His contributions to the genre as publisher were so significant that, along with the novelists H. G. Wells and Jules Verne, he is sometimes called "The Father of Science Fiction". In his honor, annual awards presented at the World Science Fiction Convention are named the "Hugos" *Wik

Gernsback's second novel, Baron Münchausen's Scientific Adventures, was serialized in Amazing in 1928, with the opening installment taking the February cover

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1904 Wendell Meredith Stanley (16 August 1904 – 15 June 1971) was an American biochemist, virologist and Nobel laureate. Stanley was born in Ridgeville, Indiana, and earned a BS in Chemistry at Earlham College in Richmond, Indiana. He then studied at the University of Illinois, gaining an MS in science in 1927 followed by a Ph.D. in chemistry two years later. His later accomplishments include writing the book "Chemistry: A Beautiful Thing" and achieving his high stature as a Pulitzer Prize nominee.
Stanley was awarded the Nobel Prize in Chemistry for 1946. His other notable awards included the Rosenburger Medal, Alder Prize, Scott Award, and the AMA Scientific Achievement Award. He was also awarded honorary degrees by many universities both American and foreign, including Harvard, Yale, Princeton and the University of Paris. Most of the conclusions Stanley had presented in his Nobel-winning research were soon shown to be incorrect (in particular, that the crystals of mosaic virus he had isolated were pure protein, and assembled by autocatalysis)
Stanley married Marian Staples (1905-1984) in 1929 and had three daughters (Marjorie, Dorothy and Janet), and a son, (Wendell M. Junior). Stanley Hall at UC Berkeley (now Stanley Biosciences and Bioengineering Facility) and Stanley Hall at Earlham College are named in his honor. *Win

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1905 Marian Adam Rejewski (16 August 1905 – 13 February 1980) was a Polish mathematician and cryptologist who in 1932 solved the plugboard-equipped Enigma machine, the main cipher device used by Germany. The success of Rejewski and his colleagues Jerzy Różycki and Henryk Zygalski jump-started British reading of Enigma in World War II; the intelligence so gained, code-named "Ultra", contributed, perhaps decisively, to the defeat of Nazi Germany.
While studying mathematics at Poznań University, Rejewski had attended a secret cryptology course conducted by the Polish General Staff's Biuro Szyfrów (Cipher Bureau), which he joined full-time in 1932. The Bureau had achieved little success reading Enigma and in late 1932 set Rejewski to work on the problem. After only a few weeks, he deduced the secret internal wiring of the Enigma. Rejewski and his two mathematician colleagues then developed an assortment of techniques for the regular decryption of Enigma messages. Rejewski's contributions included devising the cryptologic "card catalog," derived using his "cyclometer," and the "cryptologic bomb."
Five weeks before the German invasion of Poland in 1939, Rejewski and his colleagues presented their results on Enigma decryption to French and British intelligence representatives. Shortly after the outbreak of war, the Polish cryptologists were evacuated to France, where they continued their work in collaboration with the British and French. They were again compelled to evacuate after the fall of France in June 1940, but within months returned to work undercover in Vichy France. After the country was fully occupied by Germany in November 1942, Rejewski and fellow mathematician Henryk Zygalski fled, via Spain, Portugal and Gibraltar, to Britain. There they worked at a Polish Army unit, solving low-level German ciphers. In 1946 Rejewski returned to his family in Poland and worked as an accountant, remaining silent about his cryptologic work until 1967. *Wik

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1907 Dura Kurepa (16 Aug 1907, 2 Nov 1993)The topics which Kurepa investigated are very varied but lie mostly within topology, set theory and number theory. He published over 200 papers but this number rises to over 700 items if we include books, articles and reviews. He was fascinated by the continuum hypothesis and the axiom of choice. Perhaps best known is his work on trees and partitions, especially Aronszajn and Suslin trees. His book The Theory of Sets written in Serbo-Croatian and published in 1951 illustrates his interests in that particular area. After introducing the fundamental concepts and elementary operations in Chapter 1, he looks at cardinal numbers in the second chapter, then partially ordered sets and ordinal numbers in the third. Chapter 4 is on topological and metric spaces, with the fifth and final chapter on limiting processes in analysis, measure theory, Borel and Souslin sets.
In number theory he made many contributions, but perhaps his most famous is his open problem on the left factorial function. In 1971 he published his definition of !n, the left factorial function, defined by
!n = 0! + 1! + 2! + 3! + ... + (n-1)!.
Kurepa conjectured that the greatest common divisor of !n and n! was 2 for all n > 1. There are many equivalent forms of the conjecture, but one of the most natural was given by Kurepa in the same 1971 paper, namely that !n is not divisible by n for any n > 2. If the left factorial conjecture is false we certainly know that it will fail for n > 1000000.*SAU
The same left factorial notation is more commonly used for the subfactorial used in derangements. Many mathematicians simply use "factorial sum" for Kurepa's !n. It is interesting that no one seems to have picked up on the use of an inverted exclamation point as suggested by G. Chrystal in his "Algebra, an Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges", (1889 (pg 25))

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1920 Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, celebrated for his invention of dynamic programming in 1953, and important contributions in other fields of mathematics. A Bellman equation, also known as a dynamic programming equation, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. Almost any problem which can be solved using optimal control theory can also be solved by analyzing the appropriate Bellman equation. The Bellman equation was first applied to engineering control theory and to other topics in applied mathematics, and subsequently became an important tool in economic theory. The "Curse of dimensionality", is a term coined by Bellman to describe the problem caused by the exponential increase in volume associated with adding extra dimensions to a (mathematical) space.*Wik

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1958  Anne Geneviève L'Huillier ( born 16 August 1958, ) is a French physicist, and professor of atomic physics at Lund University in Sweden.

