Noam Elkies

Noam Elkies: The Mathematician Who Broke a 200-Year-Old Conjecture at 21

In 1988, a 21-year-old graduate student found a counterexample to a conjecture Leonhard Euler had proposed in 1769 and that had survived two centuries of scrutiny by some of the greatest mathematicians in history. Noam Elkies did not just disprove Euler's sum of powers conjecture for fourth powers; he did it as a side project, in the middle of a career that would soon make him Harvard's youngest tenured professor, a chess grandmaster-caliber solver, and, by the account of his own choir director, a musician comparable to Bach.

A Child Fluent in Three Languages of Genius

Elkies was born in New York City on August 25, 1966, to an engineer father and a piano-teacher mother, and spent eight formative years living in Israel before the family returned to New York. He began composing music at age six and later trained in Juilliard's pre-college program during his teenage years, developing in parallel as a pianist and bass-baritone even as his mathematical gifts became impossible to miss. At fourteen he competed in the 1981 International Mathematical Olympiad and won a gold medal with a perfect score of 42, and at sixteen years and four months he became a Putnam Fellow — one of the top scorers in the notoriously difficult Putnam Mathematical Competition — a feat he would go on to repeat, becoming a three-time Putnam Fellow overall. He graduated from Stuyvesant High School in 1982 at fifteen, and in 1982 also presented a project at the Science Talent Search titled "Improved Lower Bound on the Greatest Element of a Sum-Distinct Set of Fixed Order," an early sign of the number-theoretic instincts that would define his career.

Columbia, Harvard, and a Conjecture Two Centuries Old

Elkies earned his bachelor's degree from Columbia University in 1985, graduating valedictorian, then moved to Harvard for his doctorate, completed in 1987 under the supervision of Benedict Gross and Barry Mazur, two of the leading figures in modern number theory. That same year, he proved that elliptic curves over the rational numbers are supersingular at infinitely many primes, a significant result in arithmetic geometry. But it was his 1988 discovery that made headlines well beyond mathematics departments: he found a counterexample disproving Euler's sum of powers conjecture for fourth powers, a claim that had stood unchallenged since 1769. The finding demonstrated that a specific equation involving four fourth powers summing to a fourth power was solvable, something Euler's conjecture had implied was impossible — overturning, with a single explicit example, two hundred years of mathematical consensus.

Youngest Full Professor in Harvard's History

Elkies joined Harvard's Society of Fellows as a junior fellow from 1987 to 1990, and in 1993, at the age of twenty-six, he was granted tenure, making him the youngest full professor in Harvard University's history. He has remained on the Harvard mathematics faculty since, continuing to produce results in number theory and arithmetic geometry, including co-developing the Schoof–Elkies–Atkin algorithm with A. O. L. Atkin, a widely used method for counting points on elliptic curves that underpins parts of modern cryptography. He has continued setting records in the field of elliptic curves for decades: he has discovered curves with exceptionally high rank, including one with rank at least 28 and, in August 2024, working with Zev Klagsbrun, a curve with rank at least 29 — extending a research thread he opened as a young mathematician across nearly four decades. In 1994 he was an invited speaker at the International Congress of Mathematicians in Zürich, one of the field's most selective honors, and he was elected to the National Academy of Sciences in 2017.

A Second Career at the Chessboard

Elkies's mathematical talent found an unusual second outlet in chess, though not primarily as a tournament player. He holds the National Master title from the United States Chess Federation, but his deeper renown in the chess world comes from problem composition and solving: he won the World Chess Solving Championship in 1996, a competition that tests not tournament strength but the specific, highly mathematical skill of finding forced solutions to constructed chess positions. One of his own composed chess problems was significant enough to be included in Mark Dvoretsky's widely respected endgame manual, a standard reference among serious chess students.

The Musician Alongside the Mathematician

Elkies's musical life ran continuously alongside his mathematics rather than trailing behind it. He formerly accompanied the Harvard Glee Club as a pianist, and its director offered a comparison rarely extended to any living musician, describing Elkies as resembling "a Bach or a Mozart" and praising his "gifted musicality, superior musicianship and sight-reading ability." He also rings bells at Harvard's Lowell House, an old campanology tradition, and has composed music throughout his adult life alongside his mathematical output.

Why Noam Is Called a Genius

Elkies's case for the word rests on an unusually concentrated overlap of achievements that are individually rare and almost never found together: a perfect-score Mathematical Olympiad gold medal at fourteen, a three-time Putnam Fellowship, a counterexample that overturned a 200-year-old conjecture at twenty-one, and tenure at Harvard at twenty-six, alongside a genuine competitive chess record and a musical ability that his own choir director compared unprompted to Bach and Mozart. That combination is not one narrow technical skill dressed up as broad brilliance — it spans pure mathematical insight, competitive problem-solving under pressure, and trained musical craft, each independently verified by people with the standing to judge it: Harvard's tenure committee, the World Chess Solving Championship judges, and a professional choir director. The honest complication is that "genius" in mathematics research is not usually a solitary flash; Elkies's major results, including the Euler counterexample, emerged from deep training under supervisors like Gross and Mazur and from a research tradition built by generations of number theorists before him, and his chess and music achievements, while genuinely elite, remain amateur-adjacent compared to full-time professionals in either field. What is not in dispute is the rare density of independently verified excellence across domains that typically demand a lifetime's specialization each.

Legacy

Elkies remains an active Harvard professor decades after his early records were set, still producing new results in elliptic curve theory into the 2020s and still training the next generation of number theorists. His Euler counterexample remains a fixture in number theory courses as a lesson in how a conjecture's long survival is no guarantee of its truth, and his Schoof–Elkies–Atkin algorithm continues to do quiet, unglamorous work inside cryptographic systems that depend on elliptic curves — a mathematician's insight from a graduate seminar now embedded in infrastructure most of its users will never know exists.

Achievements

• Multiple-time Putnam Fellow
• Youngest person to earn tenure at Harvard, age 26 (1993)
• Disproved a conjecture of Euler on sums of powers
• Internationally recognized composer of chess problems