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Topics referred to by the same term
perfect square is an element of algebraic structure that is equal to the square of another element. Square number, a perfect square integer. Perfect Square
Perfect_square
Mathematical problem
square into n {\displaystyle n} squares of one or two different sizes. A "perfect" squared square is a square such that each of the smaller squares has
Squaring_the_square
Product of an integer with itself
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with
Square_number
2004 video by R.E.M.
Perfect Square is a 2004 concert film of the alternative rock band R.E.M., filmed on July 19, 2003, at the Bowling Green in Wiesbaden, Germany. It was
Perfect_Square
Two-dimensional packing problem
{\displaystyle n} unit squares is known when n {\displaystyle n} is a perfect square, one less than a perfect square, or two less than a perfect square, as well as
Square_packing
A perfect rectangle is a rectangle that can be divided into squares of different sizes. If a perfect rectangle is specifically a square, it is analogously
Perfect_rectangle
(Mathematical) decomposition into a product
any field, where either –1, 2 or –2 is a square. In a finite field, the product of two non-squares is a square; this implies that the polynomial x 4 +
Factorization
Data
A most-perfect magic square of order n is a magic square containing the numbers 1 to n2 with two additional properties: Each 2 × 2 subsquare sums to 2s
Most-perfect_magic_square
Integer side lengths of a right triangle
c ± b {\displaystyle c\pm b} is a perfect square (for both choices of sign), and the other is twice a perfect square (for both choices of sign). Hence
Pythagorean_triple
Polynomial equation of degree 4
has a pair of folded perfect squares, one on each side of the equation. The two perfect squares balance each other. If two squares are equal, then the
Quartic_equation
American entertainment company
††† - Yen Press has the rights to series' digital release due to being a Square Enix title. 2001 Nights A, A Prime A.D. Police: Dead End City Abara Act-Age
Viz_Media
Algorithms for calculating square roots
S {\displaystyle S} . Since all square roots of natural numbers, other than of perfect squares, are irrational, square roots can usually only be computed
Square_root_algorithms
Positive integer that is an integer power of another positive integer
a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a
Perfect_power
Mathematical proof technique
divides a2 + b2. Prove that a2 + b2/ab + 1 is a perfect square. Fix some value k that is a non-square positive integer. Assume there exist positive integers
Vieta_jumping
Quadrilateral with four right angles
squaring.net. The lowest number of squares need for a perfect tiling of a rectangle is 9 and the lowest number needed for a perfect tilling a square is
Rectangle
Mathematical identity of polynomials
difference of two squares is one squared number (the number multiplied by itself) subtracted from another squared number. Every difference of squares may be factored
Difference_of_two_squares
Product of a number by itself
to squaring is quadratic. The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often
Square_(algebra)
Four basic unsolved problems about prime numbers
one prime between consecutive perfect squares? Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely
Landau's_problems
Number equal to the sum of its proper divisors
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number
Perfect_number
Triangle whose side lengths and area are integers
Heronian triangles with perfect square sides are connected to the Perfect cuboid problem. The existence of a solution to the Perfect cuboid problem is equivalent
Heronian_triangle
Number whose square is a given number
the square root of numbers having many digits. It was known to the ancient Greeks that square roots of positive integers that are not perfect squares are
Square_root
unlike the flags of Switzerland and the Vatican City, it is not a perfect square. In 1830, the flag, at that time non-officially, consisted of three
