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Gelfond-Schneider Theorem – zeta function
Gelfond-Schneider Theorem
If a is a complex number that is also algebraic, and if b is an irrational number, then a b is a transcendental number.
greatest common divisor
The greatest common divisor of two natural numbers a and b is the largest natural number c such that c divides a and c divides b.
imaginary number
By definition, the square root of –1, i.e., i 2 = –1.
Cf. complex number.
irrational number
A real number which is not a rational number, i.e., which cannot be expressed as a ratio of integers. Examples: p (the ratio of the circumference of a circle to its diameter) and the square root of 2 (ratio of the length of the diagonal of a square to the length of one side). The irrational numbers are uncountably infinite.
Related article: Irrationality of the Square Root of 2
Related MiniText: Number -- What Is How Many?
mutually prime
Two integers are mutually prime if they have no common factors larger than 1 or -1.
number
There is no precise mathematical definition of the word “number.” There are however precise definitions of the terms “natural number,” “rational number,” “real number,” “complex number,” and other less commonly used kinds of number. When a mathematician speaks about numbers she usually has one of these cases in mind and she should, at the outset, make it clear to which type of number she is referring. The naive, inborn concept of number that is shared to some degree by all humans is a matter for philosophical rather than strictly mathematical inquiry, and it may be noted that there has historically been strong opposition to the introduction of new generalizations of established concepts of number.
partition
General: A partition of a set X is a collection of subsets of X such that every element of X is in exactly one of the subsets. Such a partition is given by (and gives rise to) an equivalence relation on X. For example, division modulo 3 partitions the set of natural numbers into three subsets, each containing all those numbers leaving remainders of 0, 1, or 2 respectively when divided by 3.
Algebra: A partition of a matrix is a division of the matrix into conformable submatrices.
Analysis: A partition of a space is a collection of pairwise disjoint regions of the space whose union is the entire space. For example, a partition of an interval [a, b] of the real line is given by a finite set of points {xi} such that a = x1 < x2 < . . . < xn = b which divide the interval into disjoint subintervals.
Number Theory: Given a positive integer n, a partition of n is a set of positive integers whose sum is n. For example, 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1 are the four possible partitions of the number 4.
Peano axioms
The axiom system developed by Giuseppe Peano to formalize arithmetic. The key to his method is the introduction of a “successor operation” S on numbers; if a is a number, then Sa is the successor of that number. In this way Peano reduced arithmetic to the conceptually primitive operation of counting.
The axioms are:- 0 is a number.
- If a is a number, then Sa is a number.
- If a and b are numbers, then a + 0 = a, and a + Sb = S(a + b).
- If a and b are numbers, then a × 0 = 0, and a × Sb = a × b + a.
- (Induction Principle.) For any formula f, if we have f(0) and if f(a) always implies f(Sa), then we have f(a) for all numbers a.
perfect number
A natural number n whose distinct divisors, including 1 but not including n itself, sum to n.
Example: 6 is perfect since its divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. The first three perfect numbers are 6, 28, and 496. Any number of the form 2n – 1(2n – 1) is perfect provided that 2n – 1 is a Mersenne prime. Such numbers are called Euclid numbers. Euler proved that all even perfect numbers are Euclid numbers. It is not known whether there are infinitely many perfect numbers, or if there are any odd perfect numbers.
prime number
Any natural number greater than 1 that is evenly divisible only by itself and 1. There are infinitely many prime numbers. The number of primes less than a given number n is denoted p(n), and approaches the value n/lnn for sufficiently large n.
Related article: Fundamental Theorem of Arithmetic
proper factor
See factor.
Pythagorean triple
An ordered triple (a,b,c) of natural numbers satisfying a2 + b2 = c2. The triples (3,4,5) and (5,12,13) are the first of infinitely many examples.
rational exponent
An exponent of the form p/q, with p and q integers and q not zero. Evaluated as the qth root of the base, raised to the pth power, or equivalently, as the qth root of the pth power of the base. For a negative base, this operation is not defined except when q is odd. Irrational roots may be considered as limits of sequences of rational roots.
Cf. laws of exponents.
relatively large
A set A of natural numbers is called relatively large if the number of elements of A is greater than the least element of A.
relatively prime
Two natural numbers a and b are relatively prime if their greatest common divisor is 1.
Riemann Hypothesis
The conjecture that the zeta function has no non-trivial zeros off of the line Re(z) = 1/2.
root
An nth root of a real or complex number x is a number which when multiplied by itself n times yields x.
Of a polynomial p: A number x such that p(x) = 0.
Roth’s Theorem
(Klaus Friedrich Roth, 1955) Given a real algebraic number a, consider the least upper bound m(a) of all numbers m for which there are infinitely many rational numbers p/q such that
Then for all a, m(a) = 2. This result improves on earlier theorems of Joseph Liouville, Axel Thue, and Carl Ludwig Siegel regarding the approximation of irrational numbers by rational numbers.
transcendental number
A number which is not an algebraic number, i.e., that is not the root of any polynomial with rational coefficients. It is known that e and p (pi) are transcendental. Since the algebraic numbers are countable and the set of all real numbers is uncountable, this means that the set of transcendental numbers is uncountably large as well.
Wilson's Theorem
If p is a prime number then (p-1)!  -1 modulo p and conversely.
zeta function
The function z(s) given by:
This function gives series representations of many significant numbers, e.g., z(2) = p2/6, and z(4) = p4/90.
Cf. Riemann Hypothesis.
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