She leads an attosecond (10^-18) physics group which studies the movements of electrons in real time, which is used to understand the chemical reactions on the atomic level. Her experimental and theoretical research are credited with laying the foundation for the field of attochemistry. In 2003 she and her group beat the world record for the shortest laser pulse, of 170 attoseconds.

L'Huillier became a member of the Royal Swedish Academy of Sciences in 2004. She has received various physics awards including the Wolf Prize in Physics in 2022 and the Nobel Prize in Physics in 2023.

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DEATHS

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1705 Jakob Bernoulli . (27 December 1654 – 16 August 1705) He was so fascinated with the way the logarithmic spiral reproduces itself in its involute, its evolute, and its caustics of reflection and refraction, that he requested it be engraved on his tombstone, together with the inscription Eadem mutata resurgo (Though changed, I will arise the same). *VFR (the spiral on his tombstone is not logarithmic, but Archimedian... perhaps he is spinning in his grave even yet.)
 He was one of the first to fully utilize differential calculus and introduced the term "integral" in integral calculus. Jacob Bernoulli's first important contributions were a pamphlet on the parallels of logic and algebra (1685), work on probability in 1685 and geometry in 1687. His geometry result gave a construction to divide any

triangle into four equal parts with two perpendicular lines.(a nice exercise to try) 

----A short digression about this quadrisection construction -----------

He gave a general algebraic solution which required finding a root of a polynomial of degree 8 and worked this out numerically for one scalene triangle.The question of whether Bernoulli’s polynomial equation of degree 8 has the needed root in all cases is not answered completely.

 Leonhard Euler wrote a 22 page paper in 1779 in which he gives a complete solution using trigonometry.

Euler states his solution in a theorem which we paraphrase.

Theorem 1. (Euler 1779) Given a scalene triangle ∆ABC with AB the side of middle length, there is a quadrisection XP and Y Q intersecting in a point O in the interior of the triangle so that X and Y lie on side AB and triangle XOY is one of the 4 areas of the quadrisection. The other areas of the quadrisection are quadrangles.  Euler does not claim that the triangular portion of a quadrisection must lie on the side of middle length. Also, he does not appear to discuss whether there is more than one quadrisection of a triangle, except to note that an equilateral triangle has 3 quadrisections. In fact, we will see there are lots of triangles with quadrisections where the triangular portion lies on the shortest side, but no triangles having a quadrisection with the triangular portion on the longest side.

Revisiting the quadrisection problem of Jacob Bernoulli. Carl Eberhart 

------------------------End Digression------------------

By 1689 he had published important work on infinite series and published his law of large numbers in probability theory. He published five treatises on infinite series (1682 - 1704). He was the first of the Bernoulli family of mathematicians. *TIS
Jacob Bernoulli’s Ars Conjectandi from 1713 is the first major book on the theory of probability and statistics. It is because of its 300 years anniversary that 2013 was named the international year of Statistics. The exhibited copy is in fact a first edition! It is showing the proof of the law of large numbers, one of the results for which it is famous. * University of Copenhagen Dept of Math Sciences

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1899 Robert Wilhelm Eberhard Bunsen (30 March 1811[?) – 16 August 1899) was a German chemist. He investigated emission spectra of heated elements, and discovered caesium (in 1860) and rubidium (in 1861) with the physicist Gustav Kirchhoff. The Bunsen–Kirchhoff Award for spectroscopy is named after Bunsen and Kirchhoff.

Bunsen also developed several gas-analytical methods, was a pioneer in photochemistry, and did early work in the field of organic arsenic chemistry. With his laboratory assistant Peter Desaga, he developed the Bunsen burner, an improvement on the laboratory burners then in use. *Wik

His Bunsen burner was created for use in flame tests of various metals and salts because its nonluminous flame did not interfere with the colored flame given off by the test material.*TiS

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1920 Sir Joseph Norman Lockyer (17 May 1836, 16 Aug 1920) British astronomer who in 1868 discovered and named the element helium that he found in the Sun's atmosphere before it had been detected on Earth. He also applied the name chromosphere for the sun's outer layer. Lockyer discovered, together with Pierre J. Janssen, the prominences (red flames) that surround the solar disk. He was also interested in the classification of stellar spectra and developed the meteoric hypothesis of stellar evolution. His works include the books Contributions to Solar Physics (1873), The Sun's Place in Nature (1897) and Inorganic Evolution (1900). *TIS

A solar prominence (also known as a filament when viewed against the solar disk) is a large, bright feature extending outward from the Sun's surface. Prominences are anchored to the Sun's surface in the photosphere, and extend outwards into the Sun's hot outer atmosphere, called the corona.

