Pat'sBlog

Monday, 26 October 2026

#26 Repunit and Residual... History and Etymology of Math Terms

Repunit The term repunit, for a number made up of all unit digits, was created by Albert Beiler in the 1960's as a contraction of repeated unit. It still does not appear in most dictionaries, and is therefore difficult to trace... more to come, I hope.

One of the most famous puzzle masters of the early 1900's was Henry Ernest Dudeney. In his puzzle book,The Canterbury Puzzles, of 1907 he poses a problem about Repunit factoring, without using the term repunit. The problem is posed thus:

"It used to be told at St Edmondsbury," said Father Peter on one occasion, "that many years ago they were so overrun with mice that the good abbot gave orders that all the cats from the country round should be obtained to exterminate the vermin. A record was kept, and at the end of the year it was found that every cat had killed an equal number of mice, and the total was 1,111,111 mice. How many cats do you suppose there were?"
Later the problem provides that there is more than one cat, and each cat kills more than one mouse, in fact each cat killed more mice than there were cats.

I will begin to discuss the answer shortly, so those who wish to solve the problem first might stop reading and take a few moments to work on a solution.
Dudeney points out that for the solution to be unique, there must be only two factors of 1,111,111. In truth, the factors of 1,111,111 are 239 and 4649, so there must have been 239 cats who each caught 4649 mice.

In the solutions in the second edition in 1919, Dudeney discusses the general problem of finding solutions of repunits (again without using the term) and gives some interesting tables and remarks which I have tried to recopy accurately

Lucas, in his L'Arithmetique Amusante, gives a number of curious tables which he obtained from an arithmetical treatise, called the Talkhys, by Ibn Albanna, an arabian mathematician and astronomer of the first half of the thirteenth century. In the Paris National Library are several manuscriptes dealing with the Talkhys, and a commentary by Alkalacadi, who died in 1486. Among the tables given by Lucas is one giving all the factors of numbers of the above form (repunits) up to n=18 (eighteen ones in a row). It seems almost inconceivable that Arabians of that date could find the factors where n=17 as given in my introduction [On page 18 of the introduction he gives the factors of 11,111,111,111,111,111 as 2,071,723 and 5,363,222,357.]. But I read Lucas as stating that they are given in Talkhys, though an eminent mathematician reads him differently, and suggest to me that they were discovered by Lucas himself. This can, of course, be settled by an examination of the Talkhys, but this has not been possible during the war.(WWI)
The difficulty lies wholly with those cases where n is a prime number. If n=2, we get the prime 11. The factors when n=3, 5, 11, and 13 are respectively (3x37), (41 x 271), (21,649x513,239), and (53 x 79 x 265,371,653)[Dudeney used a raised dot for multiplication which I have replaced with an x for my convenience]. I have given in these pages the factors where n=7 and n=17. The factors where n=19, 23, and 37 are unknown, if there are any.^* [emphasis added]

In a footnote He points out that "Mr. Oscar Hoppe, of New York, informs me that, after reading my statement in the introduction [where he says n=19 is prime], he was led to investigate the case of n=19, and after long and tedious work he succeeded in probing the number to be a prime. He submitted his proof to the London Mathematical Society, and a specially appointed committee of that body accepted the proof as final and conclusive. He refers me to the Proceedings of the Society for 14th February, 1918.

Dudeney also points out three "curious series of factors" that he thought would "doubtless interest the reader." They are shown here: n=2 --> 11
n=6 --> 11 x 111 x 91
n=10 --> 11 x 11,111 x 9091
n=14 --> 11 x 1,111,111 x 909,091
Can you guess n=18?

Or the same numbers can be written as
n=6 --> 111 x 1,001
n=10 --> 11,111 x 100,001
n=14--> 1,111,111 x 1,000,000,001

For the cases where n is a multiple of 4, we get
n=4 --> 11 x 101
n=8 --> 11 x 101 x 10,001
n=12--> 11 x 101 x 100,010,001
n=16--> 11 x 101 x 1,000,100,010,001

Residual Sit back, stay right there, and I will tell you the origin of residual. Wait! I just did. The common re prefix means back and sid is from the Latin sedere which means to sit, so the literal meaning of residual is one who sits back or, more appropriately, stays seated. In statistics we use it in the same sense as residue, that which remains (stays seated) when something else is taken away; what remains from the observed amount when the predicted amount is removed. Another closely related word is residence. Other words drawn from the sedere root include sedentary [one who sits around a lot], sediment [stuff that settles], and sedative [something that keeps you from moving around].

On This Day in Math - October 26

There are no foolish questions and no man becomes a fool until he has stopped asking questions.
Charles P Steinmetz

The 299th day of the year. If a cubic cake was cut with 12 straight cuts, it can produce a maximum of 299 pieces.... a good day to "let 'em eat cake."

299= 150^2 - 149^2 = 18^2 - 5^2

There are 299 composite numbers less than 1000 which are products of two primes. (This counts non-distinct semi-primes such as 9=3x3 0r 2 = 5x5},

EVENTS
1639 (OS) Jeremiah Horrocks makes his first written prediction of the 1639 Transit of Venus, in a letter to William Crabtree to alert him – that Venus would pass across the Sun’s disk on 24 November. Although they notified several other acquaintances of the impending event, these two men seem to have been the only recorded astronomers to make measurements of the passage. *Allan Chapman, Jeremiah Horrocks and the transit of Venus of 1639

The friends, followers of the new astronomy of Johannes Kepler, were self-taught mathematical astronomers who had worked methodically to correct and improve Kepler's Rudolphine tables by observation and measurement. In 1639, Horrocks was the only astronomer to realise that a transit of Venus was imminent; others became aware of it only after the event when Horrocks's report of it was circulated. Although the friends both died within five years of making their observations, their ground-breaking work was influential in establishing the size of the Solar System; for this and their other achievements Horrocks and Crabtree, along with their correspondent William Gascoigne, are considered to be the founding fathers of British research astronomy.

Horrocks produced several drafts of a Latin treatise Venus in sole visa (Venus seen on the Sun) based on his observations, which he presumably intended to publish, but he died suddenly from unknown causes on 3 January 1641, aged 22.

