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  2. General number field sieve - Wikipedia

    en.wikipedia.org/wiki/General_number_field_sieve

    Now the product of the factors a − mb mod n can be obtained as a square in two ways—one for each homomorphism. Thus, one can find two numbers x and y, with x 2 − y 2 divisible by n and again with probability at least one half we get a factor of n by finding the greatest common divisor of n and xy.

  3. Prime-counting function - Wikipedia

    en.wikipedia.org/wiki/Prime-counting_function

    For x > 1 let π 0 (x) = π(x) − ⁠ 1 / 2 ⁠ when x is a prime number, and π 0 (x) = π(x) otherwise. Bernhard Riemann, in his work On the Number of Primes Less Than a Given Magnitude, proved that π 0 (x) is equal to [9] Riemann's explicit formula using the first 200 non-trivial zeros of the zeta function

  4. List of logarithmic identities - Wikipedia

    en.wikipedia.org/wiki/List_of_logarithmic_identities

    ln(r) is the standard natural logarithm of the real number r. Arg(z) is the principal value of the arg function; its value is restricted to (−π, π]. It can be computed using Arg(x + iy) = atan2(y, x). Log(z) is the principal value of the complex logarithm function and has imaginary part in the range (−π, π].

  5. Logarithm - Wikipedia

    en.wikipedia.org/wiki/Logarithm

    Here M(x, y) denotes the arithmetic–geometric mean of x and y. It is obtained by repeatedly calculating the average (x + y)/2 (arithmetic mean) and (geometric mean) of x and y then let those two numbers become the next x and y. The two numbers quickly converge to a common limit which is the value of M(x, y). m is chosen such that

  6. Logarithmic mean - Wikipedia

    en.wikipedia.org/wiki/Logarithmic_mean

    The area interpretation allows the easy derivation of some basic properties of the logarithmic mean. Since the exponential function is monotonic, the integral over an interval of length 1 is bounded by x and y. The homogeneity of the integral operator is transferred to the mean operator, that is (,) = (,).

  7. Log–log plot - Wikipedia

    en.wikipedia.org/wiki/Log–log_plot

    A log–log plot of y = x (blue), y = x 2 (green), and y = x 3 (red). Note the logarithmic scale markings on each of the axes, and that the log x and log y axes (where the logarithms are 0) are where x and y themselves are 1. Comparison of linear, concave, and convex functions when plotted using a linear scale (left) or a log scale (right).

  8. How to Make Butternut Squash Soup 10x Better, According ... - AOL

    www.aol.com/lifestyle/butternut-squash-soup-10x...

    For this soup, we went with sweet-and-spicy pumpkin seeds, which take all of two minutes to prep and 10 minutes to roast. Luke tosses a few handfuls of the seeds in a small mixing bowl with a few ...

  9. Natural logarithm - Wikipedia

    en.wikipedia.org/wiki/Natural_logarithm

    The natural logarithm of x is generally written as ln x, log e x, or sometimes, if the base e is implicit, simply log x. [2] [3] Parentheses are sometimes added for clarity, giving ln(x), log e (x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.