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In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function).

The first such distribution found is π(N) ~ N/log(N), where π(N) is the prime-counting function (the number of primes less than or equal to N) and log(N) is the natural logarithm of N. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N). Consequently, a random integer with at most 2n digits (for large enough n) is about half as likely to be prime as a random integer with at most n digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(101000) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(102000) ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log(N).

INFO: Full form for PP is Piecewise Polynomial in Maths category
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INFO: Full form for SCV is Schwarz Critical Value in Maths category
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INFO: Full form for SCV is Squared Coefficient of Variation in Maths category
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INFO: Full form for PCIC is Piecewise Convex Information Criterion in Maths category
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INFO: Full form for QP is Quasi Polynomial in Maths category
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INFO: Full form for QP is Qualitative Probability in Maths category
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INFO: Full form for QP is Quadrature Polynomial in Maths category
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INFO: Full form for QP is Quadratic Programming in Maths category
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INFO: Full form for LMDS is Localization Using Multi-dimensional Scaling in Maths category
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INFO: Full form for MO is Mathematical Operation in Maths category
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INFO: Full form for MNE is Modified Neville Extrapolation in Maths category
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INFO: Full form for MSD is Mean Squared Displacement in Maths category
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INFO: Full form for MPE is Mean Predictive Error in Maths category
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INFO: Full form for MPE is Mathematics of Planet Earth in Maths category
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Mathematics (from Greek: μάθημα, máthēma, 'knowledge, study, learning') includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition.

Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of such by mathematical proof. When mathematical structures are good models of real phenomena, mathematical reasoning can be used to provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.

INFO: Full form for MPE is Multiplication Property of Equality in Maths category
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INFO: Full form for MFP is Multi-dimensional Fixed Partition in Maths category
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INFO: Full form for NRA is Nested Relational Algebra in Maths category
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INFO: Full form for NPID is Nonlinear Proportional Integral Differential in Maths category
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INFO: Full form for MMF is Multi Mode Function in Maths category
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INFO: Full form for GASP is Genetic Algorithm Sample Packer in Maths category
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