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The following table attempts to list the oldest-known Paleolithic and Paleo-Indian sites where hominin tools have been found. It includes sites where compelling evidence of hominin tool use has been found, even if no actual tools have been found. Stone tools preserve more readily than tools of many other materials.
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function: is a Dirichlet character of modulus (where is a positive integer) if for all integers and : [1]
In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.
Primitive root modulo m: A number g is a primitive root modulo m if, for every integer a coprime to m, there is an integer k such that g k ≡ a (mod m). 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.
For a primitive () th root x, the number () / is a primitive th root of unity. If k does not divide λ ( n ) {\displaystyle \lambda (n)} , then there will be no k th roots of unity, at all. Finding multiple primitive k th roots modulo n
The oldest known Oldowan tools have been found at Nyayanga on the Homa Peninsula in Kenya and are dated to ~2.9 million years ago (Ma). [10] The Oldowan tools were associated with Paranthropus teeth and two butchered hippo skeletons. [10] Early Oldowan tools are also known from Gona in Ethiopia (near the Awash River), and are dated to about 2.6 ...
If the order of a is actually equal to φ(n), and therefore as large as possible, then a is called a primitive root modulo n. This means that the group U(n) is cyclic and the residue class of a generates it. The order of a (mod n) also divides λ(n), a value of the Carmichael function, which is an even stronger statement than the divisibility ...
Carmichael calls an element a for which () is the least power of a congruent to 1 (mod n) a primitive λ-root modulo n. [3] This is not to be confused with a primitive root modulo n , which Carmichael sometimes refers to as a primitive φ {\displaystyle \varphi } -root modulo n .)