Flag_of_Belgium
Unique positive real number which when multiplied by itself gives 2
lemma can be used to show that two identical perfect squares can never be added to produce another perfect square. Suppose the contrary that 2 {\displaystyle
Square_root_of_2
Classical averages studied in ancient Greece
product is a perfect square, esp. a natural perfect square number. However, the mean is irrational when the product is not a perfect square, esp. when the
Pythagorean_means
Method for solving quadratic equations
{\displaystyle x^{2}+10x+28.} This quadratic is not a perfect square, since 28 is not the square of 5: ( x + 5 ) 2 = x 2 + 10 x + 25. {\displaystyle (x+5)^{2}\
Completing_the_square
Park in Fountain Valley, California, US
effort to create the park. The park derives its name from the near-perfect square of land that it occupies, bounded by Edinger and Warner Avenues on the
Mile_Square_Regional_Park
Greatest integer less than or equal to square root
{y}}} run forever on each input y {\displaystyle y} which is not a perfect square. Algorithms that compute ⌊ y ⌋ {\displaystyle \lfloor {\sqrt {y}}\rfloor
Integer_square_root
Figurate number
to just check if 24x + 1 is a perfect square. For non-generalized pentagonal numbers, in addition to the perfect square test, it is also required to check
Pentagonal_number
Set of positive integers such that the product of any two plus one is a perfect square
\ldots ,a_{m}\}} such that a i a j + 1 {\displaystyle a_{i}a_{j}+1} is a perfect square for any 1 ≤ i < j ≤ m . {\displaystyle 1\leq i<j\leq m.} A set of m
Diophantine_quintuple
Measure of readability
syllables in three 10-sentence samples, estimate the count's square root (from the nearest perfect square), and add 3. A 2010 study published in the Journal of
SMOG
Square of numbers with equal row, column and diagonal totals
of squares. Except for n ≤ 5, the enumeration of higher-order magic squares is still an open challenge. The enumeration of most-perfect magic squares of
Magic_square
Geographical quadrant
[clarification needed] Originally, the District of Columbia was a near-perfect square but contained more than one settlement; the Capitol was to be the center
Quadrants_of_Washington,_D.C.
Numbers obtained by adding the two previous ones
{\displaystyle 5x^{2}+4} or 5 x 2 − 4 {\displaystyle 5x^{2}-4} is a perfect square. This is because Binet's formula, which can be written as F n = ( φ
Fibonacci_sequence
Concept in number theory
intuitively that there are more positive integers than perfect squares, because every perfect square is already positive and yet many other positive integers
Natural_density
Natural number
senary 100006 (decimal 1296). It is the 64th quarter square. It is the smallest four-digit perfect square: 1024 = 322. 1024 is the smallest number with exactly
1024_(number)
Type of motion which combines translation and rotation with respect to a surface
with the surface. The construction of a specific surface allows even a perfect square wheel to roll with its centroid at constant height above a reference
Rolling
Natural number
triangular numbers. Three is the only prime which is one less than a perfect square. Any other number which is n 2 {\displaystyle n^{2}} − 1 for some integer
3
Function whose actual domain of definition may be smaller than its apparent domain
) {\displaystyle f(n)} is only defined if n {\displaystyle n} is a perfect square (that is, 0 , 1 , 4 , 9 , 16 , … {\displaystyle 0,1,4,9,16,\ldots }
Partial_function
In mathematics, when is n!+1 a square
of n {\displaystyle n} such that n ! + 1 {\displaystyle n!+1} is a perfect square, where n ! {\displaystyle n!} is the factorial. Only three values of
Brocard's_problem
Number raised to the third power
minimum perfect cube, since the cube of a negative integer is negative. For example, (−4) × (−4) × (−4) = −64. Unlike perfect squares, perfect cubes do
Cube_(algebra)
Formula that provides the solutions to a quadratic equation
\end{aligned}}} Because the left-hand side is now a perfect square, we can easily take the square root of both sides: x + b 2 a = ± b 2 − 4 a c 2 a .
