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Integer side lengths of a right triangle
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known
Pythagorean_triple
Relation between sides of a right triangle
the 8th and 5th century BC, contains a list of Pythagorean triples and a statement of the Pythagorean theorem, both in the special case of the isosceles
Pythagorean_theorem
Can one split the integers into two sets such that every Pythagorean triple spans both?
Boolean Pythagorean triples problem is a problem from Ramsey theory about whether the positive integers can be colored red and blue so that no Pythagorean triples
Boolean Pythagorean triples problem
Boolean_Pythagorean_triples_problem
formulas for generating Pythagorean triples have been developed. Euclid's, Pythagoras' and Plato's formulas for calculating triples have been described here:
Formulas for generating Pythagorean triples
Formulas_for_generating_Pythagorean_triples
Mathematical tree of integer right triangles
primitive Pythagorean triples is a mathematical tree in which each node represents a primitive Pythagorean triple and each primitive Pythagorean triple is represented
Tree of primitive Pythagorean triples
Tree_of_primitive_Pythagorean_triples
Relation between the side lengths and altitude of a right triangle
In geometry, the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theorem or the upside down Pythagorean theorem) is as follows: Let
Inverse_Pythagorean_theorem
17th-century conjecture proved by Andrew Wiles in 1994
{\displaystyle y} , and z {\displaystyle z} ; these solutions are known as Pythagorean triples (with the simplest example being 3, 4, 5). Around 1637, Fermat wrote
Fermat's_Last_Theorem
Right triangle with a feature making calculations on the triangle easier
the Pythagorean theorem, but that "there is no evidence that they used it to construct right angles". The following are all the Pythagorean triple ratios
Special_right_triangle
Polynomial equation whose integer solutions are sought
equation of degree two that has been studied. Its solutions are the Pythagorean triples. This is also the homogeneous equation of the unit circle. In this
Diophantine_equation
Philosophical system based on the teachings of Pythagoras
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans
Pythagoreanism
Set of integers, the lengths of the sides of a triangle with a 60° angle
Similar to a Pythagorean triple, an Eisenstein triple (named after Gotthold Eisenstein) is a set of integers which are the lengths of the sides of a triangle
Eisenstein_triple
Locus of the zeros of a polynomial of degree two
transforms a Pythagorean triple into another Pythagorean triple, only one of the two cases is sufficient for producing all primitive Pythagorean triples up to
Quadric
Babylonian clay tablet of numbers in Pythagorean triples
table relates to a Pythagorean triple, that is, a triple of integers ( s , l , d ) {\displaystyle (s,l,d)} that satisfies the Pythagorean theorem, s 2 + l
Plimpton_322
Topics referred to by the same term
vegetarianism before the nineteenth century Pythagorean theorem Pythagorean triple Pythagorean prime Pythagorean trigonometric identity Table of Pythagoras
Pythagorean
Generalization of golden and silver ratios
\theta } is a positive integer, as it is with some Pythagorean triangles. For a primitive Pythagorean triple, a2 + b2 = c2, with positive integers a < b <
Metallic_mean
Texts belonging to the Śrauta ritual
of the Pythagorean theorem, both in the case of an isosceles right triangle and in the general case, as well as lists of Pythagorean triples. In Baudhayana
Shulba_Sutras
Number used to approximate the square root of 2
b, c (necessarily satisfying the Pythagorean theorem a2 + b2 = c2), then (a,b,c) is known as a Pythagorean triple. As Martin (1875) describes, the Pell
Pell_number
Hypotenuse of right triangle from its sides
In mathematics, Pythagorean addition is a binary operation on the real numbers that computes the length of the hypotenuse of a right triangle, given its
Pythagorean_addition
Triangle containing a 90-degree angle
integers, the triangle is called a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple. The relations between the sides
Right_triangle
Natural number
