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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 ...
For instance, the first counterexample must be odd because f(2n) = n, smaller than 2n; and it must be 3 mod 4 because f 2 (4n + 1) = 3n + 1, smaller than 4n + 1. For each starting value a which is not a counterexample to the Collatz conjecture, there is a k for which such an inequality holds, so checking the Collatz conjecture for one starting ...
Essentially different solutions The sudoku symmetry group. The precise structure of the sudoku symmetry group can be expressed succinctly using the wreath product (≀). The possible row (or column) permutations form a group isomorphic to S 3 ≀ S 3 of order 3! 4 = 1,296.
Kaprekar's routine. In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with a number, sorts the digits into descending and ascending order, and calculates the difference between the two new numbers. As an example, starting with the number 8991 in ...
For example, 11 can be 2-split into 2+9, 3+8, 4+7, and 5+6. The respective products are 18, 24, 28, and 30 and the players put a tick mark beside each of these products in their tables (Table 1). When they are done, some numbers have no tick marks, some have one, and some have more than one. Sue now looks at her sum and all its 2-splits.
The product of two variables ranging from 90-99 will result in a 4-digit number. The first step is to find the ones-digit and the tens digit. Subtract both variables from 100 which will result in 2 one-digit number. The product of the 2 one-digit numbers will be the last two digits of one's final product.
In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers. if and only if n is not of the form for nonnegative integers a and b . The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as ) are. 7, 15, 23, 28, 31, 39 ...
Some sequences have alternate names: 4n+1 are Pythagorean primes, 4n+3 are the integer Gaussian primes, and 6n+5 are the Eisenstein primes (with 2 omitted). The classes 10 n + d ( d = 1, 3, 7, 9) are primes ending in the decimal digit d .