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Posts Tagged ‘Ronald Graham

“No part of mathematics is ever, in the long run, ‘useless’.”*…

The number 1 can be written as a sum of distinct unit fractions, such as 1/2 + 1/3 + 1/12 + 1/18 + 1/36…

Number theorists are always looking for hidden structure. And when confronted by a numerical pattern that seems unavoidable, they test its mettle, trying hard — and often failing — to devise situations in which a given pattern cannot appear.

One of the latest results to demonstrate the resilience of such patterns, by Thomas Bloom of the University of Oxford, answers a question with roots that extend all the way back to ancient Egypt.

“It might be the oldest problem ever,” said Carl Pomerance of Dartmouth College.

The question involves fractions that feature a 1 in their numerator, like 1/2, 1/7 or 1/122. These “unit fractions” were especially important to the ancient Egyptians because they were the only types of fractions their number system contained: With the exception of a single symbol for 23, they could only express more complicated fractions (like 3/4) as sums of unit fractions (1/2 + 1/4).

The modern-day interest in such sums got a boost in the 1970s, when Paul Erdős and Ronald Graham asked how hard it might be to engineer sets of whole numbers that don’t contain a subset whose reciprocals add to 1. For instance, the set {2, 3, 6, 9, 13} fails this test: It contains the subset {2, 3, 6}, whose reciprocals are the unit fractions 1/2, 1/3 and 1/6 — which sum to 1.

More exactly, Erdős and Graham conjectured that any set that samples some sufficiently large, positive proportion of the whole numbers — it could be 20% or 1% or 0.001% — must contain a subset whose reciprocals add to 1. If the initial set satisfies that simple condition of sampling enough whole numbers (known as having “positive density”), then even if its members were deliberately chosen to make it difficult to find that subset, the subset would nonetheless have to exist.

“I just thought this was an impossible question that no one in their right mind could possibly ever do,” said Andrew Granville of the University of Montreal. “I didn’t see any obvious tool that could attack it.”…

Bloom, building on work by Ernie Croot, found that tool. The ubiquity of ways to represent whole numbers as sums of fractions: “Math’s ‘Oldest Problem Ever’ Gets a New Answer,” by Jordana Cepelewicz (@jordanacep) in @QuantaMagazine.

* “No part of mathematics is ever, in the long run, ‘useless.’ Most of number theory has very few ‘practical’ applications. That does not reduce its importance, and if anything it enhances its fascination. No one can predict when what seems to be a most obscure theorem may suddenly be called upon to play some vital and hitherto unsuspected role.” – C. Stanley Ogilvy, Excursions in Number Theory

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As we recombine, we might send carefully-calculated birthday greetings to Ulugh Beg (or, officially, Mīrzā Muhammad Tāraghay bin Shāhrukh); he was born on this date in 1394. A Timurid sultan with a hearty interest in science and the arts, he is better remembered as an astronomer and mathematician.

The most important observational astronomer of the 15th century, he built the great Ulugh Beg Observatory in Samarkand between 1424 and 1429– considered by scholars to have been one of the finest observatories in the Islamic world at the time and the largest in Central Asia. In his observations he discovered a number of errors in the computations of the 2nd-century Alexandrian astronomer Ptolemy, whose figures were still being used. His star map of 994 stars was the first new one since Hipparchus. Among his contributions to mathematics were trigonometric tables of sine and tangent values correct to at least eight decimal places.

Ulugh Beg’s Statue in Samarkand, Uzbekistan

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“Mathematics is the art of giving the same name to different things”*…

Freeman Dyson was a translator: he turned physics into math, and those subjects into English for the general public.

Mathematics, like life, is complicated. But, for those who do mathematics, it is a source of joy. “The main thing is just astonishment that there’s such a rich world out there—a wonderful, abstract, very beautiful, simple world,” [John] Conway said. “It’s like Pizarro standing on the shores of the Pacific or whatever. . . . I can sit here in this chair and go on a voyage of exploration. A very different voyage of exploration, but, still, there are things to be discovered, things to be seen, that you can quite easily be the first person ever to see.”

So many of us now sit in our rooms, bound in space while time drips away. It can be a bit of a comfort to know that, as long as you are able to sit still and think, your creative spirit can be an engine of exploration. On their journeys, these playful, curious mathematicians discovered Monsters and numbers so large that they can hardly be written down. We’re grateful for the lively stories of their expeditions, and for the thinkers who led them. They’ll be missed…

John Conway, Ronald Graham, and Freeman Dyson all explored the world with their minds. Dan Rockmore (@dan_rockmore) celebrates “Three Mathematicians We Lost in 2020.”

Special bonus: an interview with an heir to Dyson– that’s to say, an important mathematician who’s also a gifted “translator”– Steven Strogatz.

* Henri Poincaré

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As we share their amazement, we might we might spare a thought for Max Born; he died on this date in 1970.  A German physicist and Nobel Laureate, he coined the phrase “quantum mechanics” to describe the field in which he made his greatest contributions.  But beyond his accomplishments as a practitioner, he was a master teacher whose students included Enrico Fermi and Werner Heisenberg– both of whom became Nobel Laureates before their mentor– and  J. Robert Oppenheimer.

Less well-known is that Born, who died in 1970, was the grandfather of Australian phenom and definitive Sandy-portrayer Olivia Newton-John.

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