Memorial to Jeremiah Horrocks in St Michael's Church, Hoole. The Latin is taken from Horrocks's report of the 1639 transit and reads "Ecce gratissimum spectaculum et tot votorum materiem": "oh, most grateful spectacle, the realization of so many ardent desires"

1675 Leibniz first use of a calculus symbol was in a letter on October 26. He wrote that he would use w as the differential of the variable y. *Florian Cajori, The History of Notations of the Calculus

The integral symbol was first used by Gottfried Wilhelm Leibniz (1646-1716) on October 29, 1675, in an unpublished manuscript, Analyseos tetragonisticae pars secunda:

Utile erit scribi $\int$ pro omnia, ut $\int$ l = omn. l, id est summa ipsorum l. [It will be useful to write $\int$ for omn. so that $\int$ l = omn. l, or the sum of all the l's.]

Prior to this he would write "Omnia" or the shortened omn for "everything" or, the sum of all.

Two weeks later, on Nov. 11, in Methodi tangentium inversae exempla, he first placed dx after the integral symbol, replacing

\frac{�}{�}

1676 Newton, through the intermediary of Oldenburg, wrote Leibniz concerning his work on the calculus. An anagram contained the statement of the problem of integrating differential equations. *VFR

The foundations of these operations is evident enough, in fact; but because I cannot proceed with the explanation of it now, I have preferred to conceal it thus: 6accdae13eff7i3l9n4o4qrr4s8t12ux. On this foundation I have also tried to simplify the theories which concern the squaring of curves, and I have arrived at certain general Theorems.

The anagram expresses, in Newton's terminology, the fundamental theorem of the calculus: "Data aequatione quotcunque fluentes quantitates involvente, fluxiones invenire; et vice versa", which means "Given an equation involving any number of fluent quantities to find the fluxions, and vice versa."

1738 Theophilus Grew, Mathematician advertises his availability for all manner of mathematical instruction in the Philadelphia Gazette.

"Forasmuch as Mathematical Learning is (and has been in all Ages) promoted in most Parts of the World especially in all great Towns, and generally pursued by the Gentry and those of the first Rank, as a necessary Qualification; it is to be hoped that this flourishing City will follow the Example and give it such Encouragement as it justly deserves.
In order to which there will be taught this Winter, over against Mr. James Steel’s in Second-Street, Philadelphia; Reading, Writing, Vulgar Arithmetick, Decimal Arithmetick, Accompts, Euclid’s Elements, Practical Geometry, Mensuration, Gauging, Surveying, Algebra, Trigonometry, Geography, Navigation, Astronomy, Dialing, Projection of the Sphere, the Use of Globes, Maps, Quadrants, Scales, sliding Rules, and all other Instruments for the Mathematical Service, by Theophilus Grew, Mathematician."

Grew would go on to become the first professor of mathematics at the Univ of Pennsylvania when it was founded. *Natl. Archives

1818 Thomas Jefferson writes Nathaniel Bowditch to offer him the Math Professorship at the newly forming University of Virginia

I have stated that where men of the 1st. order of science in their line can be found in our country, we shall give them a willing preference. we are satisfied that we can get from no country a professor of higher qualifications than yourself for our Mathematical department, and we entertain the hope and with great anxiety that you will accept of it. the house for that Professorship will be ready at midsummer next or soon after, when we should wish that school to be opened. I know the prejudices of every state against the climates of all those South of itself: but i know also that the candid traveller advancing Southwardly, to a certain degree at least, sees that they are more prejudices, and that the real advantages of climate are in the middle & temperate states, and especially when above their tide waters.

*Letters of Thomas Jefferson, http://etext.lib.virginia.edu (While little known today, Bowditch was considered on of the first world class mathematicians from the "new world." The figures known as Lissajous figures, are more fittingly often referred to as Bowditch curves.)

1843 John T Graves replies to Hamilton about the invention of Quaternions,

"There is something in the system which gravels me. I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties."
"If with your alchemy you can create three pounds of gold, why should you stop there?

,
Graves is credited by Hamilton with being a critical inspiration in the Quaternions, and would quickly go on to liberty to create imaginaries himself and create the "Octaves", an eight dimensional normed division algebra. Why should you stop there indeed? *Joan Baez Rankin Lecture of September 17, 2008 Glascow

1847 William Whewell wrote to Aubrey De Vere expressing dismay at the influence of Carlyle's pessimism among his friends and in society. *@GalileosBalls, Twitter

1893 Williamina Fleming, a "computer" at the Harvard Observatory, examining a photograph taken three months earlier, recognized a dozen prominent hydrogen lines, a spectrum unique to a nova. The first Nova to be detected by spectral photography. Fleming also is recognized for the discovery in 1888 of the Horsehead Nebula.

The Harvard Calculators, Fleming stahding

1893 Karl Pearson’s ﬁrst statistical publication. *VFR In Pearson' s first published statistical paper of 26 October 1893, he introduced the method of moments as a means of curve fitting asymmetrical distributions. One of his aims in developing the method of moments was to provide a general method for determining the values of the parameters of a frequency distribution. *StatProb web site

1896 Comptes Rendus publishes, "Extension of the Reimann-Roch Theorem to Algebraic Surfaces. A note by M. M. Noether, presented by M. Hermite *Mathematical Intellignecer vol 8 #4

1946 A one-page handwritten letter in German from Albert Einstein to Polish American physicist Ludwik Silberstein is dated Oct. 26, 1946. It would not be known of until sometime in 2021 when it was put up for auction by an anonymous collector. The letter, handwritten by Albert Einstein in which he writes out his famous E = mc² equation, sold at auction for more than $1.2 million on 21 May, 2021. There are only three other known examples of Einstein writing the world-changing equation in his own hand. This fourth example is the only one known in a private collection. In 1935 Silberstein claimed that A. Einstein's theory was flawed, in need of a revision. In response, Einstein and Nathan Rosen published a Letter to the Editor in which they pointed out a critical flaw in Silberstein's reasoning. Unconvinced, Silberstein took the debate to the popular press.

1960 Saga, a silent shoot-em-up Western playlet made on the TX-0 computer, was run on CBS' special for MIT's 100th anniversary. The TX-0 was the first general purpose transistorized computer. The program for Saga comprised 4,096 words of magnetic core storage. The 13,000 lines of code choreographed the movements of each object. A line of direction was written for each action, even if it went wrong. This led to the high point of the show where sheriff put his gun in the holster of the robber resulting in a never ending loop.
Doug Ross explained the rule-based diagram: If the robber drank from alcohol, his judgement would start to decline, but the program would remain logical.*CHM

TX-0 computer circuitry used Philco surface-barrier transistors, which were encapsulated in plug-in vacuum tubes for testing and easy removal.

