Leo Moser

Leo Moser is an Austrian-born Canadian mathematician known for his work in number theory. Over the course of his career, Moser contributed to over one hundred research publications, primarily in the fields of number theory, graph theory, and algebra. Moser made a number of contributions to mathematics, including Moser's polygon notation and Moser's worm problem—a geometry problem defined by Moser that remains unsolved.

Moser was also known for his teaching and communication of mathematics, and he was described by Max Wyman (Former President of the University of Alberta) as "one of the best teachers of mathematics the University of Alberta has had" and "the most popular of the lecturers... sponsored by the Mathematical Association of America." During his career, Moser was the only mathematician to be invited to take part in the program for two consecutive years. Moser lectured at more than one hundred Canadian and American Universities and gave talks at high schools.

Moser was born in Austria in 1921, and his family emigrated to Canada when he was a child. Growing up in Winnipeg, Moser attended the University of Manitoba to study mathematics before moving to the University of Toronto for his Master's. He then completed a Ph.D. at The University of North Carolina at Chapel Hill. Moser spent most of his career at the University of Alberta. He married his wife, Eva, in 1946 and had four children, Barbara, Melanie, David, and Sheryl. Moser was affected by a serious heart condition for much of his life and passed away at forty-eight years old in 1970.

Leo Moser was born in Vienna, Austria, on April 11, 1921. His parents, Laura Feurstein and Robert Moser, married in Austria before Leo was born and emigrated to Canada with the family settling in Winnipeg. Leo was young when they emigrated, attending primary and secondary school there. Laura and Robert Moser had another son, William, on September 5, 1927. Robert Moser would also go on to become a mathematician, citing his older brother, Leo, as the biggest influence why, stating:

Leo was my first mathematical mentor. When I was a teenager he was already showing me the pleasures of mathematics. Leo's enthusiasm for mathematics was boundless, his creativity was ingenious, and creating problems was his specialty.

After finishing secondary school, Moser remained in Winnipeg, attending the University of Manitoba to study mathematics. He graduated with a B.Sc. in 1944, transferring to the University of Toronto, where he studied for his Master's degree, graduating in 1945.

For his doctoral studies, Moser moved to America and attended the University of North Carolina at Chapel Hill to work with Alfred Brauer. Under Brauer's supervision, Moser began research in number theory, publishing a number of papers before submitting his dissertation. One of these papers, "A theorem on the distribution of primes," was published in 1949 and appeared in the American Mathematical Monthly with Moser, giving a short proof stating that for any positive integer (r) there is a prime (p) satisfying:

The paper went on to indicate how the proof could be modified to give a proof of Bertrand's theorem. Moser submitted his dissertation in 1951, titled "On Sets of Integers which Contain No Three in Arithmetical Progression and on Sets of Distances Determined by Finite Point Sets."

After a short period working at the Texas Technical College, Moser moved back to Canada, taking a position at the University of Alberta in 1951, where he would remain for the rest of his career, becoming a professor in the mathematics department. Ambikeshwar Sharma described visiting Moser at the University of Alberta in Edmonton shortly after:

the department was small and had no separate building to itself, Leo Moser was the centre of activity discussing problems with any one who cared to listen... In the faculty lounge in the department, he would discuss problems or tell anecdotes to his students. Anyone who came in and wanted to listen was welcome. ... he had started a book of problems in which visitors and anyone who has a problem could put his problem in

Professor Moser was known throughout the mathematical community as a significant researcher and excellent lecturer. His lectures were said to be delivered with vigor, humor, and clarity. During the academic year 1962-63, Moser traveled on a lecture tour for the Mathematical Association of America (MAA), becoming the only mathematician during his career to be invited to take part in the MAA's lecture program for two consecutive years. Between the MAA lecture program and a similar one sponsored by the American National Science Foundation, Moser lectured at more than one hundred Canadian and American Universities and gave talks to high school students, high school teachers, and professional mathematicians.

While working at The University of Alberta, Moser had a number of doctoral students, including the following:

• John Moon (1962)
• Mangesh Murdeshwar (1964)
• Harvey Abbott (1965)
• Marilyn Faulkner (1966)
• Robert MacLeod (1966)
• James Riddell (1967)

Mathematicians typically use factorial or exponentiation to express very large numbers. However, these functions lead to lengthy notations when dealing with extremely large numbers. To solve the problem, a number of mathematicians have offered alternative notations extensible to extremely large numbers. Hugo Steinhaus developed a form of notation where:

Examples of the Steinhaus notation.

• Triangle(n) = nⁿ
• Square(n) = n inside n triangles
• Circle(n) = n inside n squares

These are typically written as the variable (e.g., n) inside the shape.

Moser proposed an extension to Steinhaus's notation using an increasing sequence of polygons to deal with larger numbers. Moser's new polygon notation added additional levels above the square with polygons that have increasing numbers of sides, i.e., pentagon, hexagon, etc.

Using the Steinhaus-Moser notation Square(2), 2 inside 2 triangles, is equal to:

Square(3) is equal to 3 inside 3 triangles or (3^3)^(3^3)^(3^3)^(3^3), which is equal to 3^(3^85) a number with over 17 duodecillian digits.

