Bhargava's factorial function
In mathematics, Bhargava's factorial function, or simply Bhargava factorial, is a certain generalization of the the factorial function developed by the Fields Medal winning mathematician Manjul Bhargava as part of his thesis in Harvard University in 1996. The Bhargava factorial has the property that many number-theoretic results involving the ordinary factorials remain true even when the factorials are replaced by the Bhargava factorials. Using an arbitrary infinite subset S of the set Z of integers, Bhargava associated a positive integer with every positive integer k, which he denoted by k !S, with the property that if we take S = Z itself, then the integer associated with k, that is k !Z, would turn out to be the ordinary factorial of k. 
Motivation for the generalization
The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example,
- For any positive integers k and l, (k + l)! is a multiple of k! l!.
- Let f(x) be a primitive integer polynomial, that is, a polynomial in which the coefficients are integers and are relatively prime to each other. If the degree of f(x) is k then the greatest common divisor of the set of values of f(x) for integer values of x is a divisor of k!.
- Let a0, a1, a2, . . . , an be any n + 1 integers. Then the product of their pairwise differences is a multiple of 0! 1! ... n!.
- Let Z be the set of integers and n any integer. Then the number of polynomial functions from Z to Z/nZ is given by .
Bhargava posed to himself the following problem and obtained an affirmative answer: In the above theorems, can one replace the set of integers by some other set S (a subset of Z, or a subset of some ring) and define a function on S which assigns a value in S denoted by k!S for each k in S such that the statements obtained from the theorems given earlier by replacing k! by k!S remain true?
Hindu horary astrology
Praśna śāstra is the Indian (Sanskrit) terminology for that branch of astrology known in the Western world as horary astrology. In Indian traditions, it is considered as one of the six important branches of astrology the other branches being gola (study of the celestial sphere, observational astronomy), ganita (mathematics and computational schemes of astronomy), jataka (natal astrology), muhurta (Hellenistic catarchic astrology, or electional astrology) and nimitta (interpretation of omens). The techniques and ideas of horary astrology had their origins in the group of Old Greek astrological methods known as katarche (beginnings). These Greek ideas were introduced to India through the Yavanajataka of Sphujidhvaja.
Sphujidhvaja begins with rules for reconstructing the lost horoscope of the client from the horoscope of the time of his query. Thereafter he discusses methods of determining the subject of the querist's question before he puts it. The following chapter answers the question of whether what the querist is thinking of will in fact occur; and then there come discussions of lost or stolen objects, of sickness and death, of the sex of unborn or unseen children, of dinner, of various aspects of sleep, and of the subjects of dreams. The final two chapters in this section give elaborate rules for reconstructing names from astrological phenomena.
Ashtamangala prasnam is a certain type of practice of the prasna branch of Hindu astrology. The terminology indicates the use of eight (ashta) auspicious (mangala) objects in its practice. These objects are The practice of ashtamangala prasnam is highly popular and held in high esteem in the Indian state of Kerala. In fact, the author of Prasna Marga, an authoritative book on its practice was written by Narayanan Nambutiri, an astrologer from Edakad, Thalasseri in Kerala. Prasna Marga was written in 1649 CE.
Prasna is one of the six important branches of Hindu astrology. It deals with horary astrology in which an astrologer attempts to answer a question by constructing a horoscope for the exact time at which the question was received and understood by the astrologer. The other branches are jataka (natal astrology) which attempts to determine an individual's personality and path in life based on the horoscope of the individual, muhurta (electional astrology) in which the practitioner decides the most appropriate time for an event based on the astrological auspiciousness of that time, nimitta (interpretation of omens), gola (study of astronomy) and ganita (study of mathematics).
For a detailed accounts of the practice of this form of horary astrology, see:
- M. R. Bhat, B. P. Nair (2002). Essentials of Horary Astrology, Or, Praśnapadavī. Motilal Banarsidass Publ. pp. 77 – 94 (Chapter IV). ISBN 81-208-1012-0.
- N. E. Muthuswamy (2003). Ashtamangala Prasna: Horary Indian Astrology : a Comprehensive Book on Prasna, Kerala Hoarary Astrology. CBH Publications.
- S.C. Kursija (2013). Horary for Beginners (3rd ed.). All India Federation of Astrologers' Societies, New delhi.
- B. V. Raman (1993). Prasna Tantra: Horary Astrology (BVR Astrology Series). UBS Publishers' Distributors Pvt. Limited. ISBN 9788185674667.