BIRTHS

1796 James Curley (Irish: Séamus MacThoirealaigh (26 October 1796 – 24 July 1889) was an Irish-American astronomer. He was born at Athleague, County Roscommon, Ireland. His early education was limited, though his talent for mathematics was discovered, and to some extent developed, by a teacher in his native town. He left Ireland in his youth, arriving in Philadelphia on 10 October 1817. Here he worked for two years as a bookkeeper and then taught mathematics at Frederick, Maryland. In 1826 he became a student at the old seminary in Washington, DC, intending to prepare himself for the Catholic priesthood, and at the same time taught one of its classes. The seminary, however, which had been established in 1820, was closed in the following year and he joined the Society of Jesus on 29 September 1827. After completing his novitiate he again taught in Frederick and was sent in 1831 to teach natural philosophy at Georgetown University. He also studied theology and was ordained priest on 1 June 1833. His first Mass was said at the Visitation Convent, Georgetown, where he afterwards acted as chaplain for fifty years.He spent the remainder of his life at Georgetown, where he taught natural philosophy and mathematics for forty-eight years. He planned and superintended the building of the Georgetown Observatory in 1844 and was its first director, filling this position for many years. One of his earliest achievements was the determination of the latitude and longitude of Washington, D.C. in 1846. His results did not agree with those obtained at the Naval Observatory, and it was not until after the laying of the first transatlantic cable in 1858 that his determination was found to be nearer the truth. *Wik

1846 Lewis Boss (26 Oct 1846; 12 Oct 1912) American astronomer best known for his compilation of two catalogs of stars (1910, 1937). In 1882 he led an expedition to Chile to observe a transit of Venus. About 1895 Boss began to plan a general catalog of stars, giving their positions and motions. After 1906, the project had support from the Carnegie Institution, Washington, D.C. With an enlarged staff he observed the northern stars from Albany and the southern stars from Argentina. With the new data, he corrected catalogs that had been compiled in the past, and in 1910 he published the Preliminary General Catalogue of 6,188 Stars for the Epoch 1900. The work unfinished upon his death was completed by his son Benjamin in 1937 (General Catalogue of 33,342 Stars for the Epoch 1950, 5 vol.)*TIS

1849 Georg Frobenius (26 Oct 1849; 3 Aug 1917) German mathematician who made major contributions to group theory, especially the concept of abstract groups (with Ludwig Stickleberger) and the theory of finite groups of linear substitutions (with Issai Schur), that later found important uses in the theory of finite groups as it applies to quantum mechanics. He also contributed to means of solving linear homogenous differential equations. The fact so many of Frobenius's papers read like present day text-books on the topics which he studied is a clear indication of the importance that his work, in many different areas, has had in shaping the mathematics which is studied today.*TIS

1877 Max Mason (26 Oct 1877; 23 Mar 1961) American mathematical physicist, educator, and science administrator. During World War I he invented several devices for submarine detection - several generations of the Navy's "M," or multiple-tube, passive submarine sensors. This apparatus focused sound to ascertain its source. To determine the direction from which the sound came, the operator needed only to seek the maximum output on his earphones by turning a dial. The final device had a range of 3 miles. Mason's special interest and contributions lay in mathematics (differential equations, calculus of variations), physics (electromagnetic theory), invention (acoustical compensators, submarine-detection devices), and the administration of universities and foundations. *TIS

1885 Niels Erik Norlund (26 Oct 1885 in Slagelse, near Soro, Sjaelland, Denmark - 4 July 1981 in Copenhagen, Denmark) In 1907 he was awarded a gold medal for an essay on continued fractions and his resulting two publications were in 1908: Sur les différences réciproques; and Sur la convergence des fractions continues both published in Comptes Rendus de l'Academie des Sciences. These publications in the most prestigious French journal earned Norlund an international reputation despite still being an undergraduate. In the summer of 1910 he earned a Master's degree in astronomy and in October of that year he successfully defended his doctoral thesis in mathematics Bidrag til de lineaere differentialligningers Theori. In the same year he published the 100-page paper Fractions continues et différences réciproques as well as Sur les fractions continues d'interpolation, a paper on Halley's comet, and an obituary of his teacher Thorvald Thiele. Norlund's sister Margrethe married Niels Bohr whose brother, Harald, was also an outstanding mathematician. In 1955 Norland reached retirement age. That mathematics was his first love now became clear, for once he gave up the responsibilities of the Geodesic Institute he returned to mathematics research. He published Hypergeometric functions in 1955 which was reviewed by Arthur Erdélyi, "This is one of those rare papers in which sound mathematics goes hand in hand with excellent exposition and style; and the reader is both instructed and delighted. It is likely to become the standard memoir on the generalized hypergeometric series ... " The paper Sur les fonctions hypergéométriques d'ordre supérieur (1956) gives a very full, rigorous and classical treatment of some integrals from generalized hypergeometric function theory.*SAU

1902 Henrietta Hill Swope(26 October 1902; Saint Louis, Missouri - 24 November 1980; Pasadena, California)was an American astronomer. She was the eldest child of Gerard and Mary Dayton (Hill) Swope; her mother was the daughter of Thomas Hill, president of Harvard University, 1862-1868. She received her A.B. from Barnard College in 1926 and her A.M. from Radcliffe College in 1928. In 1936, while assistant at the Harvard Observatory (1928-1942), she was a member of the expedition sent jointly by the Harvard Observatory and the Massachusetts Institute of Technology to study the solar eclipse in Soviet Central Asia. During World War II she was staff member of the M.I.T. Radiation Laboratory and then served as a mathematician in the Hydrographic Office of the U.S. Department of the Navy. From 1947 to 1952 she taught astronomy at Barnard College and in 1952 was appointed assistant, later research fellow, at the Mt. Wilson and Palomar Observatories in California. After her retirement in 1968, she continued to work at the Observatories.
HHS was a member of the American Astronomical Society; she received the AAS Annie Jump Cannon Prize in 1968 for her research on photometry and variable stars. She was responsible for developing a new yardstick for measuring the universe: calibrating distance by determining the brightness of stars. She received the Distinguished Alumna Award of Barnard College in 1975 and the Barnard Medal of Distinction in 1980.
The Swope Telescope at the Las Campanas Observatory in Chile is named in her honor, as is asteroid 2168 Swope.