smallest integer-sided right triangle, making part of the smallest Pythagorean triple (3, 4, 5). 5 is the first safe prime and the first good prime. 11
5
Four integers where the sum of the squares of three equals the square of the fourth
zero (thus allowing Pythagorean triples to be included) with the only condition being that d > 0. In this setting, a Pythagorean quadruple (a, b, c, d)
Pythagorean_quadruple
Tiling by squares of two sizes
tiling is 13, based on the Pythagorean triple (5,12,13). By overlaying a square grid of side length c onto the Pythagorean tiling, it may be used to generate
Pythagorean_tiling
Characterization by prime factors of sums of two squares
squares, counted by the sum of squares function; for instance, every Pythagorean triple a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} gives a second representation
Sum_of_two_squares_theorem
t2, 1 + t2) values in the above formulae are proportional to the Pythagorean triple (2pq, q2 − p2, q2 + p2). For example, for n = 3 terms, π 2 = arctan
List of trigonometric identities
List_of_trigonometric_identities
Natural number
25=3^{2}+4^{2}} . In other words, the numbers 3, 4 and 5 form the smallest Pythagorean triple. In Ezekiel's vision of a new temple: The number twenty-five is of
25_(number)
Historical development of geometry
expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians." They make use of Pythagorean triples, which are
History_of_geometry
Unique positive real number which when multiplied by itself gives 2
{\displaystyle b^{2}+b^{2}=a^{2}} Here, (b, b, a) is a primitive Pythagorean triple, and from the lemma a is never even. However, this contradicts the
Square_root_of_2
Natural number
the first 7 primes with themselves. It is the hypotenuse of two Pythagorean triples: 3052=2072+2242=1362+2732. 306 = 2 × 32 × 17. It is the 17th oblong
300_(number)
Branch of pure mathematics
dated c. 1800 BC. It is a broken clay tablet that contains a list of Pythagorean triples, that is, integers ( a , b , c ) {\displaystyle (a,b,c)} such that
Number_theory
Development of mathematics in South Asia
simple Pythagorean triples, such as: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (12, 35, 37), as well as a statement of the Pythagorean theorem
Indian_mathematics
Relates the tangent of half of an angle to trigonometric functions of the entire angle
{1}{2}}(\eta +\theta )\,.} Furthermore, using double-angle formulae and the Pythagorean identity 1 + tan 2 1 2 α = 1 / cos 2 1 2 α {\textstyle 1+\tan ^{2}{\tfrac
Tangent_half-angle_formula
Dutch computer scientist
solvers to resolve mathematical conjectures such as the Boolean Pythagorean triples problem, Schur's theorem number 5, and Keller's conjecture in dimension
Marijn_Heule
Indian mathematician and astronomer (598–668)
value stated. Also, if m and x are rational, so are d, a, b and c. A Pythagorean triple can therefore be obtained from a, b and c by multiplying each of them
Brahmagupta
1890 BC). All these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the most ancient and widespread
History_of_mathematics
Natural number
{\displaystyle 61=5^{2}+6^{2}} , it is the hypotenuse of the primitive Pythagorean triple ( 11 , 60 , 61 ) {\displaystyle (11,60,61)} . There are sixty-one
61_(number)
Natural number
numbers 1 through n such that every Pythagorean triple is multicolored, i.e. where the Boolean Pythagorean triples problem becomes false. The 200-terabyte
7825
90° angle (π/2 radians)
to confirm if an angle is a true right angle. It is based on the Pythagorean triple (3, 4, 5) and the rule of 3-4-5. From the angle in question, running
Right_angle
Numbers obtained by adding the two previous ones
triangle with integer sides, or in other words, the largest number in a Pythagorean triple, obtained from the formula ( F n F n + 3 ) 2 + ( 2 F n + 1 F n + 2
Fibonacci_sequence
Natural number
sequence. 265 is the 7th number to be the hypotenuse for two separate Pythagorean Triples. The other two values would be 23 and 264 or 96 and 247. 265 is the
265_(number)
Topics referred to by the same term
Power point tracking, a solar energy charging technology Primitive Pythagorean triple, three integers that form a right triangle Probabilistic polynomial-time
PPT
and was later reduced to 850 megabytes. 2016 – Solving the Boolean Pythagorean triples problem required the generation of 200 terabytes of proof. 2017 –
