338 Mersenne prime

that it is because of Mersenne’s efforts that Galileo’s

work became known outside of Italy.

Mersenne played an important role in 17th-century

science, not only for his contribution to number theory

and mechanics, but also for his service as a channel of

communication between mathematicians: scholars

would write to Mersenne for the sole purpose of hav-

ing their ideas disseminated. Letters from over 78 dif-

ferent correspondents, including P

IERRE DE

F

ERMAT

(1601–65), Galileo, and Christiaan Huygens were dis-

covered in his monastery cell after his death.

In 1644 Mersenne published Cogitata physico-mathe-

matica (Physico-mathematical thoughts), his famous text

on number theory. Mersenne was particularly interested

in finding a formula that would generate all the prime

numbers. Although he failed in this effort, his work did

lead him to consider those prime numbers pfor which 2p

– 1 is also prime, now called the M

ERSENNE PRIME

s.

These numbers have proved to be of significant impor-

tance in several different branches of number theory.

Mersenne died in Paris, France, on September 1,

1648.

Mersenne prime A

PRIME

number of the form 2n– 1

is called a Mersenne prime. For example, 23– 1 = 7

and 27– 1 = 127 and are Mersenne primes. These num-

bers were studied by French philosopher and mathe-

matician M

ARIN

M

ERSENNE

(1588–1648) in his

attempts to find a formula that would generate all

prime numbers. Although he failed in this pursuit,

primes of this form are today named in his honor.

Note that if nfactors as n= ab, then the quantity 2n

– 1 also factors: 2ab –1 = (2a–1)(2a(b–1) + 2a(b–2) +…+ 2a

+ 1). Thus in order for 2n–1 to be prime, it must be the

case that nis prime. However, not every prime number

nleads to a Mersenne prime. For example, although n=

11 is prime, 211 – 1 = 2047 = 23 ×89 is not. The first

few Mersenne primes are 3, 7, 31, 127, 8191, 131071,

524287, 2147483647, … corresponding to the prime

values nequal to 2, 3, 5, 7, 13, 17, 19, 31,…

Only 40 Mersenne primes are currently known, yet

despite their scarcity, they still remain a fruitful source

of large prime numbers. Almost certainly, when a news-

paper proclaims that a new “largest” prime has been

found, it turns out to be of the form 2n– 1. For exam-

ple, the largest known prime as of the year 2004 is the

Mersenne prime with n= 20,996,011. It is a prime

number over 6 million digits long. Mersenne primes are

intimately connected with

PERFECT NUMBER

s.

See also

DIFFERENCE OF TWO CUBES

.

midpoint A point on a line segment dividing the

length of that segment into two equal parts is called the

midpoint of the segment. If two points in a plane have

C

ARTESIAN COORDINATES

P= (x1, y1) and Q= (x2, y2),

then the midpoint Mof the segment connecting Pto Q

has coordinates . Similarly, for

two points P= (x1,y1,z1) and Q= (x2,y2,z2) in three-

dimensional space, the coordinates of the midpoint M

of the line segment connecting them are given by:

.

A line through the midpoint of a line segment and

PERPENDICULAR

to that segment is called a perpendicu-

lar bisector. A study of

EQUIDISTANT

points shows that

the three perpendicular bisectors of the three sides of

any triangle meet at a single point (called the circum-

center of the triangle). The three

MEDIAN

s

OF A TRIAN

-

GLE

are also

CONCURRENT

.

The circle-midpoint theorem asserts that if one

draws a circle Cin the plane and selects a point P

anywhere in the plane, then all the midpoints of line

segments connecting Pto points on the circle form a

circle of half the original radius. This can be seen

valid as follows:

Assume the circle has radius rand is positioned

about the origin of a Cartesian coordinate sys-

tem. Then any point Qon the circle has coor-

dinates Q= (rcosθ, rsinθ), for some value θ. If

the coordinates of Pare given by P= (a,b),

then the coordinate of the midpoint Mis

. As θvaries, this

describes a circle of radius with center

(,).

See also

BISECTOR

;

CIRCLE THEOREMS

.

midrange See

STATISTICS

:

DESCRIPTIVE

.

Möbius, August Ferdinand (1790–1868) German

Topology, Astronomy Born on November 17, 1790,

b

–

2

a

–

2

r

–

2