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1995 Thomas Brooke Benjamin​, FRS (15 April 1929 – 16 August 1995) was an English mathematical physicist and mathematician, best known for his work in mathematical analysis and fluid mechanics, especially in applications of nonlinear differential equations. *Wik

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2013 David Rees FRS (29 May 1918 – 16 August 2013) was a British professor of pure mathematics at the University of Exeter, having been head of the Mathematics / Mathematical Sciences Department at Exeter from 1958 to 1983. During the Second World War, Rees was active on Enigma research in Hut 6 at Bletchley Park.

Rees won a scholarship to Sidney Sussex College, Cambridge, supervised by Gordon Welchman and graduating in summer 1939. On completion of his education, he initially worked on semigroup theory; the Rees factor semigroup is named after him. He also characterised completely simple and completely 0-simple semigroups, in what is nowadays known as Rees's theorem. The matrix-based semigroups used in this characterisation are called Rees matrix semigroups.

Later in 1939, Welchman drafted Rees into Hut 6, Bletchley Park, for the war effort. He was credited with the first decode using the Herivel tip. He was subsequently seconded to the Enigma Research Section, where the Abwehr Enigma was broken, and later to the Newmanry, where the Colossus computer was built.

After the war, Rees was appointed an assistant lecturer at Manchester University in 1945 and a full lecturer at University of Cambridge in 1948. In 1949, he was a Fellow of Downing College.

At the behest of Douglas Northcott he switched his research focus to commutative algebra. In 1954, in a joint paper with Northcott, Rees introduced the Northcott–Rees theory of reductions and integral closures, which has subsequently been influential in commutative algebra. In 1956 he introduced the Rees decomposition of a commutative algebra.

In 1958, Rees and his family moved to Exeter, where he had been appointed to the Chair of Pure Mathematics. In 1959, he was awarded a DSc by the University of Cambridge.

According to Craig Steven Wright, Rees was the third part of the Satoshi team that created Bitcoin.*Wik

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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

Fabian Franklin's Beautiful Proof of the Pentagonal Theorem

    

*Wik

On Aug 16, 1878 Charles Hermite wrote to J J Sylvester at Johns Hopkins concerned about his accepting a Math Chair in America and questioning the ability of the American people to contribute to research-level mathematics. Only three years later he would be reading the paper of Fabian Franklin, a young assistant mathematics instructor at Johns Hopkins, before the French Academy. The paper was on a short, purely graphic, proof of Euler's theorem on pentagonal numbers. Hans Rademacher called this proof “the first major achievement of American mathematics.”

Some background for students: Pentagonal numbers are named for the ways of arranging dots into pentagons, much like the square numbers or triangular numbers. The true or pure pentagonal numbers are 1, 5, 12, 22, 35, 51, 70, 92,...

*Wik

You can get them by using the formula \(  \frac{3n^2-n}{2} \) with n a positive integer.
But for what we are doing today, we need to also include the generalized pentagonal numbers. They are obtained from the formula given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4..., producing the sequence 0, 1, 2, 5, 7, 12, 15, 22, 26, 35,...

One of the amazing things about this sequence is that it shows up in relation to finding the sum of the divisors of n, and finding the number of partitions of n. As Euler used it for the Pentagonal Number theorem, it was written out as \( 1 − n − n^2 + n^5 + n^7 − n^{12} − n^{15} + n^{22} + n^{26} − n^{35} − etc\) Notice all these numbers are the same as the ones in the generalized pentagonal numbers, hence, the pentagonal number theorem.

All that is beautiful math, but today I focus on a detail of the theorem that led to the graphic proof. What the terms of the Pentagonal Number theorem really say, and a question about them. If we look, for instance, in the partitions of five, they are {5},{4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1,1}, and {1,1,1,1,1} These can be divided into two sets, can you figure out how some are different than the others? Look at the first three. Now look at the last four. Each of the last four have repeats of one or more numbers. The first three are all distinct. Of the three with distinct digits, two of them have an even number of integers in their composition, {4,1} and {3,2}. The other, {5}, has an odd number of integers. And if you look at the exponent of X⁵ in the theorem, you see that it is positive, and that's what the theorem really says. If you look at all the partitions of any number, this polynomial gives you the the number of even distinct partitions (an even number of integers in it) minus the number of odd distinct partitions. So 5 has one more even than odd, and 7 does as well, but 12 has one more odd than even. but what first aroused Franklin's curiosity, was why there were so many missing exponents, why there were so many like X³ and X⁴ and X⁶. These would all have an equal number of odd and even partitions, hence the terms with zero for a coefficient which simply did not appear. what would explain this?

Franklin's insight was that for numbers like 6, the numbers could be matched up into odd and even pairs that offset each other in the count. For example, the distinct partitions of 6 are {6}, {5,1}, {4,2}, {3,2,1} the other seven partitions of 6 all have a repeated value. In his plan, the first two were matched together, and the last two were converted to each other, and he even had a graphic plan to show it always would work.