List of long mathematical proofs
List_of_long_mathematical_proofs
Branch of mathematics
of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples, which are
Geometry
Matrix B such that B² equals a given matrix A
particular, if ( r , s , t ) {\displaystyle \ (r,s,t)\ } is any Pythagorean triple, then 1 t [ r s s − r ] {\displaystyle \ {\frac
Square_root_of_a_matrix
American mathematician
Math. Soc. 24 (2011), 603–648. Arxiv with Hee Oh: Almost Prime Pythagorean Triples in Thin Orbits. J. Reine Angew. Math. 667 (2012), 89–131. Arxiv Homepage
Alex_Kontorovich
Index of articles associated with the same name
number Pythagorean triples are sets of three integers such that the sum of the squares of the first two equals the square of the third. A Pythagorean prime
Sum_of_squares
Conjecture in number theory
a Pythagorean triple, were considered by L. Jesmanowicz in the 1950s. J. Jozefiak proved that there are an infinite number of primitive Pythagorean triples
Beal_conjecture
Type of spline curve
In mathematics, a Pythagorean hodograph curve or PH curve is a curve defined by a polynomial parametric equation for which the speed (the derivative of
Pythagorean_hodograph_curve
When generating a primitive automedian triangle from a primitive Pythagorean triple using the Euclidean parameters m , n {\displaystyle m,n} , then m
Automedian_triangle
of Z {\displaystyle \mathbb {Z} } along all integers. Pythagorean triple A Pythagorean triple is three positive integers a, b, c such that a2 + b2 =
Glossary_of_number_theory
Natural number
topological spaces with four points. It is the hypotenuse of a primitive Pythagorean triple: 2332 = 1052 + 2082. Sloane, N. J. A. (ed.). "Sequence A005384 (Sophie
233_(number)
Three deities that are worshipped as one
triple deity is a deity with three apparent forms that function as a singular whole. Such deities may sometimes be referred to as threefold, tripled,
Triple_deity
Triangle with integer side lengths
A Pythagorean triangle is right-angled and Heronian. Its three integer sides are known as a Pythagorean triple or Pythagorean triplet or Pythagorean triad
Integer_triangle
different positive integers is the hypotenuse of a Pythagorean triple, while any hypotenuse of a Pythagorean triple is a contraharmonic mean of two different positive
Contraharmonic_mean
Indian sage and mathematician
written around 800 BC.. It contains a list of Pythagorean triples and a statement (without proof) of the Pythagorean theorem, both in the special case of the
Baudhayana
Complex numbers with unit norm and both real and imaginary parts rational numbers
set of such points turns out to be closely related to primitive Pythagorean triples. Consider a primitive right triangle, that is, with integer side
Group of rational points on the unit circle
Group_of_rational_points_on_the_unit_circle
Particular mapping that projects a sphere onto a plane
{m^{2}-n^{2}}{m^{2}+n^{2}}}\right)} which gives Euclid's formula for a Pythagorean triple. The pair of trigonometric functions (sin x, cos x) can be thought
Stereographic_projection
Unspecified value mentioned by Plato
which is remarkable for also being the sum of the cubes for the Pythagorean triple (3, 4, 5): 33 + 43 + 53 = 63. Such considerations tend to ignore the
Plato's_number
Partial results found before the complete proof
For n equal to 2, the equation has infinitely many solutions, the Pythagorean triples.) A solution (a, b, c) for a given n leads to a solution for all
Proof of Fermat's Last Theorem for specific exponents
Proof_of_Fermat's_Last_Theorem_for_specific_exponents
Topics referred to by the same term
Pythagorean tree may refer to: Tree of primitive Pythagorean triples Pythagoras tree (fractal) This disambiguation page lists articles associated with
Pythagorean_tree
Branch of mathematical combinatorics
problem related to Ramsey theory. Another large example is the Boolean Pythagorean triples problem. Theorems in Ramsey theory are generally one of the following
Ramsey_theory
Property of geometry, also used to generalize the notion of "distance" in metric spaces
a/3, it generates a right triangle that is always similar to the Pythagorean triple with sides 3, 4, 5. Now consider a triangle whose sides are in a geometric
Triangle_inequality
