1976

Using 1,200 hours of computer time, Kenneth Appel

and Wolfgang Haken prove the

FOUR

-

COLOR

THEOREM

.

1978

Ronald Rivest, Adi Shamir, and Leonard Adleman

develop the RSA public-key encryption system.

1979

Benoit Mandelbrot discovers the M

ANDELBROT SET

. He

later founds the field of

FRACTAL

geometry.

Under the instigation of Pope John Paul II, the Roman

Catholic Church opens its files on the Galilean trials.

The Church reverses its 17th-century condemnation of

the scholar in 1992.

1994

A

NDREW

W

ILES

, with the assistance of Richard Taylor,

proves F

ERMAT

’

S LAST THEOREM

.

1998

Thomas Hales uses computer methods to establish

J

OHANNES

K

EPLER

’s 1611 conjecture that the cubic

close packing and the hexagonal close packing of

spheres are the densest packings of spheres. Mathe-

maticians are unable to verify the proof without the

aid of a computer.

1999

Hales proves the “honeycomb conjecture,” which states

that any partition of the plane into regions of equal area

has perimeter equal to the design of a honeycomb.

Yasumasa Kanada of the University of Tokyo computes

πto 206,158,430,000 decimal places.

2000

Michael Hutchings, Frank Morgan, Manuel Ritoré,

and Antonio Ros prove the outstanding “double bub-

ble” conjecture in the theory of

SOAP BUBBLES

. They

establish that the design of minimal surface area that

encloses two fixed volumes is indeed the “double bub-

ble” configuration one observes in nature.

2003

Tomás Oliveira e Silva verifies that G

OLDBACH

’

S CON

-

JECTURE

holds true for all even numbers between four

and 6 ×1016.

Michael Shafer discovers that the 6,320,430-digit num-

ber 220,996,011 – 1 is

PRIME

. It is the largest prime and

the 40th M

ERSENNE PRIME

known to this date.

2004

Martin Dunwoody announces to the mathematical

community that he may have proved the Poincaré con-

jecture. Mathematicians await the details of his proof.

Appendix I 545