Pythagorean Triples before and after Pythagoras

Agarwal, Ravi P.

doi:10.3390/computation8030062

Cited by 8 publications

(15 citation statements)

References 29 publications

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“…The converse, i.e., any Pythagorean triple is necessarily of the form (40) also holds. For the proof and history of this result see, Agarwal [14]. From ( 18), (32), and (40) the following relations hold ( ) ( ) ( )…”

Section: S S S Smentioning

confidence: 94%

“… The n-th biquadratic number is the sum is the sum of the first n Haüy rhombic dodecahedral numbers. Indeed, from (2), ( 11 ∑  Fermat's Last Theorem confirms that a fourth power cannot be the sum of two other fourth powers, for details see Agarwal [14]. In 1770, Edward Waring (1736-1798, England) conjectured (known as Waring's problem) that every positive integer can be expressed as the sum of at most 19 fourth powers, and every integer larger than 13,792 can be expressed as the sum of at most 16 fourth powers.…”

Section: Biquadratic Numbers (Bc) Nmentioning

confidence: 97%

“…Thus the equation 2 [14]).  To find the sum of the reciprocals of all generalized pentagonal numbers we need the following well-known result, e.g., see Andrews et al [27], page 536, and Efthimiou [28]…”

Section: Generalized Pentagonal Numbers (Centered Hexagonal Numbers Hex Numbers) (Gp) Nmentioning

confidence: 99%

“…This system generates infinite number of solutions ( )  To find all tetrakaidecagonal numbers which are also decagonal numbers, we need to find integer solutions of the equation ( ) (5,14), corresponding to (22) the system is…”

Section: Tetrakaidecagonal Numbers (Tet) Nmentioning

confidence: 99%

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Pythagoreans Figurative Numbers: The Beginning of Number Theory and Summation of Series

Agarwal¹

2021

JAMP

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In this article we shall examine several different types of figurative numbers which have been studied extensively over the period of 2500 years, and currently scattered on hundreds of websites. We shall discuss their computation through simple recurrence relations, patterns and properties, and mutual relationships which have led to curious results in the field of elementary number theory. Further, for each type of figurative numbers we shall show that the addition of first finite numbers and infinite addition of their inverses often require new/strange techniques. We sincerely hope that besides experts, students and teachers of mathematics will also be benefited with this article.

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Section: S S S Smentioning

confidence: 94%

Section: Biquadratic Numbers (Bc) Nmentioning

confidence: 97%

Section: Generalized Pentagonal Numbers (Centered Hexagonal Numbers Hex Numbers) (Gp) Nmentioning

confidence: 99%

Section: Tetrakaidecagonal Numbers (Tet) Nmentioning

confidence: 99%

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Pythagoreans Figurative Numbers: The Beginning of Number Theory and Summation of Series

Agarwal¹

2021

JAMP

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“…In this work, we leverage on the gaps found in the literature towards providing a novel proposal for semi-prime factorization. While there are several properties of Pythagorean triples, new patterns based on these properties are yet to be researched in the context of semi-prime factorization [15]. The main contributions of the paper are envisaged via the key features of our proposed factorization method, as listed below: i.…”

Section: Introductionmentioning

confidence: 99%

New Semi-Prime Factorization and Application in Large RSA Key Attacks

Overmars¹,

Venkatraman²

2021

JCP

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Semi-prime factorization is an increasingly important number theoretic problem, since it is computationally intractable. Further, this property has been applied in public-key cryptography, such as the Rivest–Shamir–Adleman (RSA) encryption systems for secure digital communications. Hence, alternate approaches to solve the semi-prime factorization problem are proposed. Recently, Pythagorean tuples to factor semi-primes have been explored to consider Fermat’s Christmas theorem, with the two squares having opposite parity. This paper is motivated by the property that the integer separating these two squares being odd reduces the search for semi-prime factorization by half. In this paper, we prove that if a Pythagorean quadruple is known and one of its squares represents a Pythagorean triple, then the semi-prime is factorized. The problem of semi-prime factorization is reduced to the problem of finding only one such sum of three squares to factorize a semi-prime. We modify the Lebesgue identity as the sum of four squares to obtain four sums of three squares. These are then expressed as four Pythagorean quadruples. The Brahmagupta–Fibonacci identity reduces these four Pythagorean quadruples to two Pythagorean triples. The greatest common divisors of the sides contained therein are the factors of the semi-prime. We then prove that to factor a semi-prime, it is sufficient that only one of these Pythagorean quadruples be known. We provide the algorithm of our proposed semi-prime factorization method, highlighting its complexity and comparative advantage of the solution space with Fermat’s method. Our algorithm has the advantage when the factors of a semi-prime are congruent to 1 modulus 4. Illustrations of our method for real-world applications, such as factorization of the 768-bit number RSA-768, are established. Further, the computational viabilities, despite the mathematical constraints and the unexplored properties, are suggested as opportunities for future research.

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Pythagorean Figurative Numbers

Agarwal

2024

Mathematics Before and After Pythagoras

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Pythagorean Triples before and after Pythagoras

Cited by 8 publications

References 29 publications

Pythagoreans Figurative Numbers: The Beginning of Number Theory and Summation of Series

Pythagoreans Figurative Numbers: The Beginning of Number Theory and Summation of Series

New Semi-Prime Factorization and Application in Large RSA Key Attacks

Pythagorean Figurative Numbers

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