Computer program for the Boolean satisfiability problem
Solver Competition. Cube-and-Conquer was used to solve the Boolean Pythagorean triples problem. Cube-and-Conquer is a modification or a generalization of
SAT_solver
square on the hypotenuse Pythagorean triple – a set of three positive integers that can occur in the Pythagorean theorem Pythagorean quadruple - a set of
List of things named after Pythagoras
List_of_things_named_after_Pythagoras
Cuboid whose edges and face diagonals have integer lengths
generated with Saunderson's parametric formula. Let (u, v, w) be a Pythagorean triple (that is, u2 + v2 = w2.) Then the edges a = u | 4 v 2 − w 2 | , b
Euler_brick
Taylor, 1995) Burr–Erdős conjecture (Choongbum Lee, 2017) Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor W. Marek, 2016) Sensitivity
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Generalization of Fermat's Last Theorem and of Catalan's conjecture,
and n and with c = am + bn), with m=n=k=2 (for the infinitely many Pythagorean triples), and e.g. 7 5 + 393 3 = 7792 2 {\displaystyle 7^{5}+393^{3}=7792^{2}}
Fermat–Catalan_conjecture
Number, approximately 2.41421
because of its connections to the square root of 2, almost-isosceles Pythagorean triples, square triangular numbers, Pell numbers, the octagon, and six polyhedra
Silver_ratio
City-state in ancient Sumer
mathematics, including the Plimpton 322 tablet that contains patterns of Pythagorean triples. Larsa is found (as UD.UNUG) on Proto-cuneiform lexical lists from
Larsa
Natural number
2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, Z equal to Y+1) ordered by increasing Z; then sequence gives
41_(number)
Number of dots in a centred dot square
is the hypotenuse of a Pythagorean triple (3-4-5, 5-12-13, 7-24-25, ...). This is exactly the sequence of Pythagorean triples where the two longest sides
Centered_square_number
Algorithm for computing greatest common divisors
factorization is helpful in many applications, such as deriving all Pythagorean triples or proving Fermat's theorem on sums of two squares. In general, the
Euclidean_algorithm
Subfield of computer science and logic
Oliver; Marek, Victor W. (2016). "Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer". Theory and Applications of Satisfiability
Automated_reasoning
322 tablet, created around 1900–1600 BC, which gives a table of Pythagorean triples and represents some of the most advanced mathematics prior to Greek
History_of_algebra
Theorem on rational values of the sine
Other mathematicians have given new proofs in subsequent years. Pythagorean triples form right triangles where the trigonometric functions will always
Niven's_theorem
Circles related to a point in the plane
Three mutually tangent circles of radii in ratios 4:4:1 yield a 3-4-5 Pythagorean triple triangle
Tangent_circles
conjecture Sato–Tate conjecture Langlands program modularity theorem Pythagorean triple Pell's equation Elliptic curve Nagell–Lutz theorem Mordell–Weil theorem
List_of_number_theory_topics
Square matrix with ones on the main diagonal and zeros elsewhere
Retrieved 2020-08-14. Mitchell, Douglas W. (November 2003). "87.57 Using Pythagorean triples to generate square roots of I 2 {\displaystyle I_{2}} ". The Mathematical
Identity_matrix
Mathematical treatise by Euclid
Proposition 29 gives Euclid's formula for producing all fundamental Pythagorean triples. Additionally, this book classifies irrational lengths into thirteen
Euclid's_Elements
Academic journal
colorings, Euler numbers, continued fractions, Stirling numbers, Pythagorean triples, Ramsey theory, Lucas-Bernoulli numbers, quadratic residues, higher-order
Fibonacci_Quarterly
Quartic diophantine equation in 4 variables
(a+b)^{4}+(c+d)^{4}+(a+b+c+d)^{4}} it can be seen it is a special Pythagorean triple, ( a 2 + a b + b 2 ) 2 + ( c 2 + c d + d 2 ) 2 = ( ( a + b ) 2 + (
Jacobi–Madden_equation
Triangle whose side lengths and area are integers
obtained by joining a Pythagorean triangle and its mirror image along a side of the right angle. Starting with the Pythagorean triple 3, 4, 5 this gives
Heronian_triangle
Writing medium, especially for writing in cuneiform
in cuneiform script. Believed to have been written c. 1800 BCE, this table lists two of the three numbers in what are now called Pythagorean triples.
