Centered square number
Encyclopedia
In elementary number theory, a centered square number is a centered figurate number
that gives the number of dots in a square
with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given city block distance
of the center dot on a regular square lattice
. While centered square numbers, like figurate number
s in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics
for their elegant geometric and arithmetic properties.
The figures for the first four centered square numbers are shown below:
In other words, a centered square number is the sum of two consecutive square number
s. The following pattern demonstrates this formula:
The formula can also be expressed as
that is, n th centered square number is half of n th odd square number plus one, as illustrated below:
Like all centered polygonal numbers, centered square numbers can also be expressed in terms of triangular number
s:
where
is the nth triangular number. This can be easily seen by removing the center dot and dividing the rest of the figure into four triangles, as below:
The difference between two consecutive octahedral number
s is a centered square number (Conway and Guy, p.50).
All centered square numbers are odd, and in base 10 one can notice the one's digits follows the pattern 1-5-3-5-1.
All centered square numbers and their divisors have a remainder of one when divided by four. Hence all centered square numbers and their divisors end with digits 1 or 5 in base 6
, 8
or 12
.
All centered square numbers except 1 are the third term of a Leg-Hypotenuse Pythagorean triple
(for example, 3-4-5, 5-12-13).
. Unlike regular square number
s, which are never prime, quite a few of the centered square numbers are prime. The first few centered square primes are:
Figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions...
that gives the number of dots in a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given city block distance
Taxicab geometry
Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their coordinates...
of the center dot on a regular square lattice
Square lattice
In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice. It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group is known symbolically as p4m.Two...
. While centered square numbers, like figurate number
Figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions...
s in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...
for their elegant geometric and arithmetic properties.
The figures for the first four centered square numbers are shown below:
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Relationships with other figurate numbers
The nth centered square number is given by the formulaIn other words, a centered square number is the sum of two consecutive square number
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
s. The following pattern demonstrates this formula:
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The formula can also be expressed as
that is, n th centered square number is half of n th odd square number plus one, as illustrated below:
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Like all centered polygonal numbers, centered square numbers can also be expressed in terms of triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
s:
where
is the nth triangular number. This can be easily seen by removing the center dot and dividing the rest of the figure into four triangles, as below:
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The difference between two consecutive octahedral number
Octahedral number
In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres...
s is a centered square number (Conway and Guy, p.50).
Properties
The first few centered square numbers are:- 1, 5, 1313 (number)13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....
, 2525 (number)25 is the natural number following 24 and preceding 26.-In mathematics:It is a square number, being 5² = 5 × 5. It is the smallest square that is also a sum of two squares: 25 = 3² + 4²...
, 4141 (number)41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...
, 6161 (number)61 is the natural number following 60 and preceding 62.-In mathematics:It is the 18th prime number. The previous is 59, with which it comprises a twin prime. Sixty-one is a cuban prime of the form p = / , x = y + 1....
, 8585 (number)85 is the natural number following 84 and preceding 86.-In mathematics:85 is an octahedral number, a centered triangular number, a centered square number, a decagonal number, and a Smith number....
, 113113 (number)113 is the natural number following 112 and preceding 114.-In mathematics:One hundred [and] thirteen is the 30th prime number, following 109 and preceding 127, a Sophie Germain prime, a Chen prime and a Proth prime as it is a prime number of the form 7 × 24 + 1...
, 145145 (number)145 is the natural number following 144 and preceding 146.- In mathematics :* Although composite, 145 is a pseudoprime.* Given 145, the Mertens function returns 0.* 145 is a pentagonal number and a centered square number....
, 181, 221, 265, 313313 (number)313 is an integer following 312 and preceding 314. 313 is a:* prime number* twin prime with 311* centered square number* long prime* pythagorean prime* regular prime* palindromic prime in both decimal and binary.* truncatable prime* happy number...
, 365365 (number)365 is the natural number following 364 and preceding 366.-Mathematics:365 is a semiprime centered square number...
, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, … .
All centered square numbers are odd, and in base 10 one can notice the one's digits follows the pattern 1-5-3-5-1.
All centered square numbers and their divisors have a remainder of one when divided by four. Hence all centered square numbers and their divisors end with digits 1 or 5 in base 6
Senary
In mathematics, a senary numeral system is a base- numeral system.Senary may be considered useful in the study of prime numbers since all primes other than 2 and 3, when expressed in base-six, have 1 or 5 as the final digit...
, 8
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
or 12
Duodecimal
The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written as 'A', 'T' or 'X', and the number eleven as 'B' or 'E'...
.
All centered square numbers except 1 are the third term of a Leg-Hypotenuse Pythagorean triple
Pythagorean triple
A Pythagorean triple consists of three positive integers a, b, and c, such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are pairwise coprime...
(for example, 3-4-5, 5-12-13).
Centered square prime
A centered square prime is a centered square number that is primePrime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. Unlike regular square number
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
s, which are never prime, quite a few of the centered square numbers are prime. The first few centered square primes are:
- 5, 13, 41, 61, 113, 181, 313, 421, 613, 761, 1013, 1201, 1301, 1741, 1861, 2113, 2381, 2521, 3121, 3613, … .