Here are two partitions of the number 33. The first is a partition into {9,8,7,5,4} The right diagonal which has 3 dots in it, and the bottom row which has 4 dots in it are the key. Since 3 is less than four (and four then, is the smallest number in the partition, we can move the 3 dots in the diagonal to make a new row in the bottom, and reduce the top three rows by one. So the matching partition is {8,7,6,5,4,3} at right. Note that we have changed the odd permutation into an even one. And we can go back by moving the three on the bottom row in the right partition to return to the left. These two always match. But he realized that there might be a situation in which this didn't work. For instance if the number of dots in the diagonal and the number of dots in the bottom row are the same (they share a corner dot) then you couldn't move either one. This can only happen if the bottom row is equal to the diagonal, or if the diagonal is one more than the bottom row. Here are examples of numbers that can't be matched to another. The top row is made up of numbers that have them equal. If you try to shift either one to make and odd partition even, or vice versa it just won't work. And the bottom sets show numbers that can be arranged with a partition that has the lowest row one more than the diagonal. They won't work either.

Now look at these numbers, count the dots, what do you notice. These are all the numbers in the Pentagonal Theorem Polynomial. And try as you may, you can't find another partition of any of these numbers that can't be transformed from odd to even or even to odd. If this partition has an odd number of rows, then the coefficient of that power in the polynomial is positive (5 in top row and 7 in bottom row are both positive coefficients since there is one odd partition that can't be matched to this even partition. The distinct partitions of 7 are {7}, {6,1}. {5,2}, {4,3} and {4,2,1} . Draw the diagrams, 7 can be transformed to {6,1} by dropping the last dot (a diagonal of one) to be a second row. {5,2} and be transformed in the same way to make {4,2,1} but the {4,3} has no match, so 3 evens, 2 odds makes the coefficient one.

I had never seen this before until I read a paper "The Pentagonal Number Theorem and All That" by Dick Koch from 2016. Here is the link, with thanks to John Golden@mathhombre

On This Day in Math - August 15

 

voyager exits solar system 1990, see Events(2006)

The 227th day of the year; 227 is a prime number, but it can also be written as the sum of the sum and the product of the first four primes: (2 + 3 + 5 + 7)+(2 x 3 x 5 x 7) = 227. In a similar way, the first two primes work (2+3)+(2x3)=11 is prime. Can you find another? (Ben Vitale has found all the cases under 1000 for which p = (a + b + c + … ) + (a * b * c …) He even found another way to express 227. His blog also has lots of other number curiosities, so give it a look. Much fun.

The harmonic sequence, or sum of the reciprocals of the integers grows to infinity, but slowly. It takes the 227th term (1/227) to finally push it over the value 6.(And don't even think about trying to get to seven!)

A beauty about six primes, 227 + 251 + 257 = 233 + 239 + 263, and if you square each one, 227^2 + 251^2 + 257^2 = 233^2 + 239^2 + 263^2 *Prime Curios.

227 is also the largest odd day number of the year which can NOT be expressed as a prime added to twice a square. There are three others you might find, and three others larger than 366. Observe that these are all seven prime.[OEIS gives ten numbers that include 1, 5779, and 5993. These last two are composite. and that seems to be ALL of them that exist. if there are more, they are larger than 10^13.)
This problem is based on an original conjecture by C Goldbach that all ODD COMPOSITE numbers could be written as twice a square plus a prime. The last two show he was wrong.

227 is the 7^2 prime number *Prime Curios

The harmonic sequence, or sum of the reciprocals of the integers grows to infinity, but slowly. It takes the 227th term (1/227) to finally push it over the value 6.(And don't even think about trying to get to seven!)

A beauty about six primes, 227 + 251 + 257 = 233 + 239 + 263, and if you square each one, 227^2 + 251^2 + 257^2 = 233^2 + 239^2 + 263^2 *Prime Curios.

227 is a palindrome in base eight (343)

The number (7 * 10^227+71)/3 Forms a prime with the digit 2 followed by 225 digits of 3, then ending in 57
Students might try a few starting with 3, ,4, etc.

22/7 is a common approximation for Pi in middle school.

There are 227 composite days in a year. *Prime Curios

See More Math Facts for every Year Day here.

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EVENTS

310 BC "Agathocles, who was already at the point of being overtaken and surrounded, gained unhoped for safety as night closed in. On the next day there occurred such an eclipse of the Sun that utter darkness set in and the stars were seen everywhere; wherefore Agathocles' men, believing that the prodigy portended misfortune for them, fell into even greater anxiety about the future. After they had sailed for six days and the same number of nights, just as day was breaking, the fleet of the Carthaginians was unexpectedly seen far away." From: Diodorus Siculus (Greek historian, 1st century BC), Library of History. Agathocles was a tyrant who had made his escape, with a fleet of sixty ships, from a blockade at Syracuse harbor by the Carthaginians. Quoted in Historical Eclipses and Earth's Rotation, by F Richard Stephenson, Cambridge University Press, 1997,

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1665 Robert Hooke writes to Boyle in Oxford about his newly devised reflecting quadrant (also called octant), "My quadrant does to admiration for taking angles, so that thereby we are able from hence to tell the true distance between (St.) Paul's and any other church steeple in the city.... within the quantity of twelve foot." *Lisa Jardine, Ingenious Pursuits, pg 152

Two men independently developed the octant around 1730: John Hadley (1682–1744), an English mathematician, and Thomas Godfrey (1704–1749), a glazier in Philadelphia. While both have a legitimate and equal claim to the invention, Hadley generally gets the greater share of the credit. This reflects the central role that London and the Royal Society played in the history of scientific instruments in the eighteenth century.

