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• The study of those sometimes subtle and far-reaching relationships is number theory, sometimes referred to as higher arithmetic
•  Number theorists

  scrutinize the properties of integers, the natural numbers you know as -1, -2, 0, 1, 2 and so forth. It's part theoretical and part experimental, as mathematicians seek to discover fascinating and even unexpected mathematical interactions.
• They uphold their theories with axioms (previously established statements presumed to be true) and theorems (statements based on other theorems or axioms).
• Mathematician Pierre de Fermat asked the same question about cubes and, in 1637, he claimed to have worked out a mathematical proof that, via line after line of painstaking logic, showed beyond any doubt that no, the sum of two cubes can't be a cube.
• Fermat merely wrote, "I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain" 
• 1993, with the aid of computational math undiscovered in Fermat's time, English mathematician Andrew Wiles succeeded in proving the 356-year-old theorem.
• Brown University mathematics professor Joseph H. Silverman lays out five basic steps in number theory:

  • Accumulate mathematical or abstract data.
  • Examine the data and search for patterns or relationships.
  • Formulate a conjecture (typically in the form of an equation) to explain these patterns or relationships.
  • Test the conjecture with additional data.
  • Devise a proof showing the conjecture to be correct. The proof should start with known facts and end with the desired result.
• Fermat's Last Theorem, therefore, was really a conjecture for 356 years and only became a true theorem in 1993.
•  Euclid's Proof of Infinite Primes (which proves that prime numbers are limitless), have remained a solid model of mathematical reasoning since 300 B.C. Still other number theory conjectures, both old and new, remain unproofed.
• THEOREMS ON NUMBER THEORY

  1. Analytic Number Theory
  2. Algebraic Number Theory
  3. Baker's Theorem
  4. Chinese Remainder Theorem
  5. Pentagonal Number Theorem
  6. Six Exponentials Theorem
• Analytic Number Theory

  In terms of its tools, as the study of the integers by means of tools from real and complex analysis
• Analytic Number Theory


  In terms of its concerns, as the study within number theory of estimates on size and density, as opposed to identities.
• Some subjects generally considered to be part of analytic number theory, e. g., sieve theory, are better covered by the second rather than the first definition: some of seive theory, for instance, uses little analysis, yet it does belong to analytic number theory.
• Examples of problems in analytic number theory:

  1. Prime number theorem
  2. Goldbach conjecture (or the twin prime conjecture, or the Hardy – Littlewood conjectures)
  3. Waring problem
  4. Riemann hypothesis
•  Some  of the most important tools of analytic number theory are the circle method, sieve methods and L – functions (or, rather, the study of their properties).
•  The theory of modular forms ( and, more generally, automorphic forms) also occupies an increasingly central place in the toolbox of analytic number theory. 
• Analytic number theory is the branch of the number theory that uses methods from mathematical analysis to prove theorems in number theory.
• Its major proofs include that of Dirichlet's theorem on arithmetic progressions, stating the existence of infinitely many primes in arithmetic progressions of the form a + nb, where a and b are relatively prime. The proof of the prime number theorem based on the Riemann zeta function is another important proof. (analytic number)
• Multiplicative number theory deals with the distribution of the prime numbers, applying the Dirichlet series as generating functions.
• Analytic number theory

  is the branch of number theory which uses real and complex analysis to investigate various properties of integers and prime numbers.
• Examples of topics falling under analytic number theory include Dirichlet L-series, the Riemann zeta function , the totient function , and the prime number theorem
• Algebraic Number Theory

  is any complex number that is a solution to some polynomial equation f(x) = 0 with rational coefficients. Fields of algebraic numbers are also called algebraic number fields or shortly number fields.
• analytic and algebraic  number theory can and do overlap: the former is defined by its methods, the latter by its objects of study.
• Number fields are often studied as extensions of smaller number fields: a field L is said to be an extension of a field K if L contains K.
• Their classification was the object of the programme of class field theory, which was initiated in the late 19th century (partly by Kronecker and Eisentein) and carried out largely in 1900 – 1950.
•   An example of an active area of research in algebraic number theory is Iwasawa theory.
• The Langlands program, one of the main current large – scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non – abelian extension of number fields.
• Algebraic number theory is a branch of number theory in which the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients.
• An algebraic number field is any finite (and therefore algebraic) field extension of the rational numbers. These domains contain elements analogous to the integers, the so-called algebraic integers.
• In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers.
• The result, proved by Alan Baker (1966, 1967a, 1967b), subsumed many earlier results in transcendental number theory and solved a problem posed by Alexander Gelfond nearly fifteen years earlier.
• Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1.
• Recently G. Wüstholz proved a theorem in transcendence which includes and greatly extends many classical results.
•  Chinese remainder theorem 

  states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.
• The earliest known statement of the theorem is by the Chinese mathematician Sunzi in Sunzi Suanjing in the 3rd century AD.
• The Chinese remainder theorem is widely used for computing with large integers, as it allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers
• The Chinese remainder theorem (expressed in terms of congruences) is true over every principal ideal domain. It has been generalized to any commutative ring, with a formulation involving ideals.
• The earliest known statement of the theorem, as a problem with specific numbers, appears in the 3rd-century book Sunzi Suanjing by the Chinese mathematician Sunzi
•  What amounts to an algorithm for solving this problem was described by Aryabhata (6th century) Special cases of the Chinese remainder theorem were also known to Brahmagupta (7th century), and appear in Fibonacci's Liber Abaci (1202).  The result was later generalized with a complete solution called Dayanshu in Qin Jiushao's 1247 Mathematical Treatise in Nine Sections , ShushuJiuzhang) which was translated into English in early 19th century by British missionary Alexander Wylie.
• The notion of congruences was first introduced and used by Gauss in his DisquisitionesArithmeticae of 1801. Gauss illustrates the Chinese remainder theorem on a problem involving calendars, namely, "to find the years that have a certain period number with respect to the solar and lunar cycle and the Roman indiction.