Quadratic_formula
Mathematical problem of square numbers which are also square-pyramidal
a perfect square, would have N = 8, yielding a total of (14 × 14 = ) 196 cannonballs. The only numbers that are simultaneously triangular and square pyramidal
Cannonball_problem
Natural number
expressed as (8+9)2. 289 is a perfect square being equal to 172. It is also the 7th number to only have 3 factors because it is a square of a prime number. 289
289_(number)
Treatise by Archimedes
contains accurate approximations to the square root of 3 (one larger and one smaller) and other larger non-perfect square roots; however, Archimedes gives no
Measurement_of_a_Circle
Number that is not a ratio of integers
the golden ratio φ, and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like any real
Irrational_number
Visualization of the prime numbers
than the square spiral used by Ulam, and are spaced so that one perfect square occurs in each full rotation. (In the Ulam spiral, two squares occur in
Ulam_spiral
"Top Music Videos". Billboard. Vol. 115, no. 46. November 15, 2003. Perfect Square: "Top Music Video". Billboard. Vol. 116, no. 14. April 3, 2004. When
R.E.M._discography
Unofficial holiday
2081. Square Root Days fall upon the same nine dates each century. Notably, May 5, 2025, which also coincided with Cinco de Mayo, is a perfect Square Root
Square_Root_Day
Square of a triangular number
JSTOR 2688231. Stroeker, R. J. (1995), "On the sum of consecutive cubes being a perfect square", Compositio Mathematica, 97 (1–2): 295–307, MR 1355130. Toeplitz, Otto
Squared_triangular_number
Natural number
only square number that is the sum of two consecutive, positive cubes: 3 2 = 9 = 1 3 + 2 3 {\displaystyle 3^{2}=9=1^{3}+2^{3}} If an odd perfect number
9
Words can be read horizontally and vertically
9-square in 1897; and the 10-square in 2023. Here are examples of English word squares up to order eight: The following is one of several "perfect" nine-squares
Word_square
Integer that is a perfect square modulo some integer
an integer q is a quadratic residue modulo n if it is congruent to a perfect square modulo n; that is, if there exists an integer x such that x 2 ≡ q (
Quadratic_residue
Natural number
{\displaystyle n^{n}} , where n = 4 {\displaystyle n=4} . 256 is a perfect square (162). 256 is the only 3-digit number that is zenzizenzizenzic. It is
256_(number)
2003 video by R.E.M.
the Rain" (Buck, Mills, Stipe) (David Weir) – 4:38 Recorded at Trafalgar Square on April 29, 2001: "Imitation of Life" (Buck, Mills, Stipe) "Losing My Religion"
In View: The Best of R.E.M. 1988–2003
In_View:_The_Best_of_R.E.M._1988–2003
Figurate number
hdl:1893/29437. Beldon, Tom; Gardiner, Tony (2002). "Triangular Numbers and Perfect Squares". The Mathematical Gazette. 86 (507): 423–431. doi:10.2307/3621134
Triangular_number
Open space surrounded by covered walks or open galleries
neighboring site by Abbot Richbold (784–804) the cloister was made a perfect square, against the south flank of the new church, precisely as in the plan
Cloister
Arithmetic function
{\displaystyle n} is the characteristic function of the squares: ∑ d | n λ ( d ) = { 1 if n is a perfect square, 0 otherwise. {\displaystyle \sum _{d|n}\lambda
Liouville_function
Shape with four equal sides and angles
subdivision with distinct smaller squares is called a perfect squared square. Another variant of squaring the square called "Mrs. Perkins's quilt" allows
Square
Song by R.E.M.
concert that it is his favorite R.E.M. song. The song is featured on the Perfect Square DVD from 2003, and opens with lyrics from "Chorus and the Ring" followed
Country_Feedback
Crocheted block assembled to make larger textiles
colors. The small blocks can be sewn or drawn together, so as to make a perfect square, this joining being done on the wrong side. The idea is to have the
Granny_square
Traditional Japanese wrapping cloth
chirimen, cotton, rayon, and nylon. The cloth is typically square, although not a perfect square, to facilitate the wrap. Sizes vary, from 45 by 45 centimetres
Furoshiki
Unsolved problem in mathematics
{n(n-1)}{2}}n^{n}(n-1)^{n-1}t^{n-1}(1-t),} and this is not in general a perfect square. Solutions for alternating groups must be handled differently for odd
Inverse_Galois_problem
Mathematical proof expressed visually
sum of all positive odd numbers up to 2n − 1 is a perfect square—more specifically, the perfect square n2—can be demonstrated by a proof without words.