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1768 Lagrange, in a letter to D’Alembert, expressed his difficulty in solving the problem: Given a nonsquare positive integer n, to find a square integer x² such that nx² +1 shall also be a square. *VFR
In the same letter, he showed that x^2/3 could be expanded in a trigonometric series. D'Alembert had often used the function as an example that could not be so expanded. *Mathematical thought from ancient to modern times, Volume 2 , Morris Kline

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1771 Benjamin Franklin writes to John Canton to share the news of Priestley's discovery that, unlike animals, a plant seemed to survive after months. He would later be inspired to place a life animal under the glass with the plant and realize that the animal survived longer. *Steven Johnson, The Invention of Air

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1951 The Soviet Union issued a postage stamp with a portrait of Sofya Kovalevskaya. *VFR

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1994  Microsoft Corp. decided to work to incorporate an Internet browser into its upcoming Windows 95 operating system in an effort to catch up to the Internet bandwagon it had missed. On August 15, Windows 95 programmer Benjamin Slivka sent an e-mail to his coworkers suggesting a World Wide Web browser as a feature for Windows 95. Microsoft has faced legal challenges for the way it bundled the result of the project - Internet Explorer - with Windows software. *CHS 

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2006 Voyager 1, the most distant man-made object, reached 100 astronomical units from the sun - meaning 100 times more distant from the sun than is Earth - about 15,000 million km (9,300 million miles) from the sun. At such great distance, the sun is a mere point of light, so solar energy is not an option, but having a nuclear power source, Voyager 1 continues to beam back information. The spacecraft, launched nearly 30 years earlier, on 5 Sep 1977, had flown beyond the outer planets and reached the heliosheath, the outer edge of our solar system, where the sun's influence wanes. Voyager 1 continues traveling at a speed of about one million miles per day and could cross into interstellar space before 10 years later.

***Voyager 1 is back in contact with mission control after being off-line for five months.

On 14 November 2023, the spacecraft suddenly began transmitting unintelligible data instead of its usual engineering and science report.  The spacecraft still appeared to be receiving commands and operating normally, allowing the team to diagnose the problem lay in Voyager 1’s software and develop a fix.

On 20 April, the team received the first understandable message back from the spacecraft in five months.

Even once all its scientific systems are shut down, Voyager will continue transmitting a locator signal back to Earth, which will remain in range of the Deep Space Network until 2036. At that point, the Voyagers will sail beyond our sight, but will continue to travel ever outward into the depths of the Universe.  BBC Sky at Night

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BIRTHS

1720 Jean-Baptiste Le Roy (15 August 1720;Paris, France - 21 January 1800, Paris) Son of the renowned clockmaker Julien Le Roy, Jean-Baptiste Le Roy was one of four brothers to achieve scientific prominence in Enlightenment France; the others were Charles Le Roy (medicine and chemistry), Julien-David Le Roy (architecture), and Pierre Le Roy(chronometry). Elected to the Académie Royale des Sciences in 1751 as adjoint géomètre, Le Roy played an active role in technical as well as administrative aspects of French science for the next half-century. He was elected pensionnaire mécanicies in 1770 and director of the Academy for 1773 and 1778, and became both a fellow of the Royal Society and a member of the American Philosophical Society in 1773.
Le Roy’s major field of inquiry was electricity, a subject on which European opinion was much divided at mid-century. The most prominent controversy engaged the proponents of the Abbé Nollet’s doctrine of two distinct streams of electric fluids (outflowing and inflowing) and the partisans of Benjamin Franklin’s concept of a single electric fluid. This debate intensified in France in 1753 with an attack on Franklin’s views by Nollet. Le Roy, later a friend and correspondent of Franklin, defended his single-fluid theory and offered considerable experimental evidence in support thereof. He played an important role in the dissemination of Franklin’s ideas, stressing particularly their practical applications, and published many memoirs on electrical machines and theory in the annual Histoires and Mémoires of the Academy and in the Journal de Physique.
A regular contributor to the Encyclopédie, Le Roy wrote articles dealing with scientific instruments. The most important of these included comprehensive treatments of “Horlogerie,” “Télescope,” and “Électrométre” (in which Le Roy claimed priority for the invention of the electrometer). He also promoted the use of lightning rods in France, urged that the Academy support technical education, and was active in hospital and prison reform. After the Revolutionary suppression of royal academies, Le Roy was appointed to the first class of the Institut National (section de mécanique) at its formation in 1795. *Encycopedia.com

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1795 Émile Léger (Born: 15 Aug 1795 in Lagrange-aux-Bois, France; Died: 15 Dec 1838 in Paris, France)Léger only published four mathematical papers but one contains possibly the first mention of what today is a well known fact about the Euclidean algorithm,

Léger deeply explored the connection between the Euclidean algorithm and the expansion of real numbers into continued fractions. Since the Euclidean algorithm essentially underlies the continued fraction expansion of a rational number a/b  he was interested in:

The number of steps the algorithm takes to terminate.

