239 (two hundred
ndthirty-nine) is the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
following
238
__NOTOC__
Year 238 ( CCXXXVIII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Pius and Pontianus (or, less frequently, year 991 ''Ab urbe condita''). The denomination 238 ...
and preceding
240.
Properties
239 is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
. The next is 241, with which it forms a pair of
twin prime
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime' ...
s; hence, it is also a
Chen prime
In mathematics, a prime number ''p'' is called a Chen prime if ''p'' + 2 is either a prime or a product of two primes (also called a semiprime). The even number 2''p'' + 2 therefore satisfies Chen's theorem.
The Chen primes are named a ...
. 239 is a
Sophie Germain prime
In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +&nbs ...
and a
Newman–Shanks–Williams prime. It is an
Eisenstein prime
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form
: z = a + b\omega ,
where and are integers and
: \omega = \frac ...
with no imaginary part and real part of the form 3''n'' − 1 (with no exponentiation implied). 239 is a factor of the
repdigit
In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit".
Ex ...
1111111, with the other prime factor being 4649. 239 is also a
happy number
In number theory, a happy number is a number which eventually reaches 1 when the number is replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1^2+3^2=10, and 1^2+0^2=1. On the other hand, 4 is not a happy ...
.
239 is the smallest positive integer ''d'' such that the imaginary
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of Degree of a field extension, degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free ...
Q() has
class number = 15.
HAKMEM
HAKMEM, alternatively known as AI Memo 239, is a February 1972 "memo" ( technical report) of the MIT AI Lab containing a wide variety of hacks, including useful and clever algorithms for mathematical computation, some number theory and schemat ...
entry
HAKMEM
HAKMEM, alternatively known as AI Memo 239, is a February 1972 "memo" ( technical report) of the MIT AI Lab containing a wide variety of hacks, including useful and clever algorithms for mathematical computation, some number theory and schemat ...
(incidentally AI memo 239 of the
MIT AI Lab
Computer Science and Artificial Intelligence Laboratory (CSAIL) is a research institute at the Massachusetts Institute of Technology (MIT) formed by the 2003 merger of the Laboratory for Computer Science (LCS) and the Artificial Intelligence Lab ...
) included an item on the properties of 239, including these:
* When expressing 239 as a sum of
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
s, 4 squares are required, which is the maximum that any integer can require; it also needs the maximum number (9) of positive
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
s (23 is the only other such integer), and the maximum number (19) of fourth powers.
* 239/
169 is a
convergent of the
simple continued fraction
A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence \ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fr ...
of the
square root of 2
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...
, so that 239
2 = 2 · 169
2 − 1.
* Related to the above, = 45
°.
* 239 · 4649 = 1111111, so 1/239 = 0.0041841 repeating, with period 7.
* 239 can be written as ''b''
''n'' − ''b''
''m'' − 1 for ''b'' = 2, 3, and 4, a fact evidenced by its
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two values (0 and 1) for each digit
* Binary function, a function that takes two arguments
* Binary operation, a mathematical op ...
representation 11101111,
ternary representation 22212, and
quaternary
The Quaternary ( ) is the current and most recent of the three periods of the Cenozoic Era in the geologic time scale of the International Commission on Stratigraphy (ICS), as well as the current and most recent of the twelve periods of the ...
representation 3233.
* There are 239 primes < 1500.
* 239 is the largest integer ''n'' whose
factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times ...
can be written as the product of distinct factors between ''n'' + 1 and 2''n'', both included.
* The only solutions of the Diophantine equation ''y''
2 + 1 = 2''x''
4 in positive integers are (''x'', ''y'') = (1, 1) or (13, 239).
References
{{DEFAULTSORT:239 (Number)
Integers