# Noam Elkies

Noam Elkies 2005
Noam Elkies 2007

Noam David Elkies (born August 25, 1966 in New York City ) is an American mathematician who deals with number theory and combinatorics.

## Life

In 1981 he won a gold medal at the 22nd International Mathematical Olympiad , with the maximum possible score of 42, one of the youngest participants to have done that. Elkies won the Putnam Fellowship three times as an undergraduate student, first in 1982 at the age of 16. He did his PhD in 1987 at Harvard with Barry Mazur and Benedict Gross with Supersingular primes of a given elliptic curve over a number field . In 1990 he became an assistant professor at Harvard, where he was given a full professorship in 1993 (at age 26, setting the previous record set by lawyer Alan Dershowitz ). In 1991 Elkies received the NAS Award for Initiatives in Research . In 2017 he was elected to the National Academy of Sciences .

## mathematics

In his dissertation he proved that for every elliptic curve E over the rational numbers there are infinitely many supersingular prime numbers ("supersingular" means in this case that the E modulo p considered is a supersingular elliptic curve over the finite field , i.e. one elliptic curve with an unusually large number of endomorphisms). ${\ displaystyle \ mathbb {F} _ {p}}$

In 1988 he gave a counterexample for Euler's conjecture about power sums of integers. He claimed that if

${\ displaystyle \ sum _ {i = 1} ^ {n} a_ {i} ^ {k} = b ^ {k}}$

${\ displaystyle n \ geq k}$should be. LJ Lander and TR Parkin had already given a counterexample for k = 5 in 1966, Elkies gave one for k = 4 (in 1988 Roger Frye found a smaller solution using computer methods based on Elkie's work).

At about the same time as Tetsuji Shioda , he developed the theory of the Mordell-Weil lattice in 1990, which the Mordell-Weil group (group of rational points of an elliptic curve or Abelian variety over a global body) treats as a lattice.

Elkies also worked on numerical / algorithmic problems in number theory of elliptic curves, particularly important for cryptographic applications. With AOL Atkin , he improved Schoof's algorithm for determining the number of rational points on elliptic curves.

Elkies is a fan of puzzle games and has also worked in the field of combinatorial games. He is also known for discovering many new and interesting configurations in John Conway's game Life . In the field of combinatorics he worked a. a. via grids, spherical packings and codes.

In 1994 he was invited speaker at the International Congress of Mathematicians in Zurich ( Linearized algebra ). In 2004 he received the Levi L. Conant Prize .

In 2003, together with Henry Cohn , he developed upper bounds for the closest packing of spheres in different dimensions through families of auxiliary functions, which showed, in particular for d = 8 (E8 lattice) and d = 24 (Leech lattice), that the lattices in question came very close to the closest packing of spheres. Proof of this was published in 2016 by Maryna Viazovska .

## chess

Elkies is an active student composer and a great master in solving chess compositions. He has composed more than 40 chess studies. In 1996 he became world champion in solving chess problems and studies in Tel Aviv .

He retired from tournament chess in his early twenties, after having fulfilled the standard for a National Master (2200 Elo points) of the US Federation with around 2260 Elo points .

Noam Elkie's
Internet Mailing List, 2004
 a b c d e f G H 8th 8th 7th 7th 6th 6th 5 5 4th 4th 3 3 2 2 1 1 a b c d e f G H
Who wins?

Solution:

In order to be able to answer the question below the diagram, the history of the development of the position shown must be clarified with a retro analysis .

White is in check and apparently it is checkmate . In that case Black would have won. Since the black pawn offers check and the white king cannot escape, this check can only be countered by capturing the pawn. This pawn, however, could only be captured en passant by the white pawn f5. In this case, black would be mated and white would have won.

A necessary prerequisite for capturing en passant is the double step of the pawn to be captured in the immediately preceding move. Because of the chess rule, the g-pawn must have moved last. The outcome of the game depends on whether the pawn moved from g7 or from g6 to g5.

This question can be decided unambiguously by finding the last move by White.

## music

Elkies has been composing music and playing the piano since he was three years old. He is interested in the applications of mathematics in music. Some of his plays have been broadcast on radio stations in Israel and the United States.

2. Elkies: On . In: Math. Comput. Volume 51, 1988, pp. 825-835.${\ displaystyle A ^ {4} + B ^ {4} + C ^ {4} = D ^ {4}}$