How the sequence of remainders corresponds to the partial quotients in the continued fraction.

The structure and periodicity of the continued fraction in the case of quadratic irrationals, and how this reflects on the Euclidean algorithm's steps. PB notes

Émile Léger appears to have been the first (or second, if the work of de Lagny ... is counted) to recognise that the worst case of the Euclidean algorithm occurs when the inputs are consecutive Fibonacci numbers.  *SAU

The Euclidean algorithm is often taught to middle school students as a way to find the greatest common divisor, GCD, of two numbers.

 

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1863 Aleksei N. Krylov, (15 Aug 1863 in Visyaga, Simbirskoy [now Ulyanovskaya], Russia - 26 Oct 1945 in Leningrad, USSR [now St Petersburg, Russia]) noted for mathematics, mechanics and engineering. *VFR Krylov made many mathematical advances in his applications of mathematics to shipbuilding. In hydrodynamics, among many advances, he made significant contributions to the theory of ships moving in shallow water. In 1904 he constructed a mechanical integrator to solve ordinary differential equations, being the first in Russia to make such an instrument. He improved Fourier's method for solving boundary value problems in a 1905 paper and gave many applications. *SAU

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1865 Hantaro Nagaoka (15 Aug 1865; 11 Dec 1950) Japanese physicist who was influential in advancing physics in Japan in the early twentieth century. In 1904, he published his Saturnian model of the atom, inspired by the rings around the planet Saturn. He placed discrete, negatively charged electrons of the same tiny mass, spaced in a ring revolving around a central huge positive spherical mass at its centre. Considering the electrostatic forces, he made a mathematical analogy to Maxwell's model of the stability of the motion of Saturn's rings in a huge central gravitational field. However, Nagaoka's theory failed in other ways, and he sidelined it in 1908. *TIS

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1892 Louis Victor Pierre Raymond duc de Broglie (15 Aug 1892 -19 Mar 1987) was a French physicist best known for his research on quantum theory and for his discovery of the wave nature of electrons. De Broglie was of the French aristocracy - hence the title "duc" (Prince). In 1923, as part of his Ph.D. thesis, he argued that since light could be seen to behave under some conditions as particles (photoelectric effect) and other times as waves (diffraction), we should consider that matter has the same ambiguity of possessing both particle and wave properties. For this, he was awarded the 1929 Nobel Prize for Physics. *TIS 

As a long time teacher of mathematics, my favorite quote from de Broglie is:

There is no reason why the history and philosophy of science should not be taught in such a way as to bring home to all pupils the grandeur of science and the scope of its discoveries.

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1893 Leslie John Comrie FRS (15 August 1893 – 11 December 1950) was an astronomer and a pioneer in mechanical computation.

During World War I, despite severe deafness, he saw action in France with the New Zealand Expeditionary Force, and lost his left leg in February 1918 to a British shell. While convalescing, he started using a mechanical calculator and went on to modify commercial calculators for specific projects.

Having joined while in school in New Zealand, Comrie was eventually the first director (1920–1922) of the Computing Section of the British Astronomical Association. In 1923 he received a PhD from St John's College of the University of Cambridge. He travelled to the United States to teach at Swarthmore College and Northwestern University in 1924, where he also pioneered the teaching of numerical analysis. He returned to England to join the HM Nautical Almanac Office at the Royal Greenwich Observatory, where he became deputy superintendent in 1926.

His article On the Construction of Tables by Interpolation was published in April 1928, and described the use of punched card equipment for interpolating tables of data, in contrast to the more inefficient and error-prone methods involving mechanical devices like the pinwheel calculators under the Brunsviga brand name. In the same year, he became the first person to use punched card equipment for scientific calculations, by using Fourier synthesis to compute the principal terms in the motion of the Moon between 1935 and 2000 (improving upon the predictions of Ernest William Brown). Wallace J. Eckert, an American student of Brown at Columbia University, would later use IBM's vast computational resources to refine the predictions even further.

He was promoted to Superintendent of the Nautical Almanac Office in 1930. However, his unconventional use of machines for calculation caused tensions with his superiors, and he was suspended in August 1936.

Comrie founded in 1937 the world's first private company for scientific computing, incorporated as Scientific Computing Service, Limited. During World War II, he headed a team of 30 scientists to computerize war work, such as the creation of bombing tables for the Allies of World War II. He later used this technology to computerize British football pools.

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1896 Gerty Theresa Cori (née Radnitz; August 15, 1896 – October 26, 1957) was a Bohemian-Austrian and American biochemist who in 1947 was the third woman to win a Nobel Prize in science, and the first woman to be awarded the Nobel Prize in Physiology or Medicine, for her role in the "discovery of the course of the catalytic conversion of glycogen".

