Geometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects.
Prime quadrupletIn number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form {p,\ p+2,\ p+6,\ p+8}. This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. The first eight prime quadruplets are: {5, 7, 11, 13}, {11, 13, 17, 19}, {101, 103, 107, 109}, {191, 193, 197, 199}, {821, 823, 827, 829}, {1481, 1483, 1487, 1489}, {1871, 1873, 1877, 1879}, {2081, 2083, 2087, 2089} All prime quadruplets except {5, 7, 11, 13} are of the form {30n + 11, 30n + 13, 30n + 17, 30n + 19} for some integer n.
Gaussian integerIn number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as or Gaussian integers share many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total ordering that respects arithmetic.
Group (mathematics)In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation is an infinite group, which is generated by a single element called 1 (these properties characterize the integers in a unique way).
Sexy primeIn number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and 11 − 5 = 6. The term "sexy prime" is a pun stemming from the Latin word for six: sex. If p + 2 or p + 4 (where p is the lower prime) is also prime, then the sexy prime is part of a prime triplet.
God in JudaismGod in Judaism has been conceived in a variety of ways. Traditionally, Judaism holds that Yahweh, the god of Abraham, Isaac, and Jacob and the national god of the Israelites, delivered the Israelites from slavery in Egypt, and gave them the Law of Moses at Mount Sinai as described in the Torah. Jews traditionally believe in a monotheistic conception of God (God is only one), which is both transcendent (wholly independent of, and removed from, the material universe) and immanent (involved in the material universe).
God in ChristianityGod in Christianity is believed to be the eternal, supreme being who created and preserves all things. Christians believe in a monotheistic conception of God, which is both transcendent (wholly independent of, and removed from, the material universe) and immanent (involved in the material universe). Christian teachings on the transcendence, immanence, and involvement of God in the world and his love for humanity exclude the belief that God is of the same substance as the created universe (rejection of pantheism) but accept that God the Son assumed hypostatically united human nature, thus becoming man in a unique event known as "the Incarnation".
Existence of GodThe existence of God (or more generally, the existence of deities) is a subject of debate in theology, philosophy of religion and popular culture. A wide variety of arguments for and against the existence of God or deities can be categorized as logical, empirical, metaphysical, subjective or scientific. In philosophical terms, the question of the existence of God or deities involves the disciplines of epistemology (the nature and scope of knowledge) and ontology (study of the nature of being or existence) and the theory of value (since some definitions of God include "perfection").
Personal godA personal god, or personal goddess, is a deity who can be related to as a person instead of as an impersonal force, such as the Absolute. In the scriptures of the Abrahamic religions, God is described as being a personal creator, speaking in the first person and showing emotion such as anger and pride, and sometimes appearing in anthropomorphic shape. In the Pentateuch, for example, God talks with and instructs his prophets and is conceived as possessing volition, emotions (such as anger, grief and happiness), intention, and other attributes characteristic of a human person.
Pollard's rho algorithmPollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. It uses only a small amount of space, and its expected running time is proportional to the square root of the smallest prime factor of the composite number being factorized. The algorithm is used to factorize a number , where is a non-trivial factor. A polynomial modulo , called (e.g., ), is used to generate a pseudorandom sequence. It is important to note that must be a polynomial.
Quantum complexity theoryQuantum complexity theory is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational model based on quantum mechanics. It studies the hardness of computational problems in relation to these complexity classes, as well as the relationship between quantum complexity classes and classical (i.e., non-quantum) complexity classes. Two important quantum complexity classes are BQP and QMA.
Post-quantum cryptographyIn cryptography, post-quantum cryptography (PQC) (sometimes referred to as quantum-proof, quantum-safe or quantum-resistant) refers to cryptographic algorithms (usually public-key algorithms) that are thought to be secure against a cryptanalytic attack by a quantum computer. The problem with currently popular algorithms is that their security relies on one of three hard mathematical problems: the integer factorization problem, the discrete logarithm problem or the elliptic-curve discrete logarithm problem.
