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 having their ideas disseminated. Letters from over 78 different correspondents, including PIERRE DE FERMAT
(1601–65), Galileo, and Christiaan Huygens were discovered in his monastery cell after his death.
In 1644 Mersenne published Cogitata physico-mathematica (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 p for which 2
p
– 1 is also prime, now called the MERSENNE PRIMEs.
These numbers have proved to be of significant importance in several different branches of number theory.
Mersenne died in Paris, France, on September 1,
1648.
Mersenne prime A PRIME number of the form 2
n – 1
is called a Mersenne prime. For example, 2
3 – 1 = 7
and 2
7 – 1 = 127 and are Mersenne primes. These numbers were studied by French philosopher and mathematician MARIN MERSENNE (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 n factors as n = ab, then the quantity 2
n
– 1 also factors: 2
ab –1 = (2
a –1)(2
a(b–1) + 2
a(b–2) +…+ 2
a
+ 1). Thus in order for 2
n –1 to be prime, it must be the
case that n is prime. However, not every prime number
n leads to a Mersenne prime. For example, although n =
11 is prime, 2
11 – 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 n equal 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 newspaper proclaims that a new “largest” prime has been
found, it turns out to be of the form 2
n – 1. For example, 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 NUMBERs.
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
CARTESIAN COORDINATES P = (x 1 , y 1 ) and Q = (x 2 , y 2 ),
then the midpoint M of the segment connecting P to Q
has coordinates
. Similarly, for
two points P = (x 1 ,y 1 ,z 1 ) and Q = (x 2 ,y 2 ,z 2 ) in threedimensional 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 perpendicular 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 circumcenter of the triangle). The three MEDIANs OF A TRIANGLE are also CONCURRENT.
The circle-midpoint theorem asserts that if one
draws a circle C in the plane and selects a point P
anywhere in the plane, then all the midpoints of line
segments connecting P to points on the circle form a
circle of half the original radius. This can be seen
valid as follows:
Assume the circle has radius r and is positioned
about the origin of a Cartesian coordinate system. Then any point Q on the circle has coordinates Q = (r cosθ, r sinθ), for some value θ. If
the coordinates of P are given by P = (a,b),
then the coordinate of the midpoint M is
. As θ varies, 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
M
a r
b r
= (
cos ,
sin )
2 2
2 2
+
+
θ
θ
M
x x y y z z
=
+
+
+
(
,
,
)
1
2
1
2
1
2
2
2
2
M
x x y y
=
+
+
(
,
)
1
2
1
2
2
2
338 Mersenne prime
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 having their ideas disseminated. Letters from over 78 different correspondents, including PIERRE DE FERMAT
(1601–65), Galileo, and Christiaan Huygens were discovered in his monastery cell after his death.
In 1644 Mersenne published Cogitata physico-mathematica (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 p for which 2
p
– 1 is also prime, now called the MERSENNE PRIMEs.
These numbers have proved to be of significant importance in several different branches of number theory.
Mersenne died in Paris, France, on September 1,
1648.
Mersenne prime A PRIME number of the form 2
n – 1
is called a Mersenne prime. For example, 2
3 – 1 = 7
and 2
7 – 1 = 127 and are Mersenne primes. These numbers were studied by French philosopher and mathematician MARIN MERSENNE (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 n factors as n = ab, then the quantity 2
n
– 1 also factors: 2
ab –1 = (2
a –1)(2
a(b–1) + 2
a(b–2) +…+ 2
a
+ 1). Thus in order for 2
n –1 to be prime, it must be the
case that n is prime. However, not every prime number
n leads to a Mersenne prime. For example, although n =
11 is prime, 2
11 – 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 n equal 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 newspaper proclaims that a new “largest” prime has been
found, it turns out to be of the form 2
n – 1. For example, 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 NUMBERs.
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
CARTESIAN COORDINATES P = (x 1 , y 1 ) and Q = (x 2 , y 2 ),
then the midpoint M of the segment connecting P to Q
has coordinates
. Similarly, for
two points P = (x 1 ,y 1 ,z 1 ) and Q = (x 2 ,y 2 ,z 2 ) in threedimensional 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 perpendicular 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 circumcenter of the triangle). The three MEDIANs OF A TRIANGLE are also CONCURRENT.
The circle-midpoint theorem asserts that if one
draws a circle C in the plane and selects a point P
anywhere in the plane, then all the midpoints of line
segments connecting P to points on the circle form a
circle of half the original radius. This can be seen
valid as follows:
Assume the circle has radius r and is positioned
about the origin of a Cartesian coordinate system. Then any point Q on the circle has coordinates Q = (r cosθ, r sinθ), for some value θ. If
the coordinates of P are given by P = (a,b),
then the coordinate of the midpoint M is
. As θ varies, 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
M
a r
b r
= (
cos ,
sin )
2 2
2 2
+
+
θ
θ
M
x x y y z z
=
+
+
+
(
,
,
)
1
2
1
2
1
2
2
2
2
M
x x y y
=
+
+
(
,
)
1
2
1
2
2
2
338 Mersenne prime