1911 Shiing-shen Chern (26 Oct 1911; 3 Dec 2004) Chinese-American mathematician and educator whose researches in differential geometry include the development of the Chern characteristic classes in fibre spaces, which play a major role in mathematics and in mathematical physics. "When Chern was working on differential geometry in the 1940s, this area of mathematics was at a low point. Global differential geometry was only beginning, even Morse theory was understood and used by a very small number of people. Today, differential geometry is a major subject in mathematics and a large share of the credit for this transformation goes to Professor Chern." *TIS

1930 Walter Feit (26 Oct 1930 in Vienna, Austria - 29 July 2004 in Branford, Connecticut, USA) was an Austrian mathematician who (with John Thompson) proved one of the most important theorems about finite simple groups. In 1990 his 60th birthday was celebrated with an 'International Symposium on the Inverse Galois Problem' held in Oxford. His retirement from Yale in October 2003 was marked with the holding of a 'Conference on Groups, Representations and Galois Theory' in his honour. Feit died after a long illness at the Connecticut Hospice in Branford, Connecticut, USA. A memorial service was held on Sunday 10 October 2004 at the New Haven Lawn Club, New Haven, Connecticut. *SAU

DEATHS
1817 Aida Yasuaki Aida Yasuaki (会田安明, February 10, 1747 – October 26, 1817) a Japanese mathematician who published about 2000 works. Aida compiled Sampo tensi shinan which appeared in 1788. It is a book of geometry problems, developing formulae for ellipses, spheres, circles etc. Aida explained the use of algebraic expressions and the construction of equations. He also worked on number theory and simplified continued fraction methods due to Seki. *SAU

He made significant contributions to the fields of number theory and geometry, and furthered methods for simplifying continued fractions.

Aida created an original symbol for "equal". This was the first appearance of the notation for equal in East Asia.

1923 Charles Proteus Steinmetz (9 Apr 1865- 26 Oct 1923) German-born American inventor and electrical engineer whose theories and mathematical analysis of alternating current systems helped establish them as the preferred form of electrical energy in the United States, and throughout the world. In 1893, Steinmetz joined the newly organized General Electric Company where he was an engineer then consultant until his death. His early research on hysteresis (loss of power due to magnetic resistance) led him to study alternating current, which could eliminate hysteresis loss in motors. He did extensive new work on the theory of a.c. for electrical engineers to use. His last research was on lightning, and its threat to the new AC power lines. He was responsible for the expansion of the electric power industry in the U.S. *TIS

1957 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".

Cori was born in Prague, the capital of Bohemia within the Austro-Hungarian Empire. Growing up at a time when women were marginalized and allowed few educational opportunities, she gained admittance to medical school, where she met her future husband Carl Ferdinand Cori in an anatomy class.[4] Upon their graduation in 1920, they married. Because of deteriorating conditions in Europe, the couple emigrated to the United States in 1922. Gerty Cori continued her early interest in medical research, collaborating in the laboratory with Carl. She published research coauthored with her husband, as well as publishing singly. Unlike her husband, she had difficulty securing research positions, and the ones she obtained provided meager pay. Her husband insisted on continuing their collaboration, though he was discouraged from doing so by the institutions that employed him.

Together with her husband Carl and Argentine physiologist Bernardo Houssay, Gerty Cori received the Nobel Prize in 1947 for the discovery of the mechanism by which glycogen—a polysaccharide made from glucose—is broken down in muscle tissue into lactic acid and then resynthesized in the body and stored as a source of energy (known as the Cori cycle). They also identified the important catalyzing compound, the Cori ester. The Coris were the third ever married couple to win the Nobel Prize. In 2004, both Gerty and Carl Cori were designated a National Historic Chemical Landmark in recognition of their work in clarifying carbohydrate metabolism.

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

Gerty and Carl Cori in the lab, media photograph, undated, Washington University, St. Louis (wikipedia.org)

1968 Sergei Natanovich Bernstein (March 5, 1880 – October 26, 1968) was a Russian and Soviet mathematician. His doctoral dissertation, submitted in 1904 to the Sorbonne, solved Hilbert's nineteenth problem on the analytic solution of elliptic differential equations. Later, he published numerous works on Probability theory, Constructive function theory, and mathematical foundations of genetics. From 1906 until 1933, Bernstein was a member of the Kharkov Mathematical Society. *Wik

1970 Marcel Gilles Jozef Minnaert (12 Feb 1893; 26 Oct 1970 at age 77)
Flemish astronomer and solar physicist who was one of the pioneering solar researchers during the first half of the 20th century. Applying solar spectrophotometry, he was one of the first to make quantitative measurements of the intensity distribution inside Fraunhofer lines, and interpret from them information about the outer solar layers. His range of study also included comets, nebulae and lunar photometry. During the time he was director of the observatory at the University of Utrecht, (1937-1963) he created a modern astronomical institute to study solar and stellar spectra with resources including a solar telescope, spectrograph, photometer, and mechanical workshop. Minnaert also maintained a strong interest in the education of physics teachers, and as a university professor gave clear, enthusiastic and well-prepared lectures. *TIS

1983 Alfred Tarski (14 Jan 1902, 26 Oct 1983) Polish-born American mathematician and logician who made important studies of general algebra, measure theory, mathematical logic, set theory, and metamathematics. Formal scientific languages can be subjected to more thorough study by the semantic method that he developed. He worked on model theory, mathematical decision problems and with universal algebra. He produced axioms for "logical consequence", worked on deductive systems, the algebra of logic and the theory of definability. Group theorists study 'Tarski monsters', infinite groups whose existence seems intuitively impossible. *TIS

1984 Mark Kac (3 Aug 1914 in Krzemieniec, Poland, Russian Empire - 26 Oct 1984 in California, USA) pioneered the modern development of mathematical probability, in particular its applications to statistical physics. The method of quantization now in use involves the Feynman-Kac path integral, named after Richard Feynman and Mark Kac. He published a classic text Statistical Independence in Probability, Analysis and Number Theory in 1959. To many Kac will be remembered best for a paper he wrote for the American Mathematical Monthly in 1966. This is the famous paper Can One Hear the Shape of a Drum? and Kac received the Chauvenet Prize from the Mathematical Association of America in 1968 for the, "most outstanding expository article on a mathematical topic by a member of the Association." *SAU

1998 Kenkichi Iwasawa (11 Sept 1917 in Shinshuku-mura (near Kiryu), Gumma Prefecture, Japan - 26 Oct 1998 in Tokyo, Japan ) In the late 1960s Iwasawa made a conjecture for algebraic number fields which, in some sense, was the analogue of the relationship which Weil had found between the zeta function and the divisor class group of an algebraic function field. This conjecture became known as "the main conjecture on cyclotomic fields" and it remained one of the most outstanding conjectures in algebraic number theory until it was solved by Mazur and Wiles in 1984 using modular curves. "it is no exaggeration to say that Iwasawa's ideas have played a pivotal role in many of the finest achievements of modern arithmetical algebraic geometry on such questions as the conjecture of B Birch and H Swinnerton-Dyer on elliptic curve; the conjecture of B Birch, J Tate, and S Lichtenbaum on the orders of the K-groups of the rings of integers of number fields; and the work of A Wiles on the modularity of elliptic curves and Fermat's Last Theorem." *SAU

2010 Professor Charles Bernard Bell, Jr., (Aug 20, 1928 - Oct 26, 2010) one of the leading African American mathematicians of the twentieth century, was born in New Orleans, Louisiana on August 20, 1928. At age 19 he graduated from HBCU Xavier University in 1947 and earned his PhD. in mathematics from Notre Dame University, four years later, in 1953. From 1951 to 1955 he worked as a research engineer at Douglas Aircraft Company. After serving as an assistant professor at Xavier University for two years, he then spent a year at Stanford University as a research associate.

