Search results
Results from the WOW.Com Content Network
This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property. [2] For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin ...
That 641 is a factor of F 5 can be deduced from the equalities 641 = 2 7 × 5 + 1 and 641 = 2 4 + 5 4. It follows from the first equality that 2 7 × 5 ≡ −1 (mod 641) and therefore (raising to the fourth power) that 2 28 × 5 4 ≡ 1 (mod 641).
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
Examples m = 2, 4, 8, 16. The only character mod 2 is the principal character ,. −1 is a primitive root mod 4 =) The nonzero values of the characters mod 4 are ...
Correspondingly, a regular 3-gon and a regular 6-gon are constructible. n = 4: Similarly, ζ 4 = i, so Q(ζ 4) = Q(i), and a regular 4-gon is constructible. n = 5: The field Q(ζ 5) is not a quadratic extension of Q, but it is a quadratic extension of the quadratic extension Q(√ 5 ), so a regular 5-gon is constructible.
[3] [4] For example, if a = 2 and p = 5, then A = 31, B = 11, and n = 341 is a pseudoprime to base 2. In fact, there are infinitely many strong pseudoprimes to any base greater than 1 (see Theorem 1 of [ 5 ] ) and infinitely many Carmichael numbers, [ 6 ] but they are comparatively rare.
In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if > > are coprime integers, then for any integer , there is a prime number p (called a primitive prime divisor) that divides and does not divide for any positive integer <, with the following exceptions:
The polynomial x 2 + 2x + 2, on the other hand, is primitive. Denote one of its roots by α. Then, because the natural numbers less than and relatively prime to 3 2 − 1 = 8 are 1, 3, 5, and 7, the four primitive roots in GF(3 2) are α, α 3 = 2α + 1, α 5 = 2α, and α 7 = α + 2. The primitive roots α and α 3 are algebraically