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A primitive root modulo m exists if and only if m is equal to 2, 4, p k or 2p k, where p is an odd prime number and k is a positive integer. If a primitive root modulo m exists, then there are exactly φ ( φ ( m )) such primitive roots, where φ is the Euler's totient function.
Note that is equivalent to zero in the above equation because addition of coefficients is performed modulo 2: = + = (+) = (). Polynomial addition modulo 2 is the same as bitwise XOR. Since XOR is the inverse of itself, polynominal subtraction modulo 2 is the same as bitwise XOR too.
However, this template returns 0 if the modulus is nul (this template should never return a division by zero error). This template is not the same as the mod operator in the #expr parser function , which first truncates both operands to an integer before calculating the remainder.
Download QR code; Print/export Download as PDF; Printable version; In other projects ... This template lists various calculations and the names of their results.
Modulo operations might be implemented such that a division with a remainder is calculated each time. For special cases, on some hardware, faster alternatives exist. For example, the modulo of powers of 2 can alternatively be expressed as a bitwise AND operation (assuming x is a positive integer, or using a non-truncating definition):
This operation is also known as "bitwise xor" or "vector addition over GF" (bitwise addition modulo 2). Within combinatorial game theory it is usually called the nim-sum, as it will be called here. The nim-sum of x and y is written x ⊕ y to distinguish it from the ordinary sum, x + y. An example of the calculation with heaps of size 3, 4, and ...
In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits.It is a fast and simple action, basic to the higher-level arithmetic operations and directly supported by the processor.
Here, this is done after each addition, so that at the end of the for loop the sums are always reduced to 8 bits. At the end of the input data, the two sums are combined into the 16-bit Fletcher checksum value and returned by the function on line 13. Each sum is computed modulo 255 and thus remains less than 0xFF at all times.