In 1958, Bell became the second African American professor hired at San Diego State University (SDSU) and became one of only a handful of Black professors at a predominantly white campus in that era. He served as advisor to the nearly all white student organization, the Interfaith Council. On leave from SDSU between 1964 and 1966, Bell traveled abroad, working and studying at the Mathematics Institute of Amsterdam, the University of Madrid (Spain), the University of Vienna (Austria), the Institute of Statistics at the University of Paris (France), and the University of Erlangen (West Germany). During that time he also attended the International Mathematics Conference in Moscow, then capital of the Soviet Union.

From 1966 to 1981, Bell taught in succession at Case Western University, the University of Michigan at Ann Arbor, and the University of Washington at Seattle before returning to San Diego State University in 1981. He retired as full professor in 1992. Bell published nearly forty papers and was elected a Fellow in the Institute of Mathematical Statistics and the American Statistics Association. He also enjoyed traveling to meet fellow mathematicians in Africa, India, and Europe and welcomed the opportunity to hone his language skills in foreign countries. He once told an interviewer: “I enjoy family, I enjoy eating gumbo, and for more than four decades I enjoyed teaching mathematics and statistics and doing research in nonparametric statistics, stochastic processes and related problems and applications.” Bell’s other areas of expertise were biostatistics and signal detection which had highly sensitive military implications.

Charles B. Bell died in Los Angeles, California on October 26, 2010 at the age of 82. *Black Past

2013 Elza Furtado Gomide (August 20, 1925 – October 26, 2013) was a Brazilian mathematician and the first woman to receive a doctorate in mathematics from the University of São Paulo, in 1950, and the second in Brazil. Gomide was involved in the creation of the Society of Mathematics of São Paulo and was elected head of the department of mathematics of the University of São Paulo in 1968. (The first Brazilian woman to receive a doctorate in mathematics and the first Brazilian woman to join the Brazilian Academy of Sciences was Marília Chaves Peixoto, *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
*WM = Women of Mathematics, Grinstein & Campbell

Sunday, 25 October 2026

Almost Integers, almost incredible

I first came into teaching just as computers and programmable calculators did. It was me, Apple and Texas Instruments side by side against mathematical illiteracy. One of the things I loved doing in the classroom , and still enjoy out of it, were expressions that were "nearly" an integer, now called "almost integers" in recreational math.

I think my first introduction came a half-dozen or so years before I first became a teacher. It was the famous article by Martin Gardner in April of 1975 when he posted his "Six sensational discoveries that somehow or another have escaped public attention" in his column in Scientific American. One of his six things was his claim that e^π√163 = 262,537,412,640,768,744, a perfect integer. He credited this discovery to Ramanujan, and it has come to be known as Ramanujan's constant, but the nearness to an integer was noted in 1859 by the mathematician Charles Hermite. The number actually ends in ...743.999,999,999,999,25..., but in the age of calculators only given ten to 12 digits, it was enough to be confusing and difficult to check. You can use much smaller numbers to get similar near integers, e^π√43 comes within about 1/5000 of an integer, and my TI-84 silver Edition gives 884736744, an exact integer as the result.

The Golden Ratio, $ \phi =\frac{\sqrt{5}+1}{2} $ will produce almost integers for any power above about ten. $ \phi^{10} =122.991... $ while $ \phi^{18} =5777.9998.. $. Successive powers tend to successively overreach and undershoot an integer. It seems that the twenty-fourth power is sufficient to fool my TI which gives $ \phi^{24} =103,682 $. (I just checked this last one on Desmos online graphing calculator and it gives 103681.99999. Science marches on.

The numbers 134903170 and 196418 form a pair that introduce lots of almost integers. 134903170/196418 = 5777.999826..., these are the 45th and 27th Fibonacci numbers, but you get nearer to an integer each time you double the index of the two Fibonacci numbers. Wikipedia tells me that Fib(45*8) /Fib(27*8) will produce a string of about 30 nines after the decimal point.

An unusual one that is more of a mathematical coincidence, and not explained, is the fact that $ e^{\pi} - \pi = 19.99909997... $ The constant $ e^{\pi} $ is called Gelfond's Constant, after Aleksandr Gelfond, who proved that the number was transcendental.

For Trig Students, Sin (11) is an almost integer, (almost -1) and you can totally fool the TI-84 with $ Sin(2017*\sqrt[5]{2})$ which I'm using for my special day number next year on 2/5/2017 . The Sin(11) is nice to introduce when working with half-angle identities since $ sin^2(11)=1/2(1-cos^2(22)$.

$ 163 ( \pi - e ) $ is almost an integer, and powers of pi and e can be fun almost integers, and $ \pi^4 + \pi^5 - e^6 $ is almost zero. Students might search for such an expression that will fool their calculator.

And for Geometry class, the image at the top of the blog shows a triangle with all integer sides and segments except segment d. Ed Pegg Jr is credited with finding this gem, and showing that the segment marked d is 7.0000000857....

Know of others, ....? SHARE!

On This Day in Math - October 25

Unfortunately what is little recognized is that the most worthwhile scientific books are those in which the author clearly indicates what he does not know; for an author most hurts his readers by concealing difficulties.
E. Torricelli

The 298th day of the year; If you multiply 298 by (298 + 3) you get a palindromic number, 89,698. Can every number be similarly adjusted to make a palindrome? And this one is not just a Palindrome, it's a strobogrammatic one, rotate it 180 degrees and you get another palindrome, 86968. (Some restrict the term stobogram only to numbers that recreate themselves after rotation, and prefer ambigram for the ones that rotate to make a different number.)

6 x 298 +/- 1 are twin primes. The 55th number of the year for which this is true.