Clay_tablet
Book on algebra by Leonardo Fibonacci
topics in number theory, among them an inductive method for finding Pythagorean triples based on the sequence of odd integers, the fact that the sum of the
The_Book_of_Squares
Historical region of West Asia
322 tablet, created around 1900–1600 BC, which gives a table of Pythagorean triples and represents some of the most advanced mathematics prior to Greek
Mesopotamia
Number whose square is a given number
ISBN / Date incompatibility (help) Mitchell, Douglas W., "Using Pythagorean triples to generate square roots of I2", Mathematical Gazette 87, November
Square_root
Tree in which each node has at most three children
all primitive Pythagorean triples are described in Tree of primitive Pythagorean triples and in Formulas for generating Pythagorean triples. The root node
Ternary_tree
Product of an integer with itself
of redirect targets Power of two – Two raised to an integer power Pythagorean triple – Integer side lengths of a right triangle Quadratic residue – Integer
Square_number
Product of a number by itself
moment of inertia to the size (length). There are infinitely many Pythagorean triples, sets of three positive integers such that the sum of the squares
Square_(algebra)
Mathematical operation on invertible matrices
35\\0.35&1.06\end{pmatrix}}~.} For a richer example, start with a Pythagorean triple ( p , q , r {\displaystyle p,q,r} ) and let a = l o g ( p + r ) −
Logarithm_of_a_matrix
Topics referred to by the same term
(fractal), a plane fractal made of squares Tree of primitive Pythagorean triples Pythagorean theorem Pnytagoras, king of the Ancient Greek city state of
Pythagoras_(disambiguation)
Plimpton 322 Babylonian tablet records the oldest known examples of Pythagorean triples. 1800 BC – Egypt, Moscow Mathematical Papyrus, finding the volume
Timeline_of_mathematics
Book about right triangles by Wacław Sierpiński in 1954
Pythagorean Triangles is a book on right triangles, the Pythagorean theorem, and Pythagorean triples. It was originally written in the Polish language
Pythagorean_Triangles
Uses of the constant
{\pi }{4}}=\arctan {\frac {a}{b+c}}+\arctan {\frac {b}{a+c}},} For Pythagorean triple (a,b,c). π = arctan a + arctan b + arctan c {\displaystyle \pi
List_of_formulae_involving_π
Two numbers without shared prime factors
(July 2001), "An alternative characterisation of all primitive Pythagorean triples", Mathematical Gazette, 85: 273–275, doi:10.2307/3622017. Pommerening
Coprime_integers
American diplomat, antiquarian and writer
Plimpton's death. The artifact contains a table of numbers related to Pythagorean triples, and has been the subject of numerous studies by historians of mathematics
Edgar_James_Banks
problem (SAT) solver proves a long-standing mathematical conjecture on Pythagorean triples over the set of integers. The initial proof, 200TB long, was checked
Timeline of artificial intelligence
Timeline_of_artificial_intelligence
mathematician – Pythagorean theorem, Pythagorean triple, Pythagorean tuning, Pythagorean expectation, Pythagorean hammers, Pythagorean trigonometric identity
List_of_eponyms_(L–Z)
American engineer
the Pythagorean triples numbers? Answer: They used numbers 0 and 00 for 13 of the 15 Plimpton 322 tablet lines. Plimpton 322 Pythagorean triple "Harvey
Harvey_Jerome_Brudner
"Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean Triples, and the Babylonian Triangle Parameter Equations". Historia Mathematica
List_of_misnamed_theorems
American mathematician (1935–2020)
hulls. He also began the study of primefree sequences, the Boolean Pythagorean triples problem, the biggest little polygon, and square packing in a square
Ronald_Graham
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