Search results
Results From The WOW.Com Content Network
The number of nonresidues found will be even when m ≡ 0, 1 (mod 4), and it will be odd if m ≡ 2, 3 (mod 4). Gauss's fourth proof consists of proving this theorem (by comparing two formulas for the value of Gauss sums) and then restricting it to two primes.
Wilson's theorem. In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is (using the notations of modular arithmetic ), the factorial satisfies. exactly when n is a prime number.
In additive number theory, Fermat 's theorem on sums of two squares states that an odd prime p can be expressed as: with x and y integers, if and only if. The prime numbers for which this is true are called Pythagorean primes . For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of ...
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.
4j ≡ 2 (mod 6) We simplify by dividing by the greatest common divisor of 4,2 and 6. Division by 2 yields: 2j ≡ 1 (mod 3) The Euclidean modular multiplicative inverse of 2 mod 3 is 2. After multiplying both sides with the inverse, we obtain: j ≡ 2 × 1 (mod 3) or j ≡ 2 (mod 3) For the above to be true: j = 2 + 3k for some integer k.
In another case, testing p = 13, we have 17 (13 − 1)/2 = 17 6 ≡ 1 (mod 13), therefore 17 is a quadratic residue modulo 13. As confirmation, note that 17 ≡ 4 (mod 13), and 2 2 = 4. We can do these calculations faster by using various modular arithmetic and Legendre symbol properties.
Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones ...
The quadratic excess E ( p) is the number of quadratic residues on the range (0, p /2) minus the number in the range ( p /2, p) (sequence A178153 in the OEIS ). For p congruent to 1 mod 4, the excess is zero, since −1 is a quadratic residue and the residues are symmetric under r ↔ p − r.