298 = $ {12 \choose 1} + {12 \choose 2} + {12 \choose 3} $ This is related to the Egg Drop numbers

EVENTS
1666 William Lilly, astrologer, was called before the House of Commons to explain the embarrassing success of his 1651 prediction of the plague (of 1665) and “exorbitant ﬁre” of 1666. The House ultimately attributed the ﬁre to the papists. *W W Rouse Ball, Mathematical Recreations and Essays,6th edition, p. 390 Lilly caused much controversy in 1652 for allegedly predicting the Great Fire of London some 14 years before it happened. For this reason many people believed that he might have started the fire, but there is no evidence to support these claims. He was tried for the offense in Parliament but was found to be innocent.*Wik

In 1671, Giovanni Cassini discovered Iapetus, one of Saturn's moons. Iapetus is the third largest and one of the stranger of the 18 moons of Saturn. Its leading side is dark with a slight reddish color while its trailing side is bright. The dark surface might be composed of matter that was either swept up from space or oozed from the moon's interior. This difference is so striking that Cassini noted that he could see Iapetus only on one side of Saturn and not on the other. In Greek mythology Iapetus was a Titan, the son of Uranus, the father of Prometheus and Atlas and an ancestor of the human race. Cassini (1625-1712), first director of the Paris Royal Observatory, also discovered other moons of Saturn (Tethys, Dione, Rhea) and the major gap in its rings. *TIS

The two views of Lapetus

1713 Leibniz, in a letter to Johann Bernoulli, observed that an alternating series whose terms monotonically decrease to zero in absolute value is convergent. In a letter of January 10, 1714, he gave an incorrect proof (Big Kline, p. 461). Examination of the proof reveals that it is the one we give today, except he fails to say anything about the completeness of the reals. *VFR

1846 William Thompson (Lord Kelvin) writes to Sir George Stokes regarding the "recent proceedings about Oceanus, or Neptune, or Le Verrier. " commenting that "Cambridge is behind the rest of the world on scientific subjects.". John C. Adams, later became a fellow at Pembroke College, and he and Stokes became close friends. *The correspondence between Sir George Gabriel Stokes and Sir William Thompson, pg 2

1881 Clerk Seaton writes to the chairman of the committee on the census that he has discovered a paradox with the apportionment. Seaton had discovered the Alabama Paradox.

It seemed so easy. The 1787 US Constitution laid out simple rules for deciding how many representatives each state shall receive:
"Representatives and direct taxes shall be apportioned among the several States which may be included within this Union, according to their respective numbers, ... The number of Representatives shall not exceed one for every thirty thousand, but each State shall have at least one Representative ..."
It may have seemed easy, but for the 200+ years of US government, the question of "Who gets how many?" continues to perplex and promote controversy.
When congress discussed mathematical methods of applying this constitutional directive there were two methods of prime consideration, Jefferson's method, and Hamilton's method. Congress selected Hamilton's method and in the first use of the Presidential veto (make a note of this for extra points in History or Government class) President Washington rejected the bill. Congress submitted and passed another bill using Jefferson's method. The method used has changed frequently over the years with a method by Daniel Webster adopted in 1842, (the original 65 Representatives had grown to 223) and then replaced with Hamilton's method in 1852 (234 Representatives). In a strange "Only in America" moment in 1872, the congress reapportioned without actually adopting an official method and some analysis suggest that the difference caused Rutherford Hayes to Win instead of Samuel Tilden who would have won had Hamilton's method been used. Since 1931 the US House has had 435 Representatives with the brief exception of when Alaska and Hawaii became states. Then there was a temporary addition of one seat for each until the new apportionment after the 1960 Census. In 1941 the Huntington-Hill Method was adopted and has remained in continuous (and contentious) use ever since.
In 1880 the first of what are called the apportionment paradoxes was discovered. Here is how they state it at the Wikipedia web site:
After the 1880 census, C. W. Seaton, chief clerk of the U. S. Census Office, computed apportionment for all House sizes between 275 and 350, and discovered that Alabama would get 8 seats with a House size of 299 but only 7 with a House size of 300. In general the term Alabama paradox refers to any apportionment scenario where increasing the total number of items would decrease one of the shares. They also show a nice example (with small numbers) so you might check their site.

*pballew.net

In 1872, Chief Clerk of the Census Charles Seaton invented a simple machine that made tabulating census data easier by keeping the lines on large tallying sheets isolated and organized. The Census Office purchased Seaton's device and used it to finish the 1870 census. Seaton served as superintendent of the 1880 census, during which his device was used again.

Even with the Seaton device, the 1880 census took nearly the entire decade to tabulate and publish. Census data had become too extensive to be effectively counted by hand. Luckily, a technological advancement in data processing was on the horizon.*Wik

1904 The first K&E Pocket watch slide rule patent was approved. Prior to this time K&E sold French made Boucher designs. The patent is in the name of Elmer A. Sperry, co-inventor of the gyrocompass. The patent covers the use of the ‘S’ and ‘L’ dials
and the geared hands and dials . *Oughtred Society

1944 Max Planck writes to Hitler to plead for the life of his son, Erwin. In the note, the discoverer of the energy quantum pleads for the life of his son, who was involved in the attempted to kill Hitler three months before. Max Planck had already lost his eldest son, who was killed in the Battle of Verdun, during World War I.
Planck writes in his letter that he is ‘confident’ that the Führer will lend his ear to ‘an imploring 87-year-old’. This plea, apparently written from the Planck family’s bombed-out home in a suburb of Berlin, was ignored by the authorities. Erwin was executed on 23 January 1945, and his death certificate recorded: ‘parents unknown’. *Graham Farmelo

2001 Microsoft Releases Windows XP, the family of 32-bit and 64-bit operating systems produced by Microsoft for use on personal computers. The name "XP" stands for “Experience.” The successor to both Windows 2000 Professional and Windows ME, Windows XP was the first consumer-oriented operating system Microsoft built on the Windows NT kernel and architecture. Over 400 million copies were in use by January 2006, according to an International Data Corporation analyst. It was succeeded by Windows Vista, which was released to the general public in January 2007*CHM

2011 Scientists in California and Sweden have solved a 250-year-old mystery — a coded manuscript written by a secret society. The University of Southern California announced Tuesday, Oct 25th, that researchers had broken the Copiale Cipher — the writing used in a 105-page 18th century document from Germany.

Kevin Knight, of USC, and Beata Megyesi and Christiane Schaefer, of Uppsala University, did the work.

They used a statistical computer program to decipher part of the manuscript, which was found in East Berlin after the Cold War and is now in a private collection.

