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Leonardo of Pisa (c. 1170 – c. 1250) described this method [1] [2] for generating primitive triples using the sequence of consecutive odd integers ,,,,, … and the fact that the sum of the first n terms of this sequence is .
Animation demonstrating the smallest Pythagorean triple, 3 2 + 4 2 = 5 2. A Pythagorean triple consists of three positive integers a, b, and c, such that a 2 + b 2 = c 2. 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.
Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8.
A Young diagram representing visually a polite expansion 15 = 4 + 5 + 6. In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers.
The length of an interval of consecutive integers with property that every element has a factor in common with one of the endpoints. A059756: Sierpinski numbers: 78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, ... Odd k for which { k⋅2 n + 1 : n ∈ } consists only of composite numbers. A076336
If k is an odd integer, then 3k + 1 is even, so 3k + 1 = 2 a k ′ with k ′ odd and a ≥ 1. The Syracuse function is the function f from the set I of positive odd integers into itself, for which f(k) = k ′ (sequence A075677 in the OEIS). Some properties of the Syracuse function are: For all k ∈ I, f(4k + 1) = f(k). (Because 3(4k + 1) + 1 ...
3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1. The only partition of zero is the empty sum, having no parts. The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1. An individual summand in a partition is called a part.
Legendre's three-square theorem states which numbers can be expressed as the sum of three squares; Jacobi's four-square theorem gives the number of ways that a number can be represented as the sum of four squares. For the number of representations of a positive integer as a sum of squares of k integers, see Sum of squares function.