In 1957, Gerty Cori died after a ten-year struggle with myelosclerosis. She remained active in the research laboratory until the end of her life. She received recognition for her achievements through multiple awards and honors. *Wik

She met her husband, Carl Ferdinand Cori, while attending medical school in Prague (married 1920). They decided upon careers in medical research, took positions in America in 1922, and became U.S. citizens in 1928. They both shared the 1947 Nobel Prize in Physiology or Medicine (with Bernardo Houssay). TiS

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1897 Sylvia Winifred Skan (15 August 1897 – 10 June 1972) was an English applied mathematician. She is known for her work on aerodynamics, and in particular for the Falkner–Skan boundary layer in the fluid mechanics of airflow past a wedge-shaped obstacle, which she wrote about with V. M. Falkner in 1930, and for the associated Falkner–Skan equation.

Skan was born in Bickenhill on 15 August 1897, the oldest of five children of botanist Sidney Alfred Skan [es] and of his wife Jane Alkins. She does not appear to have earned a university degree. By 1923 she was working for the Aerodynamics Department of the National Physical Laboratory, where she carried out the entirety of her career.

As well as co-authored research papers, 17 of which listed her as first author, her works included translations of research papers from French, German and Russian into English, and a two-volume single-authored book, Handbook for Computers (1954), describing the mathematics needed for human computers. *Wik

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1905 Hermann Alexander Brück (15 August 1905 in Berlin, Germany – 4 March 2000 in Edinburgh, Scotland) was a German-born astronomer who spent the great portion of his career in the United Kingdom.

Upon graduation from Munich, Brück followed his friend Albrecht Unsöld to the Potsdam Astrophysical Observatory; Unsöld had earned his doctorate the year before, also under Sommerfeld. While there, he participated in the physics colloquium at the Humboldt University of Berlin with the physicists Max von Laue and Albert Einstein and the astronomer Walter Grotrian. With growing difficulties under National Socialism, Brück left Germany in 1936 to take a temporary research assistantship at the Vatican Observatory. In 1937 he moved to the University of Cambridge to join the circle of the modern astrophysicists around Arthur Eddington. In time, Brück became Assistant Director of the Observatories and John Couch Adams, specializing in solar spectroscopy. He taught a course in classical astronomy and started the student astronomical society, which fostered the careers of many astronomers.
In 1947, at the invitation of Éamon de Valera, Brück moved to Dublin to direct the Dunsink Observatory, which was part of the Dublin Institute for Advanced Studies, where he associated with Erwin Schrödinger. In 1950, the Observatory, along with the Royal Irish Academy, hosted the first meeting of the Royal Astronomical Society. In 1955, the International Astronomical Union held their triennial Assembly in Dublin. At this gathering, the Observatory demonstrated photoelectric equipment for photometry, which had been developed by M. J. Smyth, who had been Brück’s student in Cambridge. Also displayed was the UV solar spectroscopy which extended the Utrecht Atlas and formed part of the revised Rowland tables of the Solar spectrum; Brück’s wife, Dr. Mary Brück (née Conway), was a leading figure in this work.
In 1957, Brück moved to the University of Edinburgh. With his vision and drive, he transformed the Royal Observatory into an internationally-ranked center of research. He put together a team of astronomers and engineers headed initially by P. B. Fellgett and later by V. C. Reddish *Wik

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1918 Jean Brossel ( 15 August 1918 in Périgueux , France - 4 February 2003 in France) developed with Alfred Kastler the technique of optical pumping at origin of lasers. *Arjen Dijksman ‏@materion

In his hometown of Périgueux a square is named after him.

In 1960 Brossel won the Holweck Prize and in 1977 he was elected a member of l'Académie des sciences, whose Prix Ampère he received in 1974. In 1984 he received the gold medal of CNRS.  *Wik

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DEATHS

1758 Pierre Bouguer died (16 February 1698, Croisic – 15 August 1758, Paris). In 1727 he won the prize competition of the Acad´emie Royal des Sciences on the masting of ships. In this competition Euler only received the “accessit.” *vfr

French physicist whose work founded photometry, the measurement of light intensity. He was a child prodigy, a professor at age 15, following his father, Jean Bouguer, in hydrography - the study of bodies of water, both salt and fresh. He participated on the expedition to Peru (1735-44) to measure an arc of the meridian near the equator. In 1729, he invented a photometer to compare the intensity of two light sources illuminating separate halves of translucent paper. The eye itself, he determined, could not be used as a meter, but could establish the equality of brightness of adjacent surfaces. He determined the sun was 300 times brighter than the moon. Bouguer's law gives the attenuation of a beam of light by an optically homogeneous (transparent) medium.*TIS

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 1798 Edward Waring (ca. 1736 – 15 August 1798) was an English mathematician who gave many results about decomposing numbers into sums of powers and sums of primes.*SAU He entered Magdalene College, Cambridge as a sizar and became Senior wrangler in 1757. He was elected a Fellow of Magdalene and in 1760 Lucasian Professor of Mathematics, holding the chair until his death. He made the assertion known as Waring's Problem without proof in his writings Meditationes Algebraicae. Waring was elected a Fellow of the Royal Society in 1763 and awarded the Copley Medal in 1784.