Proof_without_words
Dukedom of Spain
Sigüenza, and confessor to the King.[citation needed] The Dukedom was a perfect square of twenty-five leagues on a side, extending towards the west from the
Duke_of_Veragua
Number that remains the same when its digits are reversed
below: There are many palindromic perfect powers nk, where n is a natural number and k is 2, 3 or 4. Palindromic squares: 0, 1, 4, 9, 121, 484, 676, 10201
Palindromic_number
Natural number
composite number. 285 is the 9th square pyramidal number. That means it is the sum of a number of consecutive perfect squares starting with 1. For 285, it
285_(number)
Unsolved problem in mathematics
differences between perfect squares and perfect cubes. It asserts that a perfect square y 2 {\displaystyle y^{2}} and a perfect cube x 3 {\displaystyle
Hall's_conjecture
Term used in organic chemistry
the species C6010+, which has 50 π-electrons: 50/2 = 25, which is a perfect square. In 2011, Jordi Poater and Miquel Solà expanded Hirsch's rule to open-shell
Spherical_aromaticity
1992 single by R.E.M.
been studio-recorded. The song is also included on the 2003 live DVD Perfect Square, the 2007 live CD/DVD R.E.M. Live, and the 2009 live CD Live at the
Drive_(R.E.M._song)
Natural number
positive integers that is simultaneously two more than a perfect square and two less than a perfect cube (123 = 112 + 2 = 53 - 2). In numerology, the sequence
123_(number)
Arithmetic operation, inverse of nth power
simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it: 32 5 = 16 ⋅ 2 5 = 16 ⋅ 2 5 = 4 2 5
Nth_root
Boutique hotel in Penang, Malaysia
tiles made of encaustic clay in geometric pieces all shaped to fit to a perfect square, Glasgow cast iron works by MacFarlane's & Co. and Art Nouveau stained
Cheong_Fatt_Tze_Mansion
Integer that is both a perfect square and a triangular number
In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number, in other words
Square_triangular_number
Cuboid whose edges and face diagonals have integer lengths
(2000). "Perfect Cuboids and Perfect Square Triangles". Mathematics Magazine, 73(5), 400–401. Sharipov R.A. (2012). "Perfect cuboids and irreducible polynomials"
Euler_brick
Japanese manga series by Noriyki Konishi
April 2023. Viz Media publishes the series in North America under their Perfect Square imprint, whereas Shogakukan Asia publishes the series in English in
Yo-kai_Watch_(manga)
Organizing thought in architectural design
Lonja del Comercio building in Havana (in plan) is a perfect square and based on the classic 9 square problem that was used, among others, by Peter Eisenman
Parti_(architecture)
Number, product of consecutive integers
nth pronic number is at a radius of n and n + 1 from a perfect square, and the nth perfect square is at a radius of n from a pronic number. The nth pronic
Pronic_number
Drawing by Leonardo da Vinci, c. 1490
breadth will be found to be the same as the height, as in the case of a perfect square. — Vitruvius in De architectura, book three, chapter one 19th-century
Vitruvian_Man
Public housing in Boston, Massachusetts
Seventh street and 3 city blocks from B street to D street, forming a perfect square. The land for the West Broadway Housing Development was cleared in 1941
D_Street_Projects
Elliptic curve
) {\displaystyle (x,y)} . In other words, the differences of perfect squares and perfect cubes tend to infinity. The question of how fast was dealt with
Mordell_curve
Famous mathematical optimization problem
Thus when we have u2 + v2, we can add the middle term −2uv to get a perfect square. We have x + a b x . {\displaystyle x+{\frac {ab}{x}}.} If we regard
Regiomontanus' angle maximization problem
Regiomontanus'_angle_maximization_problem
Natural number
preceding 112. 111 is the fourth non-trivial nonagonal number, and the seventh perfect totient number. 111 is furthermore the ninth number such that its Euler
