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A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A triangle whose side lengths are a Pythagorean triple is a right triangle and ...
The graphical representation of each possible distribution would contain P copies of the symbol ε and N – 1 copies of the symbol 0. In their demonstration, Ehrenfest and Kamerlingh Onnes took N = 4 and P = 7 (i.e., R = 120 combinations). They chose the 4-tuple (4, 2, 0, 1) as the illustrative example for this symbolic representation ...
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset of integers and a target-sum , and the question is to decide whether any subset of the integers sum to precisely .[ 1 ] The problem is known to be NP-complete. Moreover, some restricted variants of it are NP-complete too ...
In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that. Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum.) This conjecture is called "weak" because if Goldbach's ...
Combination. In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple ...
Goldbach's conjecture is used when studying computation complexity. [ 36 ] The connection is made through the Busy Beaver function, where BB (n) is the maximum number of steps taken by any n state Turing machine that halts. There is a 27 state Turing machine that halts if and only if Goldbach's conjecture is false.
Description. The simplest and most common form of mathematical induction infers that a statement involving a natural number n (that is, an integer n ≥ 0 or 1) holds for all values of n. The proof consists of two steps: The base case (or initial case): prove that the statement holds for 0, or 1. The induction step (or inductive step, or step ...
Index k = 2, because 3 is placed at an index that satisfies condition of being the largest index that is still less than a[k + 1] which is 4. Index l = 3, because 4 is the only value in the sequence that is greater than 3 in order to satisfy the condition a[k] < a[l]. The values of a[2] and a[3] are swapped to form the new sequence [1, 2, 4, 3].