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  2. Table of prime factors - Wikipedia

    en.wikipedia.org/wiki/Table_of_prime_factors

    The first: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100 (sequence A001597 in the OEIS). 1 is sometimes included. A powerful number (also called squareful ) has multiplicity above 1 for all prime factors.

  3. Landau's problems - Wikipedia

    en.wikipedia.org/wiki/Landau's_problems

    Montgomery and Vaughan showed that the exceptional set of even numbers not expressible as the sum of two primes has a density zero, although the set is not proven to be finite. [9] The best current bounds on the exceptional set is E ( x ) < x 0.72 {\displaystyle E(x)<x^{0.72}} (for large enough x ) due to Pintz , [ 10 ] [ 11 ] and E ( x ) ≪ x ...

  4. Euler's totient function - Wikipedia

    en.wikipedia.org/wiki/Euler's_totient_function

    Leonhard Euler introduced the function in 1763. [7] [8] [9] However, he did not at that time choose any specific symbol to denote it.In a 1784 publication, Euler studied the function further, choosing the Greek letter π to denote it: he wrote πD for "the multitude of numbers less than D, and which have no common divisor with it". [10]

  5. Prime number theorem - Wikipedia

    en.wikipedia.org/wiki/Prime_number_theorem

    Graph of the number of primes ending in 1, 3, 7, and 9 up to n for n < 10 000. Another example is the distribution of the last digit of prime numbers. Except for 2 and 5, all prime numbers end in 1, 3, 7, or 9. Dirichlet's theorem states that asymptotically, 25% of all primes end in each of these four digits.

  6. Asymptote - Wikipedia

    en.wikipedia.org/wiki/Asymptote

    An oblique asymptote has a slope that is non-zero but finite, such that the graph of the function approaches it as x tends to +∞ or −∞. More generally, one curve is a curvilinear asymptote of another (as opposed to a linear asymptote ) if the distance between the two curves tends to zero as they tend to infinity, although the term ...

  7. Multiplicity (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Multiplicity_(mathematics)

    Graph of x 3 + 2x 2 − 7x + 4 with a simple root (multiplicity 1) at x=−4 and a root of multiplicity 2 at x=1. The graph crosses the x axis at the simple root. It is tangent to the x axis at the multiple root and does not cross it, since the multiplicity is even.

  8. Explicit formulae for L-functions - Wikipedia

    en.wikipedia.org/wiki/Explicit_formulae_for_L...

    The other terms also correspond to zeros: The dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions.

  9. Factorization - Wikipedia

    en.wikipedia.org/wiki/Factorization

    For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x 2 – 4. Factorization is not usually considered meaningful within number systems possessing division , such as the real or complex numbers , since any x {\displaystyle x} can be trivially written as ( x y ) × ( 1 / y ) {\displaystyle ...