이경수 선생님의 수학실험실

Problem 46(Goldbach's other conjecture) It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square. \(9 = 7 + 2\times1^{2}\) \(15 = 7 + 2 \times 2^{2}\) \(21 = 3 + 2 \times 3^{2}\) \(25 = 7 + 2 \times 3^{2}\) \(27 = 19 + 2 \times 2^{2}\) \(33 = 31 + 2 \times 1^{2}\) It turns out that the conjecture was false. What is the smallest..

Problem 45(Triangular, pentagonal, and hexagonal) Triangle, pentagonal, and hexagonal numbers are generated by the following formulae: Triangle \(T_{n}=\frac{n(n+1)}{2}\) 1, 3, 6, 10, 15, ... Pentagonal \(P_{n}=\frac{n(3n-1)}{2}\) 1, 5, 12, 22, 35, ... Hexagonal \(H_{n}=n(2n-1)\) 1, 6, 15, 28, 45, ... It can be verified that \(T_{285} = P_{165} = H_{143} = 40755\). Find the next triangle number ..

Problem 44(Pentagon numbers) Pentagonal numbers are generated by the formula, \(P_{n}=\frac{n(3n−1)}{2}\). The first ten pentagonal numbers are: \(1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...\) It can be seen that \(P_{4} + P_{7} = 22 + 70 = 92 = P_{8}\). However, their difference, \(70 − 22 = 48\), is not pentagonal. Find the pair of pentagonal numbers, \(P_{j}\) and \(P_{k}\), for which their s..

Problem 43(Sub-string divisibility) The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting sub-string divisibility property. Let \(d_{1}\) be the 1st digit, \(d_{2}\) be the 2nd digit, and so on. In this way, we note the following: \(d_{2}d_{3}d_{4}=406\) is divisible by 2 \(d_{3}d_{4}d_{5}=063\..

Problem 42(Coded triangle numbers) The \( n^{th} \) term of the sequence of triangle numbers is given by, \( t_{n} = \frac{1}{2} n(n+1) \) so the first ten triangle numbers are: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For example, the word value for SKY is \(1..

https://docs.python.org/3/library/itertools.html itertools — Functions creating iterators for efficient looping — Python 3.7.4 documentation docs.python.org

Problem 41(Pandigital prime) We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is also prime. What is the largest n-digit pandigital prime that exists? In Python: import math import time def isprime(n): if n == 0 or n == 1: return False else: for i in range(2, n): if n % i == 0: return False retu..

Problem 40(Champernowne's constant) An irrational decimal fraction is created by concatenating the positive integers: 0.123456789101112131415161718192021... It can be seen that the 12th digit of the fractional part is 1. If \(d_{n}\) represents the nth digit of the fractional part, find the value of the following expression. \(d_{1}\) × \(d_{10}\) × \(d_{100}\) × \(d_{1000}\) × \(d_{10000}\) × \..