In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers (for example, every number is the sum of at most 4 squares, or 9 cubes, or 19 fourth powers, etc.). The affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. *Wik

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 1912 Carlo Miranda (15 August 1912 – 28 May 1982) was an Italian mathematician, working on mathematical analysis, theory of elliptic partial differential equations and complex analysis: he is known for giving the first proof of the Poincaré–Miranda theorem for Miranda's theorem in complex analysis, and for writing an influential monograph in the theory of elliptic partial differential equations.

Tthe Poincaré–Miranda theorem is a generalization of intermediate value theorem, from a single function in a single dimension, to n functions in n dimensions. The theorem is named after Henri Poincaré — who conjectured it in 1883 — and Carlo Miranda — who in 1940 showed that it is equivalent to the Brouwer fixed-point theorem. *Wik

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1927 Bertram Borden Boltwood (27 Jul 1870, 15 Aug 1927). was an American chemist and physicist whose work on the radioactive decay of uranium and thorium was important in the development of the theory of isotopes. Boltwood studied the "radioactive series" whereby radioactive elements sequentially decay into other isotopes or elements. Since lead was always present in such ores, he concluded (1905) that lead must be the stable end product from their radioactive decay. Each decay proceeds at a characteristic rate. In 1907, he proposed that the ratio of original radioactive material to its decay products measured how long the process had been taking place. Thus the ore in the earth's crust could be dated, and give the age of the earth as 2.2 billion years.*TIS

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1953 Ludwig Prandtl (4 Feb 1875, 15 Aug 1953) German physicist who is remembered for his studies of both aerodynamics and hydrodynamics. He established the existence of the boundary layer adjoining the surface of a solid over which a fluid flows. The design of an efficient shape, weight, and mass for ships and aircraft owes much to his work, for which he is considered to be the father of aerodynamics. His made major studies on the effects of streamlining and the properties of aircraft wings. He made improvements to such constructions as wind tunnels. The Prandtl number is a dimensionless group used in the study of convection. The von Karman-Prandtl equation describes the logarithmic variation of water velocity within a channel from zero flow at the stream bed to a maximum velocity at the water surface.*TIS

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1978 Viggo Brun (13 October 1885, Lier – 15 August 1978, Drøbak) was a Norwegian mathematician.
He studied at the University of Oslo and began research at the University of Göttingen in 1910. In 1923, Brun became a professor at the Technical University in Trondheim and in 1946 a professor at the University of Oslo. He retired in 1955 at the age of 70.
In 1915, he introduced a new method, based on Legendre's version of the sieve of Eratosthenes, now known as the Brun sieve, which addresses additive problems such as Goldbach's conjecture and the twin prime conjecture. He used it to prove that there exist infinitely many integers n such that n and n+2 have at most nine prime factors (9-almost primes); and that all large even integers are the sum of two 9 (or smaller)-almost primes.
In 1919 Brun proved that the sum of the reciprocals of the twin primes converges to Brun’s constant:
1⁄3 + 1⁄5 + 1⁄5 + 1⁄7 + 1⁄11 + 1⁄13 + 1⁄17 + 1⁄19 + . . . = 1.9021605 . . .by contrast, the sum of the reciprocals of all primes is divergent. He developed a multi-dimensional continued fraction algorithm in 1919/20 and applied this to problems in musical theory.
He also served as praeses of the Royal Norwegian Society of Sciences and Letters in 1946.
It was in 1994, while he was trying to calculate Brun’s constant,
that Thomas R. Nicely discovered a famous flaw in the Intel Pentium
microprocessor. The Pentium chip occasionally gave wrong answers
to a floating-point (decimal) division calculations due to errors in five
entries in a lookup table on the chip. Intel spent millions of dollars
replacing the faulty chips.
More recently, Nicely has calculated that the value of Brun’s constant
1s 1.902160582582 _ 0.000000001620.
*Wik

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2002 Heinz Bauer (31 January 1928 – 15 August 2002) was a German mathematician.
Bauer studied at the University of Erlangen-Nuremberg and received his PhD there in 1953 under the supervision of Otto Haupt and finished his habilitation in 1956, both for work with Otto Haupt. After a short time from 1961 to 1965 as professor at the University of Hamburg he stayed his whole career at the University of Erlangen-Nuremberg. His research focus was the Potential theory, Probability theory and Functional analysis
Bauer received the Chauvenet Prize in 1980 and became a member of the German Academy of Sciences Leopoldina in 1986. Bauer died in Erlangen. *Wik

One particularly notable aspect of Bauer's work is his abstract formulation of potential theory using convex analysis and topology, which helped to unify and generalize classical results. He introduced what is now known as the Bauer maximum principle, a deep result in convex analysis and functional analysis that plays an important role in understanding harmonic and subharmonic functions on general topological spaces.

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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