Thymaridas of Paros (Greek : Θυμαρίδας; c. 400 – c. 350 BCE) was an ancient Greek mathematician and Pythagorean noted for his work on prime numbers and simultaneous linear equations.
Greek is an independent branch of the Indo-European family of languages, native to Greece, Cyprus and other parts of the Eastern Mediterranean and the Black Sea. It has the longest documented history of any living Indo-European language, spanning at least 3500 years of written records. Its writing system has been the Greek alphabet for the major part of its history; other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic, and many other writing systems.
Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Crotone, Italy. Early Pythagorean communities spread throughout Magna Graecia.
Although little is known about the life of Thymaridas, it is believed that he was a rich man who fell into poverty. It is said that Thestor of Poseidonia traveled to Paros in order to help Thymaridas with the money that was collected for him.
Paros is a Greek island in the central Aegean Sea. One of the Cyclades island group, it lies to the west of Naxos, from which it is separated by a channel about 8 kilometres wide. It lies approximately 150 km south-east of Piraeus. The Municipality of Paros includes numerous uninhabited offshore islets totaling 196.308 square kilometres (75.795 sq mi) of land. Its nearest neighbor is the municipality of Antiparos, which lies to its southwest. In ancient Greece, the city-state of Paros was located on the island.
Iamblichus states that Thymaridas called prime numbers "rectilinear" since they can only be represented on a one-dimensional line. Non-prime numbers, on the other hand, can be represented on a two-dimensional plane as a rectangle with sides that, when multiplied, produce the non-prime number in question. He further called the number one a "limiting quantity".
Iamblichus was a Syrian Neoplatonist philosopher of Arab origin. He determined the direction that would later be taken by Neoplatonic philosophy. He was also the biographer of Pythagoras, a Greek mystic, philosopher and mathematician.
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
Iamblichus in his comments to Introductio arithmetica states that Thymaridas also worked with simultaneous linear equations.In particular, he created the then famous rule that was known as the "bloom of Thymaridas" or as the "flower of Thymaridas", which states that:
The book Introduction to Arithmetic is the only extant work on mathematics by Nicomachus.
If the sum of n quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to 1/(n + 2) [this is a typo in Flegg's book - the denominator should be n-2 to match the math below] of the difference between the sums of these pairs and the first given sum.
or using modern notation, the solution of the following system of n linear equations in n unknowns,
is given by
Iamblichus goes on to describe how some systems of linear equations that are not in this form can be placed into this form.
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.
In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the n-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space.
In mathematics, a linear equation is an equation that may be put in the form
Linear algebra is the branch of mathematics concerning linear equations such as
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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), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.
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Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states:
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation, can decide whether the equation has a solution with all unknowns taking integer values.
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Thymaridas of Paros, an ancient Pythagorean already mentioned (p. 69), was the author of a rule for solving a certain set of n simultaneous simple equations connecting n unknown quantities. The rule was evidently well known, for it was called by the special name [...] the 'flower' or 'bloom' of Thymaridas. [...] The rule is very obscurely worded , but it states in effect that, if we have the following n equations connecting n unknown quantities x, x1, x2 ... xn−1, namely [...] Iamblichus, our informant on this subject, goes on to show that other types of equations can be reduced to this, so that the rule does not 'leave us in the lurch' in those cases either.
Thymaridas (fourth century) is said to have had this rule for solving a particular set of n linear equations in n unknowns:
If the sum of n quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to 1/(n + 2) of the difference between the sums of these pairs and the first given sum.