Pentagon(2), 2 inside 2 squares or 256 inside 256 triangles. This number is known as the mega, a number too large to have any physical meaning. The first triangle of 256 has 617 digits number with the exact value:

32,317,006,071,311,007,300,714,876,688,669,951,960,444,102,669,715,484,032,130,345,427,524,655,138,867,890,893,197,201,411,522,913,463,688,717,960,921,898,019,494,119,559,150,490,921,095,088,152,386,448,283,120,630,877,367,300,996,091,750,197,750,389,652,106,796,057,638,384,067,568,276,792,218,642,619,756,161,838,094,338,476,170,470,581,645,852,036,305,042,887,575,891,541,065,808,607,552,399,123,930,385,521,914,333,389,668,342,420,684,974,786,564,569,494,856,176,035,326,322,058,077,805,659,331,026,192,708,460,314,150,258,592,864,177,116,725,943,603,718,461,857,357,598,351,152,301,645,904,403,697,613,233,287,231,227,125,684,710,820,209,725,157,101,726,931,323,469,678,542,580,656,697,935,045,997,268,352,998,638,215,525,166,389,437,335,543,602,135,433,229,604,645,318,478,604,952,148,193,555,853,611,059,596,230,656

Moser's number, named after Leo Moser, is a number equal to 2 in a megagon (million-sided polygon) in Steinhaus-Moser notation.

In 1966, Moser published a list of fifty open problems in geometry titled "Poorly formulated unsolved problems of combinatorial geometry." Problem 11 in the list states:

What is the (convex) region of smallest area which will accommodate (or cover) every planar arc of length 1?

Another way of describing Moser's work problem is: what is the size and shape of the flat surface of minimum area that can be used as a hammer head, which upon striking any given planar worm will hit it simultaneously from stem to stern? In 1992, mathematicians showed that a solution to the Worm Problem of Moser is a region with an area less than 0.27524. However, the exact solution remains unsolved.

Over his career, Moser published over one hundred papers, many in journals dedicated to mathematics teaching. Moser had several papers published before submitting his dissertation:

• "On the sum of digits of powers" (1947)
• "Some equations involving Euler's totient" (1949)
• "Linked rods and continued fractions" (1949)
• "On the danger of induction" (1949)
• "A theorem on the distribution of primes" (1949)

Moser loved problem-based mathematical questions, and while number theory was his first area of attention, he also spent significant time working on combinatorial problems and problems in graph theory. Listed below is a selection of papers published over his career (some single-authored papers, some co-authored with Joachim Lambek or Max Wyman):

• "On the different distances determined by n points" (1952)
• "Note on a combinatorial formula of Mendelsohn" (1953)
• "On the distribution of Pythagorean triangles" (1955)
• "An asymptotic formula for the Bell numbers" (1955)
• "Rational analogues of the logarithm function" (1956).

Notable papers later in his career include those below:

• "An extremal problem in matrix theory" (1966), with JW Moon
• "Some packing and covering theorems" (1967), with J W Moon
• "On packing of squares and cubes" (1968), with A Meir
• "An extremal problem in graph theory" (1970), with Paul Erdős

In 1957, Moser also published a book, An Introduction to the Theory of Numbers," an expanded version of a series of lectures for graduate students covering a range of topics, including compositions and partitions, arithmetic functions, distribution of primes, irrational numbers, congruences, diophantine equations, combinatorial number theory, and geometry of numbers. The book is distributed by The Trillia Group and is freely available under a Creative Commons license.

Leo Moser married his wife, Eva, on September 10, 1946. The couple had four children, Barbara, Melanie, David, and Sheryl. Moser suffered serious heart damage early in his life and was plagued by ill health for most of his life. However, he was said to never complain or ask for special consideration or sympathy. Many people close to him were unaware of his condition. Moser was described as a warm person with a sense of humor and a large number of amusing stories. He also always had a store of puzzles, both mathematical and non-mathematical, to share and discuss with people and was said to be willing to spend time with anyone, offering help wherever it was needed. Moser had interests outside of mathematics, most notably magic and chess.

In 1941, Moser won the Bernard Freedman Trophy for the highest finish by a new entrant in the Canadian Championship. The last round game, Leo Moser vs. D. Abraham Yanofsky won the Brilliancy Prize for the tournament, and it was Yanofsky's only loss. Moser won the Alberta Open in 1958 and was crowned top Winnipeg, Edmonton player Alberta champion five times. He was known to spend many hours playing chess with high school students. He often played fifty games simultaneously, winning all but one or two.

Moser was also a fan of mathematical limericks. After a lecture at the University of Nebraska on May 4, 1963, Moser recorded a number of mathematical limericks by him and others for fellow mathematician Walter Mientka. After his passing, Mientka published a paper presenting a transcription of these limericks in June 1972. Examples include the following:

The binary system is fun For with it strange things can be done And two as you know Is a one and an oh And five is one hundred and one.

There was a young fellow named Ben Who could only count modulo ten He said when I go Past my last little toe I shall have to start over again.

Moser suffered from a serious heart condition, significantly affecting his health during his life, although few around him were aware of it. In 1967, doctors gave Moser two months to live unless he underwent major heart surgery. A partial success, the operation led to three more years of life, with Moser passing away at forty-eight years old on February 9, 1970, in Edmonton Alberta. On May 19, 1970, President of The University of Alberta and collaborator Max Wyman published a short biography of Leo Moser, describing his talents for research:

Leo Moser had mathematical talents that were unique among the many mathmeticians I have known personally or through their published work. Foremost among these talents were intuition, simplicity, ingenuity, and clarity. Above all he demonstrated an amazing ability to analyse the most complex of mathematical problems, and by intuitive heuristic arguments outline ways in which it might be attacked, and quite often to estimate accurately what the answer was likely to be. Since Leo Moser and I worked closely together on research for a period of about five years, I can give personal witness to the contribution his unique abilities made to the solution of difficult mathematical problems.

And teaching:

Leo Moser was one of the best teachers of mathematics the University of Alberta has had. Gifted with many of the attributes of a successful actor, he made of mathematics a fresh and living subject for very large classes of students just beginning their mathematical studies. At the same time, he could completely captivate smaller groups of professional mathematicians who are prone to be critical of the work of other people.