Dirichlet's theorem on arithmetic progressionsIn number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d. The numbers of the form a + nd form an arithmetic progression and Dirichlet's theorem states that this sequence contains infinitely many prime numbers.
Pafnuty ChebyshevPafnuty Lvovich Chebyshev (Пафну́тий Льво́вич Чебышёв) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshev is known for his fundamental contributions to the fields of probability, statistics, mechanics, and number theory. A number of important mathematical concepts are named after him, including the Chebyshev inequality (which can be used to prove the weak law of large numbers), the Bertrand–Chebyshev theorem, Chebyshev polynomials, Chebyshev linkage, and Chebyshev bias.
Riemann hypothesisIn mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named.
Edmund LandauEdmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Edmund Landau was born to a Jewish family in Berlin. His father was Leopold Landau, a gynecologist, and his mother was Johanna Jacoby. Landau studied mathematics at the University of Berlin, receiving his doctorate in 1899 and his habilitation (the post-doctoral qualification required to teach in German universities) in 1901. His doctoral thesis was 14 pages long.
Waring's problemIn number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers raised to the power k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909.
Twin primeA 'twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin' or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger.
G. H. HardyGodfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of population genetics. G. H. Hardy is usually known by those outside the field of mathematics for his 1940 essay A Mathematician's Apology, often considered one of the best insights into the mind of a working mathematician written for the layperson.
John Edensor LittlewoodJohn Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanujan and Mary Cartwright. Littlewood was born on 9 June 1885 in Rochester, Kent, the eldest son of Edward Thornton Littlewood and Sylvia Maud (née Ackland). In 1892, his father accepted the headmastership of a school in Wynberg, Cape Town, in South Africa, taking his family there.
Goldbach's conjectureGoldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold for all integers less than 4e18, but remains unproven despite considerable effort. On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture: Goldbach was following the now-abandoned convention of considering 1 to be a prime number, so that a sum of units would indeed be a sum of primes.
Prime number theoremIn mathematics, 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).
