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Additional info for A Treatise on the Theory of Determinants
It is an open question, however, whether 4! þ 1 ¼ 52, 5! þ 1 ¼ 112 , and 7! þ 1 ¼ 712 are the only solutions to the equation n! þ 1 ¼ m2 . 1 1. An even number can be expressed as 2n and an odd number as 2n þ 1, where n is a natural number. Two natural numbers are said to be of the same parity if they are either both even or both odd, otherwise they are said to be of opposite parity. Given any two natural numbers of the same parity, show that their sum and difference are even. Given two numbers of opposite parity, show that their sum and difference are odd.
Any 1 1 1 1 1 2 3 1 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 ................................ 2 Sequences of natural numbers 35 slight adjustment of the algorithm may change the outcome. For example, if 3an þ 1 is replaced by 3an À 1, when an is odd, three distinct cycles are generated. R. Kaprekar (kuh PREE kur). Kaprekar’s sort–reverse–subtract routine goes as follows: given a four-digit natural number larger than 1000 for which not all digits are equal, arrange the digits in descending order, subtract the result from its reverse (the number with the digits in ascending order).
2 without proof. 2 (Alternate principle of mathematical induction) Any set of natural numbers that contains the natural number m, and contains n þ 1 whenever it contains all the natural numbers between m and n, where n > m, contains all the natural numbers greater than m. The alternate principle of mathematical induction implies the well-ordering principle. In order to see this, let S be a nonempty set of natural numbers with no least element. For n . 1, suppose 1, 2, . . , n are elements S, the complement of S.