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The former are ≡ ±1 (mod 12) and the latter are all ≡ ±5 (mod 12). −3 is in rows 7, 13, 19, 31, 37, and 43 but not in rows 5, 11, 17, 23, 29, 41, or 47. The former are ≡ 1 (mod 3) and the latter ≡ 2 (mod 3). Since the only residue (mod 3) is 1, we see that −3 is a quadratic residue modulo every prime which is a residue modulo 3.
For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0. Although typically performed with a and n both being integers, many computing systems now allow other types of numeric operands.
If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in 7 + 8 = 15, but 15:00 reads as 3:00 on the clock face because clocks "wrap around" every 12 hours and the hour number starts again at zero when it reaches 12. We say that 15 is congruent to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 ...
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In algebraic number theory the n-th power residue symbol (for an integer n > 2) is a generalization of the (quadratic) Legendre symbol to n-th powers. These symbols are used in the statement and proof of cubic , quartic , Eisenstein , and related higher [ 1 ] reciprocity laws .
An example is the function that relates each real number x to its square x 2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. The input variable(s) are sometimes referred to as the argument(s) of the function.
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x 3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x 3 ≡ p (mod q) is solvable if and only if ...
The CRT says that this is the same as p ≡ 1 (mod 840), and Dirichlet's theorem says there are an infinite number of primes of this form. 2521 is the smallest, and indeed 1 2 ≡ 1, 1046 2 ≡ 2, 123 2 ≡ 3, 2 2 ≡ 4, 643 2 ≡ 5, 87 2 ≡ 6, 668 2 ≡ 7, 429 2 ≡ 8, 3 2 ≡ 9, and 529 2 ≡ 10 (mod 2521).
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