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  2. Arithmetic progression - Wikipedia

    en.wikipedia.org/wiki/Arithmetic_progression

    Proof without words of the arithmetic progression formulas using a rotated copy of the blocks. An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...

  3. Primes in arithmetic progression - Wikipedia

    en.wikipedia.org/wiki/Primes_in_arithmetic...

    In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} .

  4. Dirichlet's theorem on arithmetic progressions - Wikipedia

    en.wikipedia.org/wiki/Dirichlet's_theorem_on...

    Linnik's theorem (1944) concerns the size of the smallest prime in a given arithmetic progression. Linnik proved that the progression a + nd (as n ranges through the positive integers) contains a prime of magnitude at most cd L for absolute constants c and L. Subsequent researchers have reduced L to 5.

  5. Problems involving arithmetic progressions - Wikipedia

    en.wikipedia.org/wiki/Problems_involving...

    The sequence of primes numbers contains arithmetic progressions of any length. This result was proven by Ben Green and Terence Tao in 2004 and is now known as the Green–Tao theorem. [3] See also Dirichlet's theorem on arithmetic progressions. As of 2020, the longest known arithmetic progression of primes has length 27: [4]

  6. Formula for primes - Wikipedia

    en.wikipedia.org/wiki/Formula_for_primes

    Download as PDF; Printable version ... a formula for primes is a formula generating ... based on Dirichlet's theorem on arithmetic progressions, that linear ...

  7. Arithmetic progression topologies - Wikipedia

    en.wikipedia.org/wiki/Arithmetic_progression...

    Each residue class is an arithmetic progression, and thus clopen. Consider the multiples of each prime. These multiples are a residue class (so closed), and the union of these sets is all (Golomb: positive) integers except the units ±1. If there are finitely many primes, that union is a closed set, and so its complement ({±1}) is open.

  8. Green–Tao theorem - Wikipedia

    en.wikipedia.org/wiki/Green–Tao_theorem

    In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k {\displaystyle k} , there exist arithmetic progressions of primes with k {\displaystyle k} terms.

  9. List of theorems - Wikipedia

    en.wikipedia.org/wiki/List_of_theorems

    Absolute convergence theorem (mathematical series) Cesàro's theorem (real analysis) Hardy–Littlewood tauberian theorem (mathematical analysis) Riemann series theorem (mathematical series) Silverman–Toeplitz theorem (mathematical analysis) Śleszyński–Pringsheim theorem (continued fraction) Stolz–Cesàro theorem