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In mathematics, the Skolem problem is the problem of determining whether the values of a constant-recursive sequence include the number zero. The problem can be formulated for recurrences over different types of numbers, including integers, rational numbers, and algebraic numbers.
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
For example, S(1) = 2 and S(2) = 3. The successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known as zeration in the context of a zeroth hyperoperation: H 0 (a, b) = 1 + b. In this context, the extension of zeration is addition, which is defined as repeated succession.
This result is named after Thoralf Skolem (who proved the theorem for sequences of rational numbers), Kurt Mahler (who proved it for sequences of algebraic numbers), and Christer Lech (who proved it for sequences whose elements belong to any field of characteristic 0). Its known proofs use p-adic analysis and are non-constructive.
The sequence 2, 1, 3, 4, 7, 11, ... of Lucas numbers satisfies the same recurrence as the Fibonacci sequence but with initial conditions = and =. More generally, every Lucas sequence is constant-recursive of order 2.
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation.
For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is a 1 {\displaystyle a_{1}} and the common difference of successive members is d {\displaystyle d} , then the n {\displaystyle n} -th term of the sequence ( a n {\displaystyle a_{n ...
Shanks et al. observed that for all restricted pseudoprimes they found, the final state of the above six registers (the "signature" of n) equals the initial state 1,−1,3, 3,0,2. [22] The same occurs with ≈ 1 / 6 of all primes, so the two sets cannot be distinguished on the strength of this test alone. [ 23 ]