Research Paper:
Pythagorean Triples and Cryptographic Coding
By Subhash Kak
Oklahoma State University, Stillwater - 2010
Pythagorean Triples and Cryptographic Coding
By Subhash Kak
Oklahoma State University, Stillwater - 2010
"The earliest statement of the theorem of the square on the diagonal
(Pythagoras theorem), together with some examples, is to be found in the
geometry text of Baudhāyana (c. 800 BC). "
A Pythagorean triple (a, b, c) consists of positive integers that are the sides of a right triangle and thus a2 + b2 = c2. Given a Pythagorean triple (a, b, c), we have other similar triples that are d(a, b, c), where d > 1. A primitive Pythagorean triple (PPT) consists of numbers that are relatively prime.
Pythagorean triples have been found on cuneiform tablets of Babylon and they are important in Vedic ritual and described in early geometry books of India and in the works of Euclid and Diophantus.
The earliest statement of the theorem of the square on the diagonal (Pythagoras theorem), together with some examples, is to be found in the geometry text of Baudhāyana (c. 800 BC). In his Śulba Sūtra 1.12 and 1.13, it is stated:
The areas (of the squares) produced separately by the length and the breadth of a rectangle together equal the area (of the square) produced by the diagonal. This is observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35, and 15 and 36.
O’Conner and Robertson in their history of mathematics project take Baudhāyana to be 800 BC. Seidenberg presents various arguments for an early date for the knowledge of the Pythagoras theorem in India (for other aspects of early Indian mathematics, and for the altar ritual that provides the context in which this mathematics was used in the Śulba Sūtras, see. Van der Waerden saw a ritual origin to the discovery of Pythagoras.
The Śulba Sūtras are texts of applied geometry that provide techniques to draw altars of different shapes and sizes in a convenient manner. The word śulba means a “cord”, “rope”, or “string” and the root śulb signifies “measurement”. The cord has marks (nyañcana in Sanskrit) that indicate where the intermediate pegs are to be fixed. Thus a cord of 12 units length with nyañcana at 3 and 7, can be readily stretched to yield the right-angled triangle (3,4,5).
Read more:
http://bit.ly/1AU8Cri
A Pythagorean triple (a, b, c) consists of positive integers that are the sides of a right triangle and thus a2 + b2 = c2. Given a Pythagorean triple (a, b, c), we have other similar triples that are d(a, b, c), where d > 1. A primitive Pythagorean triple (PPT) consists of numbers that are relatively prime.
Pythagorean triples have been found on cuneiform tablets of Babylon and they are important in Vedic ritual and described in early geometry books of India and in the works of Euclid and Diophantus.
The earliest statement of the theorem of the square on the diagonal (Pythagoras theorem), together with some examples, is to be found in the geometry text of Baudhāyana (c. 800 BC). In his Śulba Sūtra 1.12 and 1.13, it is stated:
The areas (of the squares) produced separately by the length and the breadth of a rectangle together equal the area (of the square) produced by the diagonal. This is observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35, and 15 and 36.
O’Conner and Robertson in their history of mathematics project take Baudhāyana to be 800 BC. Seidenberg presents various arguments for an early date for the knowledge of the Pythagoras theorem in India (for other aspects of early Indian mathematics, and for the altar ritual that provides the context in which this mathematics was used in the Śulba Sūtras, see. Van der Waerden saw a ritual origin to the discovery of Pythagoras.
The Śulba Sūtras are texts of applied geometry that provide techniques to draw altars of different shapes and sizes in a convenient manner. The word śulba means a “cord”, “rope”, or “string” and the root śulb signifies “measurement”. The cord has marks (nyañcana in Sanskrit) that indicate where the intermediate pegs are to be fixed. Thus a cord of 12 units length with nyañcana at 3 and 7, can be readily stretched to yield the right-angled triangle (3,4,5).
Read more:
http://bit.ly/1AU8Cri
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