The book, written in symbols and Roman letters, details complicated initiation ceremonies of a society fascinated by ophthalmology. They include making mystical signs and plucking a hair from a candidate's eyebrow. The convoluted text swears candidates to loyalty and secrecy. *Associated Press,

two pages (16 & 17) from the cipher:

BIRTHS
1789 Samuel Heinrich Schwabe (25 Oct 1789; 11 Apr 1875) Amateur German astronomer who discovered the 10-year sunspot activity cycle. Schwabe had been looking for possible intramercurial planets. From 11 Oct 1825, for 42 years, he observed the Sun virtually every day that the weather allowed. In doing so he accumulated volumes of sunspot drawings, the idea being to detect his hypothetical planet as it passed across the solar disk, without confusion with small sunspots. Schwabe did not discover any new planet. Instead, he published his results in 1842 that his 17 years of nearly continuous sunspot observations revealed a 10-year periodicity in the number of sunspots visible on the solar disk. Schwabe also made (1831) the first known detailed drawing of the Great Red Spot on Jupiter.*TIS

1811 Evariste Galois born in the little village of Bourg-la-Reine, near Paris, France. *VFR (25 Oct 1811; 31 May 1832) famous for his contributions to the part of higher algebra known as group theory. His theory solved many long-standing unanswered questions, including the impossibility of trisecting the angle and squaring the circle. Galois fought a duel with Perscheux d'Herbinville on 30 May 1832, the reason for the duel not being clear but certainly linked with a love affair. Galois was wounded in the duel, and died in hospital the following day, at age 20. His funeral was held on 2 June. It was the focus for a Republican rally and riots followed which lasted for several days. He was commemorated as a revolutionary and geometrician on a French postal stamp issued on 10 Nov 1984.*TIS

Joseph Liouville’s introduction to the reprinting of the papers and letter of Évariste Galois, Journal de Mathématiques pures et appliquées, vol. 11, 1846 (Linda Hall Library)

1877 Henry Norris Russell (25 Oct 1877; 18 Feb 1957) American astronomer and astrophysicist who showed the relationship between a star's brightness and its spectral type, in what is usually called the Hertzsprung-Russell diagram, and who also devised a means of computing the distances of binary stars. As student, professor, observatory director, and active professor emeritus, Russell spent six decades at Princeton University. From 1921, he visited Mt. Wilson Observatory annually. He analyzed light from eclipsing binary stars to determine stellar masses. Russell measured parallaxes and popularized the distinction between giant stars and "dwarfs" while developing an early theory of stellar evolution. Russell was a dominant force in American astronomy as a teacher, writer, and advisor. *TIS

1884 Motonori Matuyama, ( October 25, 1884 – January 27, 1958) a Japanese geophysicist, was born Oct. 25, 1884. In 1929, Matuyama made the rather bold suggestion that the Earth's magnetic field had once been reversed from its present orientation, so that at one time, a compass needle would have pointed south instead of north. His evidence came from igneous rocks in Japan and Manchuria, which indicated by their residual magnetic fields that, when they cooled millions of years ago, the Earth's north magnetic pole had been in the vicinity of the south geographic pole. As it turns out, the same suggestion had been made once before, by a French physicist, Bernard Brunhes, in 1905. Neither man made many converts, which is not surprising, since at the time the source of the Earth’s magnetic field was poorly understood, and radiometric dating was in its infancy. However, in 1959, it was proposed once again that the Earth's magnetic field reversed itself periodically, and by 1963 the evidence for it was being used to demonstrate the reality of sea-floor spreading and lay the basis for plate tectonics.

We now know that for the past .78 million years, the Earth's magnetic field has been normal (magnetic north being north), and that before this period, from 2.6 million to .78 million years ago, the magnetic field was reversed. The period between reversals is called a "chron", and the last chron before the present one, when the field was reversed, is called the Matuyama chron, in honor of our scientist of the day. The present chron is named the Brunhes chron, after the other early proponent of the idea of geomagnetic reversals. The last reversal, which occurred .78 million years ago, is called the Brunhes-Matuyama reversal. There were other chrons and other reversals before the Brunhes and the Matuyama, and they carry the names of Carl Friedrich Gauss and William Gilbert and other pioneers in the study of the earth's magnetism. We see above a simple chart that shows the last four chrons (second image). You will note that there are short-lived mini-reversals within each chron, such as the Jaramillo event within the Matuyama chron; the second chart shows how our understanding of the complexity of Earth’s magnetic reversals progressed in just the first decade of plate tectonics.

Matuyama’s paper appeared in the Proceedings of the Imperial Academy of Japan for 1929. We have this volume in the Library’s serials collection.

Dr. William B. Ashworth, Jr., Consultant for the History of Science, Linda Hall Library and Associate Professor, Department of History, University of Missouri-Kansas City. Comments or corrections are welcome; please direct to ashworthw@umkc.edu.

I have recently seen scientific articles predicting that the Earth is nearing another flip. (2023)

1886 Lester Randolph Ford (25 Oct 1886 in Missouri, USA - 11 Nov 1967 in Charlottesville, Virginia, USA) was an American mathematician who lectured for several years in Edinburgh before moving back to the USA. He wrote some important text-books and is best known for his contributions to the Mathematical Association of America and the American Mathematical Monthly. *SAU (Ford circles are named after him. If you have never explored this idea, and the related idea of mediants, do it today)

1910 William Higinbotham (Oct 1910; 10 Nov 1994) American physicist who invented the first video game, Tennis for Two, as entertainment for the 1958 visitor day at Brookhaven National Laboratory, where he worked (1947-84) then as head of the Instrumentation Division. It used a small analogue computer with ten direct-connected operational amplifiers and output a side view of the curved flight of the tennis ball on an oscilloscope only five inches in diameter. Each player had a control knob and a button. Late in WW II he became electronics group leader at Los Alamos, New Mexico, where the nuclear bomb was developed. After the war, he became active with other nuclear scientists in establishing the Federation of American Scientists to promote nuclear n)on-proliferation.*TIS (raise your hand if you are old enough to remember "Pong")

1926 James Eells (25 Oct, 1926 in Cleveland, Ohio, USA - 14 Feb, 2007 in Cambridge, England) made substantial and deep contributions to mathematics. His most important work was probably his 1964 paper Harmonic Mappings of Riemannian Manifolds. Eells went on to publish two definitive surveys on the topic with Luc Lemaire who studied for a doctorate under Eells' supervision. These were A report on harmonic maps (1978) and Another report on harmonic maps (1988) both published in the Bulletin of the London Mathematical Society. In 1992 a selection of Eells' papers on Harmonic maps was published as a book with this title in 1992. In this book Eells points out that:-
... harmonic maps pervade differential geometry and mathematical physics: they include geodesics, minimal surfaces, harmonic functions, Abelian integrals, Riemannian fibrations with minimal fibres, holomorphic maps between Kähler manifolds, chiral models, and strings. *SAU

