1.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers

2.
Divisor function
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In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the function, it counts the number of divisors of an integer. It appears in a number of identities, including relationships on the Riemann zeta function. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities, a related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function. The sum of divisors function σx, for a real or complex number x, is defined as the sum of the xth powers of the positive divisors of n. It can be expressed in sigma notation as σ x = ∑ d ∣ n d x, the notations d, ν and τ are also used to denote σ0, or the number-of-divisors function. When x is 1, the function is called the function or sum-of-divisors function. The aliquot sum s of n is the sum of the proper divisors, and equals σ1 − n, the cases x =2 to 5 are listed in A001157 − A001160, x =6 to 24 are listed in A013954 − A013972. For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and σ0 is even, for an integer, one divisor is not paired with a distinct divisor. Similarly, the number σ1 is odd if and only if n is a square or twice a square. For a prime p, σ0 =2 σ0 = n +1 σ1 = p +1 because by definition. Also, where pn# denotes the primorial, σ0 =2 n since n prime factors allow a sequence of binary selection from n terms for each proper divisor formed, clearly,1 < σ0 < n and σ > n for all n >2. The divisor function is multiplicative, but not completely multiplicative and it follows that d is, σ0 = ∏ i =1 r. For example, if n is 24, there are two factors, noting that 24 is the product of 23×31, a1 is 3. Thus we can calculate σ0 as so, σ0 = ∏ i =12 = =4 ⋅2 =8, the eight divisors counted by this formula are 1,2,4,8,3,6,12, and 24. Here s denotes the sum of the divisors of n, that is. This function is the one used to perfect numbers which are the n for which s = n. If s > n then n is an abundant number and if s < n then n is a deficient number