Ron Graham, who has died aged 84, was one of the most productive American mathematicians of the past half century. Working in both applied and pure mathematics, often on the underpinnings of theoretical computer science and telecommunications, he spent two-thirds of his long career in industry, at Bell Labs in New Jersey, while the rest was at the University of California in San Diego.
Ron carried out pioneering research in discrete mathematics, or combinatorics, specifically in Ramsey theory, quasi-randomness, scheduling theory, and discrete and computational geometry, as well as in recreational mathematics and mathematical magic. He was also an engaging populariser of mathematics through extensive expository talks and writings.
However, his biggest claim to fame was Graham’s Number, which was named after him and which was, when the famed maths populariser Martin Gardner brought it to the attention of a wide audience in a 1977 column for Scientific American, the largest specific positive whole number to have been used in a mathematical proof, a fact acknowledged by the Guinness Book of Records in 1980.
Ron was born in Taft, California, the eldest of three children of Jane Anderson and her husband, Leo Graham. His father worked in the oil and shipbuilding industries and changed jobs so often that Ron was hardly ever in the same school two years in a row. After his parents divorced he lived in Florida with his mother.
Although he never officially completed high school, a scholarship allowed him to attend the University of Chicago for three years at the start of the 1950s. There he mastered gymnastics, but not mathematics; in his youth he was a competitive (and prize winning) trampolinist and an accomplished acrobat, unicyclist and juggler, able to comfortably keep seven balls in the air at one time. He spent a year at the University of California at Berkeley, ostensibly as an engineering student, before dropping out again and joining the US air force for a while. He finally earned a degree (in physics) from the University of Alaska Fairbanks in 1959. He then returned to Berkeley to do PhD research in number theory.
Ron joined Bell Labs (later AT&T Labs) in 1962, and over a period of 37 years rose to be chief scientist there, directing mathematical research during an era that saw the field transform how telecommunications was done. One of his greatest contributions at Bell was get to grips with the early mathematical headaches connected with the efficient routing of mobile telephone calls, which arise when a phone leaves one cell site and connects to another as its owner moves around.
Outside of his work, Ron befriended Paul Erdős, the most prolific mathematician in history, who roamed the world without the hindrance of a proper job. Over time he effectively became his travel agent and banker, as well as a close research collaborator; they wrote around 30 papers together.
In all he published about 400 papers, 100 of them with his fourth wife Fan Chung, with whom he also wrote an influential book, Erdős on Graphs: His Legacy of Unsolved Problems (1999). Having been elected president of the International Jugglers’ Association in 1972, over the years he also published a number of papers about the mathematics of juggling.
After Erdős died in 1996, Ron continued to administer cash prizes that Erdős had offered for particularly challenging maths problems. He also mentored legions of aspiring students in maths and computer science, and many in juggling and magic. “Some people think that mathematics is a serious business that must always be cold and dry; but we think mathematics is fun, and we aren’t ashamed to admit the fact,” he once said. “Why should a strict boundary line be drawn between work and play?”
Ron was elected to the National Academy of Sciences in 1985, and was president of both the American Mathematical Society (1993-94) and the Mathematical Association of America (2003-04). He was inducted as a fellow of the Association of Computing Machinery in 1999 “for seminal contributions to the analysis of algorithms”. He received the George Pólya prize in 1971, the Euler medal in 1993 and the AMS Leroy Steele prize for lifetime achievement in 2003.
With his athletic Californian demeanour Ron would not have looked out of place in the Beach Boys, but behind his seemingly relaxed outlook there was a relentless drive and work ethic. When asked about his productivity, he liked to remind people that “there are 24 hours in every day. And if that’s not enough, you’ve always got the nights.”
He is survived by Fan, whom he married in 1983, and by four children: Ché and Marc, from his first marriage, to Nancy Young, which ended in divorce, and Laura and Christy from a relationship with Joann Quinlivan.
• Ronald Lewis Graham, mathematician, born 31 October 1935; died 6 July 2020