1945 David N. Schramm (25 Oct 1945; 19 Dec 1997) American theoretical astrophysicist who was an authority on the particle-physics aspects of the Big Bang theory of the origin of the universe. He considered the nuclear physics involved in the synthesis of the light elements created during the Big Bang comprising mainly hydrogen, with lesser quantities of deuterium, helium, lithium, beryllium and boron. He predicted, from cosmological considerations, that a third family of neutrinos existed - which was later proven in particle accelerator experiments (1989). Schramm worked to evaluate undetected dark matter that contributed to the mass of the universe, and which would determine whether the universe would ultimately continue to expand. He died in the crash of the small airplane he was piloting. *TIS

DEATHS
1400 Geoffrey Chaucer died. Although rightly famous for his Canterbury Tales, he also wrote two astronomical works. *VFR In his lifetime he was far more known for his “Treatise on the Astrolabe”

1647 Evangelista Torricelli (15 Oct 1608- 25 Oct 1647) an Italian physicist and mathematician who invented the barometer and whose work in geometry aided in the eventual development of integral calculus. Inspired by Galileo's writings, he wrote a treatise on mechanics, De Motu ("Concerning Movement"), which impressed Galileo. He also developed techniques for producing telescope lenses. The barometer experiment using "quicksilver" filling a tube then inverted into a dish of mercury, carried out in Spring 1644, made Torricelli's name famous. The Italian scientists merit was, above all, to admit that the effective cause of the resistance presented by nature to the creation of a vacuum (in the inverted tube above the mercury) was probably due to the weight of air*TIS

1733 Girolamo Saccheri (5 Sep 1667, 25 Oct 1733) Italian mathematician who worked to prove the fifth postulate of Euclid, which can be stated as, "Through any point not on a given line, one and only one line can be drawn that is parallel to the given line." Euclid saw the proof was not self-evident, yet neither did he provide one; instead he accepted it as an assumption. Subsequently many mathematicians tried to prove this fifth postulate from the remained axioms - and failed. Saccheri took the novel approach of first assuming that the postulate was wrong, then followed the all consequences seeking any one contradiction that then leaves the only original postulate as the only possible solution. In the process, he came close to discovering non-Euclidian geometry, but gave up too early.*TIS

1884 Carlo Alberto Castigliano (9 November 1847, Asti – 25 October 1884, Milan) was an Italian mathematician and physicist known for Castigliano's method for determining displacements in a linear-elastic system based on the partial derivatives of strain energy.*Wik

1905 Otto Stolz (3 May 1842 in Hall (now Solbad Hall in Tirol), Austria - 25 Oct 1905 in Innsbruck, Austria) Stolz's earliest papers were concerned with analytic or algebraic geometry, including spherical trigonometry. He later dedicated an increasing part of his research to real analysis, in particular to convergence problems in the theory of series, including double series; to the discussion of the limits of indeterminate ratios; and to integration.*SAU

1933 Friedrich Heinrich Albert Wangerin (18 November 1844 – 25 October 1933) was a German mathematician. Wangerin was known for his research on potential theory, spherical functions and differential geometry. He wrote an important two volume treatise on potential theory and spherical functions. Theorie des Potentials und der Kugelfunktionen I was published in 1909 and Theorie des Potentials und der Kugelfunktionen II was published in 1921. He studied Wangerin functions.

Wangerin was also known for writing of textbooks, encyclopaedias and his historical writings. In 1904, he wrote Theorie der Kugelfunktionen und der verwandten Funktionen, insbesondere der Laméschen und Besselschen (Theorie spezieller, durch lineare Differentialgleichungen definierter Funktionen) on functions such as Lamé function and Bessel function for the Encyklopädie der mathematischen Wissenschaften. In 1909, he wrote an article on optics (Optik ältere Theorie) for the physics volume of the same encyclopaedia.

Wangerin also played an important role in the reviewing of mathematical papers. From 1869 to 1921, he was coeditor of Fortschritte der Mathematik.

Wangerin was elected to the German Academy of Scientists Leopoldina in 1883. From 1906 to 1921, he served as President of the Academy. In 1907, he was awarded an honorary degree from Uppsala University. He received many medals, including the 1922 Cothenius medal from the German Academy of Scientists Leopoldina. *Wik

1996Ennio de Giorgi (8 February 1928 – 25 October 1996) was an Italian mathematician who worked on partial differential equations and the foundations of mathematics.

De Giorgi's first work was in geometric measure theory, on the topic of the sets of finite perimeters which he called in 1958 Caccioppoli sets, after his mentor and friend. His definition applied some important analytic tools and De Giorgi's theorem for the sets established a new tool for set theory as well as his own works. This achievement not only brought Ennio immediate recognition but displayed his ability to attack problems using completely new and effective methods which, though conceived before, can be used with greater precision as shown in his research works.

De Giorgi solved Bernstein's problem about minimal surfaces for 8 dimensions in 1969 with Enrico Bombieri and Enrico Giusti, for which Bombieri won the Fields Medal in 1974.

De Giorgi solved 19th Hilbert problem on the regularity of solutions of elliptic partial differential equations. Before his results, mathematicians were not able to venture beyond second-order nonlinear elliptic equations in two variables. In a major breakthrough, De Giorgi proved that solutions of uniformly elliptic second-order equations of divergence form, with only measurable coefficients, were Hölder continuous. His proof was proved in 1956/57 in parallel with John Nash's, who was also working on and solved Hilbert's problem.*Wik

2002 René Frédéric Thom (September 2, 1923 – October 25, 2002) was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as founder of catastrophe theory (later developed by Erik Christopher Zeeman). He received the Fields Medal in 1958.*Wik

2022 Marta Cavallo Bunge (1938 – 25 October 2022) was an Argentine-Canadian mathematician specializing in category theory, and known for her work on synthetic calculus of variations and synthetic differential topology. She was a professor emeritus at McGill University
With her doctoral student Jonathon Funk, Bunge is the co-author of Singular Coverings of Toposes (Lecture Notes in Mathematics 1890, Springer, 2006). With Felipe Gago and Ana María San Luis, Bunge is the co-author of Synthetic Differential Topology (London Mathematical Society Lecture Note Series 448, Cambridge University Press, 2018).