問題 Problem 50：Consecutive prime sum The prime 41, can be written as the sum of six consecutive primes:41 = 2 + 3 + 5 + 7 + 11 + 13 This is the longest sum of consecutive primes that adds to a prime below one-hundred.The longest sum of c…

問題 Problem 49：Prime permutations The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers are permutati…

問題 Problem 48：Self powers The series, 1^1 + 2^2 + 3^3 + ... + 10^10 = 10405071317.Find the last ten digits of the series, 1^1 + 2^2 + 3^3 + ... + 1000^1000. 解答例 sum = 0 for i in range(1, 1001): sum += i ** i tempStr = str(sum) print(…

問題 Problem 47：Distinct primes factors The first two consecutive numbers to have two distinct prime factors are:14 = 2 × 7 15 = 3 × 5The first three consecutive numbers to have three distinct prime factors are:644 = 2^2 × 7 × 23 645 = 3 …

問題 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 + 2×1^2 15 = 7 + 2×2^2 21 = 3 + 2×3^2 25 = 7 + 2×3^2 27 = 19 +…

問題 Problem 45：Triangular, pentagonal, and hexagonal Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:Triangle：Tn=n(n+1)/2 1, 3, 6, 10, 15, ... Pentagonal：Pn=n(3n−1)/2 1, 5, 12, 22, 35, ... Hexagonal…

問題 Problem 44：Pentagon numbers Pentagonal numbers are generated by the formula, Pn=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 P4 + P7 = 22 + 70 = 92 = P8. However, thei…

問題 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 d1 b…

問題 Problem 42：Coded triangle numbers The nth term of the sequence of triangle numbers is given by, tn = ½n(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 numb…

問題 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 p…

問題 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 dn represents…

問題 Problem 39：Integer right triangles If p is the perimeter of a right angle triangle with integral length sides, {a,b,c}, there are exactly three solutions for p = 120.{20,48,52}, {24,45,51}, {30,40,50}For which value of p ≤ 1000, is t…

問題 Problem 38：Pandigital multiples Take the number 192 and multiply it by each of 1, 2, and 3:192 × 1 = 192 192 × 2 = 384 192 × 3 = 576 By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 192384576 the co…

問題 Problem 37：Truncatable primes The number 3797 has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: 3797, 797, 97, and 7. Similarly we can wo…

問題 Problem 36：Double-base palindromes The decimal number, 585 = 10010010012 (binary), is palindromic in both bases.Find the sum of all numbers, less than one million, which are palindromic in base 10 and base 2.(Please note that the pal…

問題 Problem 35：Circular primes The number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.There are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79,…

問題 Problem 34：Digit factorials 145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145.Find the sum of all numbers which are equal to the sum of the factorial of their digits.Note: as 1! = 1 and 2! = 2 are not sums they are not in…

問題 Problem 33：Digit cancelling fractions The fraction 49/98 is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that 49/98 = 4/8, which is correct, is obtained by cancelling the …

問題 Problem 32：Pandigital products 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, the 5-digit number, 15234, is 1 through 5 pandigital.The product 7254 is unusual, as…

問題 Problem 31：Coin sums In England the currency is made up of pound, £, and pence, p, and there are eight coins in general circulation:1p, 2p, 5p, 10p, 20p, 50p, £1 (100p) and £2 (200p). It is possible to make £2 in the following way:1×…

問題 Problem 30：Digit fifth powers Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:1634 = 1^4 + 6^4 + 3^4 + 4^4 8208 = 8^4 + 2^4 + 0^4 + 8^4 9474 = 9^4 + 4^4 + 7^4 + 4^4 As 1 = 1^4…

問題 Problem 29：Distinct powers Consider all integer combinations of ab for 2 ≤ a ≤ 5 and 2 ≤ b ≤ 5:2^2=4, 2^3=8, 2^4=16, 2^5=32 3^2=9, 3^3=27, 3^4=81, 3^5=243 4^2=16, 4^3=64, 4^4=256, 4^5=1024 5^2=25, 5^3=125, 5^4=625, 5^5=3125 If they a…

問題 Problem 28：Number spiral diagonals Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:21 22 23 24 25 20 7 8 9 10 19 6 1 2 11 18 5 4 3 12 17 16 15 14 13It can be verified t…

問題 Problem 27：Quadratic primes Euler discovered the remarkable quadratic formula: n^2+n+41 It turns out that the formula will produce 40 primes for the consecutive integer values 0≤n≤39. However, when n=40, 40^2+40+41=40(40+1)+41 is div…

問題 Problem 26：Reciprocal cycles A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given: 1/2 = 0.5 1/3 = 0.(3) 1/4 = 0.25 1/5 = 0.2 1/6 = 0.1(6) 1/7 = 0.(142857) …

問題 Problem 25：1000-digit Fibonacci number The Fibonacci sequence is defined by the recurrence relation:Fn = Fn−1 + Fn−2, where F1 = 1 and F2 = 1. Hence the first 12 terms will be:F1 = 1 F2 = 1 F3 = 2 F4 = 3 F5 = 5 F6 = 8 F7 = 13 F8 = 21…

問題 Problem 24：Lexicographic permutations A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically,…

問題 Problem 23：Non-abundant sums A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 2…

問題 Problem 22：Names scores Using names.txt (right click and 'Save Link/Target As...'), a 46K text file containing over five-thousand first names, begin by sorting it into alphabetical order. Then working out the alphabetical value for e…

問題 Problem 21：Amicable numbers Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n). If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and each of a and b are cal…