111_(number)
Probabilistic primality testing algorithm
147755149. If n is a perfect square, then step 3 will never yield a D with (D/n) = −1; this is not a problem because perfect squares are easy to detect
Baillie–PSW_primality_test
Solution of some Diophantine equation
only has solutions in N {\displaystyle \mathbb {N} } when x is not a perfect square. The Diophantine set is {2, 3, 5, 6, 7, 8, 10, 11, 12, 13, ...}. The
Diophantine_set
Place of worship in Hinduism
around which is formed a perfect square in the space available. The circle of the mandala circumscribes the square. The square is considered divine for
Hindu_temple
Probabilistic test for the primality of an integer
in this case, n+1 = (p+1)2 is a perfect square. One can quickly detect perfect squares using Newton's method for square roots. By combining a Lucas pseudoprime
Lucas_pseudoprime
Italian-French scientist (1736–1813)
an integer n which is not a perfect square, to find a number x such that nx2 + 1[verification needed] is a perfect square; and the general differential
Joseph-Louis_Lagrange
Size of a set in mathematics
perfect squares as there are square roots. However, every number is a square root, since it can be squared, but not every number is a perfect square.
Cardinality
1967 painting by David Hockney
both times by auction at Sotheby's in London). The canvas – almost a perfect square – is dominated by the strong vertical and horizontal lines of the trees
A_Bigger_Splash
Unsolved problem in mathematics
same difference between terms as the other two, the terms are all perfect squares, and the middle terms of the three arithmetic progressions themselves
Magic_square_of_squares
Simple continued fraction whose partial denominators repeat periodically
{D}}}{Q}}} where P, D, and Q are integers, D > 0 is not a perfect square (but not necessarily square-free), and Q divides the quantity P 2 − D {\displaystyle
Periodic_continued_fraction
Office skyscraper in Paris
is made of dark granite and darkened windows. Its shape is that of a perfect square prism. It is said that its architects were inspired by the black monolith
Tour_Areva
Type of figurate number
(2n-1)n=h_{n}} divisors. ∎ The sequence of numbers that are both hexagonal and perfect squares starts 1, 1225, 1413721,... OEIS: A046177. Centered hexagonal number
Hexagonal_number
Concept in graph theory
{\sqrt {4k-3}}} is an integer t, then 4 k − 3 {\displaystyle 4k-3} is a perfect square t 2 {\displaystyle t^{2}} , so k = t 2 + 3 4 {\displaystyle k={\frac
Strongly_regular_graph
Polyhedral number representing a tetrahedron
tetrahedral numbers are also perfect squares, namely: Te1 = 12 = 1 Te2 = 22 = 4 Te48 = 1402 = 19600. Like square numbers are congruent to 0
Tetrahedral_number
Numbers with a certain property involving recursive summation
Because base 6 has no other perfect digital invariants except for 1, no positive integer other than 1 is the sum of the squares of its own digits. In base
Happy_number
Type of Diophantine equation
a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root
Pell's_equation
Mechanical measurement
engine with a displacement of 2793 cubic centimeters is an example of a perfect square engine with an 84.0 mm × 84.0 mm (3.31 in × 3.31 in) bore and stroke
Stroke_ratio
Natural number
with strictly intersecting peaks. The sum of the divisors of 310 is a perfect square. 311 is a twin prime with 313, an irregular prime, an emirp, and a permutable
300_(number)
Mathematical inequality relating inner products and norms
helpful when the inequality involves fractions where the numerator is a perfect square. The real vector space R 2 {\displaystyle \mathbb {R} ^{2}} denotes
Cauchy–Schwarz_inequality
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